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MP4056
Grades 6-8
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F
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J
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P
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Milliken Publishing Co. • St. Louis, Missouri
Geometry
Grades 6–8
Pythagoras would be proud!
The theorems and principles of basic geometry
are clearly presented in this workbook, along
with examples and exercises for practice.
Author
Janice Wendling
Artist
All concepts are explained in an easy–to–
understand fashion to help students grasp
geometry and form a solid foundation for
advanced learning in mathematics.
Elizabeth Adams
Editing and Page Design
Martha Kranes
Cover Design
Gray Communications &
Marketing
Each page introduces a new concept, along with
a puzzle or riddle which reveals a fun fact.
Thought–provoking exercises encourage
students to enjoy working the pages while
gaining valuable practice in geometry.
Project Director
Kathleen Hilmes
Copyright © 1995
Milliken Publishing Company
11643 Lilburn Park Dr.
St. Louis, MO 63146
All rights reserved.
www.millikenpub.com
The purchase of this book entitles the individual purchaser
to reproduce copies by duplicating master or by any
photocopy process for single classroom use.The
reproduction of any part of this book for commercial resale
or for use by an entire school or school system is strictly
prohibited. Storage of any part of this book in any type of
electronic retrieval system is prohibited unless purchaser
receives written authorization from the publisher.
Geometry Workbook
Table of Contents
Reading Mathematics . . . . . . . . . . . . . . . . . . .1
Undefined Terms and Basic Definitions . . . . .2
Types of Angles . . . . . . . . . . . . . . . . . . . . . . .3
Complementary and Supplementary Angles . . .4
Parallel, Perpendicular, and Skew Lines . . . .5
Angles Formed by Parallel Lines . . . . . . . . . .6
Angle Sum Theorem . . . . . . . . . . . . . . . . . . . .7
Exterior Angle Theorem . . . . . . . . . . . . . . . . .8
Classifying Triangles . . . . . . . . . . . . . . . . . . . .9
The Pythagorean Theorem . . . . . . . . . . .10-11
The Converse of the Pythagorean Theorem . . .12
Congruent Figures . . . . . . . . . . . . . . . . . . . .13
Congruent Triangles . . . . . . . . . . . . . . . . . . .14
Congruent Triangles . . . . . . . . . . . . . . . . . . .15
Angle and Triangle Word Search . . . . . . . . .16
Types of Quadrilaterals . . . . . . . . . . . . . . . . .17
Properties of Parallelograms . . . . . . . . . . . . .18
Properties of Rectangles, Rhombuses,
and Squares . . . . . . . . . . . . . . . . . . . . . . . .19
Trapezoids . . . . . . . . . . . . . . . . . . . . . . . . . .20
Ratio and Proportion . . . . . . . . . . . . . . . . . . .21
Similar Figures . . . . . . . . . . . . . . . . . . . . . . .22
© Milliken Publishing Company
Trigonometric Ratios . . . . . . . . . . . . . . . . . . . . . .23
Perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . .24
Area of Rectangles and Triangles . . . . . . . . .25
Area of Trapezoids . . . . . . . . . . . . . . . . .26-27
Area of Parallelograms . . . . . . . . . . . . . .28-29
Word Search . . . . . . . . . . . . . . . . . . . . . . . . .30
Circumference . . . . . . . . . . . . . . . . . . . . . . . .31
Area of a Circle . . . . . . . . . . . . . . . . . . . . . . .32
Area of Irregular Shapes . . . . . . . . . . . . . . . .33
Area of a Shaded Region . . . . . . . . . . . . . . .34
Three–Dimensional Figures . . . . . . . . . . . . .35
Surface Area of Right Prisms . . . . . . . . . . . .36
Volume of Rectangular Prisms . . . . . . . . . . .37
Volume of Right Prisms . . . . . . . . . . . . . . . .38
Surface Area of Regular Pyramids . . . . . . . .39
Volume of Regular Pyramids . . . . . . . . . . . .40
Surface Area of Cylinders . . . . . . . . . . . . . . .41
Surface Area of Cones . . . . . . . . . . . . . . . . .42
Volume of Cylinders and Cones . . . . . . . . . .43
Crossword Puzzle . . . . . . . . . . . . . . . . . . . . .44
Answer Key . . . . . . . . . . . . . . . . . . . . . . .45-48
MP4056
Reading Mathematics
Remember:
Learning the correct meaning and use of mathematical symbols is necessary for reading
and understanding mathematics. In geometry, the order of the letters is important in some
cases, like when naming rays and angles.
Find the corresponding symbols and shade their areas to reveal a number that is neither prime
nor composite.
1.
line AB
14.
angle with vertex at C
2.
segment AB
15.
cosine of angle X
3.
angle with vertex at B
16.
is approximately equal to
4.
triangle ABC
17.
is similar to
5.
cube root
18.
greater than
6.
arc AB
19.
square root
7.
ray AB
20.
is congruent to
8.
circle with center A
21.
pi
9.
tangent of angle X
22.
is perpendicular to
10.
not equal to
23.
parallelogram ABCD
11.
is parallel to
24.
sine of angle X
12.
right angle
25.
less than
13.
measure of angle A
26.
ordered pair
© Milliken Publishing Company
1
MP4056
Undefined Terms and Basic Definitions
Geometry is based on the undefined terms point, line, and plane.
Points can be collinear (lie on the same line).
Points and lines can be coplanar (lie in the same plane).
A ray is part of a line with one endpoint.
An angle is formed by two rays that have the same endpoint.
Refer to the diagrams and determine whether the statement is true or false. If it is true, place its
corresponding letter in the puzzle to reveal the name of a famous mathematician and his
collection of books about geometry, number theory, and geometric algebra.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
CK intersects RL at O.
E
M lies in plane X.
A
R, O, K, and A are coplanar.
R
OR and OK are sides of ∠ROK. U
A, O, and B are collinear.
C
O, L, K, and M are coplanar.
H
C, O, A, and B are coplanar.
L
Plane D contains P.
P
Plane X intersects AB at O.
I
A, B, and M are coplanar.
D
Plane X contains Z.
S
J, N, F, and P are coplanar.
A
∠CBO lies in plane X.
K
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
NJ and NH are opposite rays.
ZL intersects RO at O.
GF intersects JH at N.
E
F
L
Plane E contains ∠JNF.
I
Plane D and Plane E intersect in GF. E
G, N, F, and P are coplanar.
M
Q, H, N, and F are coplanar.
E
N and P are in plane D.
I
∠GNH lies in plane D.
N
G, N, and F are collinear.
T
Plane D contains PQ.
B
NF and NG are opposite rays.
S
E
A
X
F
K
R
D
O
Q
L
C
N
J
H
Z
G
M
B
P
© Milliken Publishing Company
2
MP4056
Types of Angles
An acute angle measures between 0° and 90°.
A right angle measures exactly 90°.
An obtuse angle measures between 90° and 180°.
A straight angle measures exactly 180°.
Refer to the diagram and classify each expression as acute, right, obtuse, or straight.
1. ∠CDL __________________________
11. ∠EDA + ∠ADL _____________________
2. ∠EAD __________________________
12. ∠ECD + ∠DCL _____________________
3. ∠DAB __________________________
13. ∠ADE + ∠EDC _____________________
4. ∠CLD __________________________
14. ∠UEC + ∠CEL _____________________
5. ∠ABI ___________________________
15. ∠EDC + ∠CDL _____________________
6. ∠CLI ___________________________
16. ∠ABD + ∠DBL _____________________
7. ∠UCL __________________________
17. ∠CDE + ∠EDB _____________________
8. ∠ABL __________________________
18. ∠EDA + ∠ADB _____________________
9. ∠ECL __________________________
19. ∠EDA + ∠ADB + ∠EDC ______________
10. ∠AED __________________________
20. ∠LDB + ∠BDA + ∠ADE
______________
E
A
B
I
D
U
© Milliken Publishing Company
C
L
3
MP4056
Complementary and Supplementary Angles
Remember:
alphabetic order
complementary
supplementary
numeric order
90°
180°
ADD TO
Use a ruler to match each angle to its complement and its supplement. Each line you
draw will cross a letter. The letters without lines through them spell the answer.
Michael Jordan scored his career 20,000th point in Chicago.
In what cities did he score his 5,000th, 10,000th, and 15,000th points?
Complement
Angle
A
47°
63°
Supplement
52°
H
C
L
C
78°
38°
C
G
13°
L
M
71°
C
H
117°
K
43°
27°
A
D
L
83°
79°
19°
27°
O
H
146°
153°
T
34°
103°
H
11°
C
R
W
L
P
P
22°
13°
C
68°
L
9°
158°
N
T
68°
D
7°
I
56°
E
161°
G
G
77°
102°
128°
63°
H
167°
137°
D
12°
U
81°
I
O
77°
I
22°
____ ____ ____
O
R
173°
169°
T
A
112°
O
99°
____ ____
____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____
© Milliken Publishing Company
4
MP4056
Parallel, Perpendicular, and Skew Lines
Parallel lines are coplanar lines that never intersect.
Skew lines are noncoplanar lines (neither parallel nor intersecting).
Perpendicular lines intersect to form right angles.
Planes can also be parallel, perpendicular, or intersecting.
pa
ral
lel
pe
rpe
nd
icu
sk
lar
ew
int
ers
ec
tin
g
pa
ral
lel
pe
rpe
nd
icu
sk
lar
ew
int
ers
ec
tin
g
Determine whether the following pairs of lines are parallel, perpendicular, skew, or
intersecting (not perpencicular). The figure is a cube—all faces are squares. Place the correct
corresponding letter in the blanks below to reveal the mathematician who developed hyperbolic
geometry.
1. AB and FG
N
P
A
D
10. HC and AB
B
F
O
M
2. AG and PQ
E
B
J
I
11. GD and AB
I
T
A
X
3. FE and FG
Z
K
Q
A
12. FG and GD
L
C
F
D
4. CD and HE
O
C
V
B
13. FE and BC
H
A
E
D
5. AG and HD
L
E
M
K
14. AB and AG
R
D
L
E
6. PQ and HD
O
Y
A
R
15. GB and DE
N
Q
V
P
7. AH and HC
P
I
T
W
16. EH and DH
U
W
G
S
8. HD and JK
T
U
Y
L
17. HC and BC
K
K
I
L
9. FG and BC
H
E
O
B
18. AG and JK
D
C
Y
H
A
B
P
X
____ ____ ____ ____ ____ ____ ____
F
1
G
2
3
4
5
6
7
Q
J
H
C
____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____
8
9
10
11
12
13
14
15
16
17
18
Y
E
K
© Milliken Publishing Company
D
5
MP4056
Angles Formed by Parallel Lines
If two parallel lines are cut by a transversal, the resulting angles will either be
congruent or supplementary.
congruent angles
vertical angles
corresponding angles
alternate interior angles
alternate exterior angles
supplementary angles
adjacent angles
same–side interior angles
same–side exterior angles
Determine whether the angles listed are conguent, supplementary, or neither. If the
angles are congruent, write the corresponding letters in the top box below. If the angles
are supplements, write the corresponding letters in the lower box below.
1. ∠1 and ∠5
I
11.
∠4
and ∠7
F
21.
∠2
and ∠13
I
2. ∠1 and ∠6
G
12.
∠11
R
22.
∠4
and ∠8
W
3. ∠1 and ∠3
E
13.
∠5
and ∠6
C
23.
∠9
and ∠14
B
4. ∠5 and ∠10
O
14.
∠7
and ∠12
I
24.
∠1
and ∠13
T
5. ∠3 and ∠16
T
15.
∠10
and ∠14
N
25.
∠3
and ∠8
N
6. ∠7 and ∠11
S
16.
∠10
and ∠13
E
26.
∠3
and ∠15
O
7. ∠4 and ∠16
A
17.
∠1
and ∠14
D
27.
∠12
8. ∠6 and ∠9
T
18.
∠8
and ∠12
E
28.
∠5
and ∠9
N
9. ∠9 and ∠12
U
19.
∠2
and ∠5
L
29.
∠6
and ∠7
D
10. ∠6 and ∠10
A
20.
∠8
and ∠11
E
30.
∠4
and ∠15
Z
and ∠16
The creators of calculus were:
(congruent)
and ∠15
l
k
__ __ __ __ __ __ __ __ __
1
__ __ __ __ __ __ __
I
2
5 6
m
3 4
7 8
m || n
AND
(supplementary)
9
10
13 14
__ __ __ __ __
11 12
15 16
n
__ __ __ __ __ __
© Milliken Publishing Company
6
MP4056
Angle Sum Theorem
The three angles of a triangle add to 180°.
Find the missing angle measure for each triangle. Use the decoder to read the message
below.
DECODER
1. 25°, 65°,
_______
A = 90°
2. 42°, 120°,
_______
C = 111°
3. 37°, 90°,
_______
E = 135°
4. 23°, 46°,
_______
F = 95°
5. 50°, 105°,
_______
G = 24°
6. 136°, 20°,
_______
H = 100°
7. 15°, 30°,
_______
I
8. 30°, 45°,
_______
L = 93°
9. 52°, 35°,
_______
M = 86°
10. 67°, 13°,
_______
N = 18°
11. 74°, 100°,
_______
O = 6°
12. 9°, 110°,
_______
P = 130°
13. 24°, 26°,
_______
R = 25°
14. 10°, 84°,
_______
S = 61°
15. 17°, 25°,
_______
T = 105°
16. 40°, 45°,
_______
U = 138°
17. 53°, 35°,
_______
Y = 92°
____ ____ ____
8
10
____ ____ ____ ____ ____
7
1
4
15
____ ____
11
8
____
16
____ ____ ____ ____ ____ ____
7
1
2
5
3
1
6
9
7
12
____ ____ ____ ____ ____
1
5
3
6
____ ____ ____ ____ ____ ____ ____ ____
8
= 53°
2
6
9
10
8
____ ____ ____
7
1
5
7
____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____
4
11
© Milliken Publishing Company
14
13
9
7
14
7
7
2
8
1
5
17
MP4056
Exterior Angle Theorem
In any triangle, the measure of one exterior angle is equal to the sum of its remote
interior angles.
An exterior angle and its adjacent interior angle are supplementary.
2
∠1 + ∠2 = ∠4
∠3 + ∠4 = 180°
∠1 + ∠2 + ∠3 = 180°
3 4
1
1.
2.
y
3.
x
y
y
x = _____
y = _____
4.
x z
x = _____
z = _____
y = _____
5.
x
x = _____
6.
x
y = _____
x
z
w
z
x
y
y
x = _____
7.
x
y
y = _____
w = _____
y = _____
x = _____
z = _____
y
w = _____
y = _____
w
x = _____
z = _____
z
x = _____
z = _____
© Milliken Publishing Company
y = _____
8
MP4053
Classifying Triangles
by sides
by angles
equilateral — all three sides equal
isosceles — at least two sides equal
scalene — no sides equal
equiangular — all three angles equal
acute — three acute angles
right — one right angle
obtuse — one obtuse angle
Classify the triangles below. There will be at least 2 answers per triangle, maybe more.
Check all columns that apply in the chart. Write the letters corresponding with those
columns checked in the blanks below to reveal the “Prince of Mathematicians”.
1.
2.
3.
15
8
6
5
4
10
10
10
3
4.
6.
5.
7
7
eq
ui
la
te
ra
iso
l
sc
el
es
sc
al
en
e
eq
ui
an
gu
la
ac
r
ut
e
7
rig
ht
ob
tu
se
7.
1.
E
L
C
U
D
A
F
2.
B
R
N
O
C
I
L
3.
N
W
F
T
S
R
P
4.
A
C
I
G
E
Y
H
5.
D
R
P
I
C
S
W
6.
V
M
H
Q
E
D
G
7.
R
A
E
D
H
U
I
8.
P
S
M
L
S
N
C
4
4
8.
5
5
____ ____ ____ ____
____ ____ ____ ____ ____ ____ ____ ____ ____
© Milliken Publishing Company
9
____ ____ ____ ____ ____
MP4056
The Pythagorean Theorem
In a right triangle, the square of the hypotenuse is equal to the sum of the
squares of the legs.
c
a
The hypotenuse is opposite the right angle.
The legs form the right angle.
b
2
2
2
Pythagorean Theorem: a + b = c
Solve for the missing side. Match your answer in the decoder to find the special name
for three integers whose lengths form a right triangle.
1.
2.
6
3.
5
8
15
8
12
6.
5.
4.
8
4
10
12
3
9
7.
7
© Milliken Publishing Company
24
10
MP4056
8.
9.
10.
15
10
4
17
26
5
11.
12.
13.
16
15
15
20
25
12
A
E
G
H
17
5
15
3
I
L
10
24
N
9
O
20
P
8
R
S
T
Y
25
12
13
6
____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____
9
6
2
8
3
5
13
7
4
3
12
____ ____ ____ ____ ____ ____ ____
2
© Milliken Publishing Company
7
1
9
11
10
4
11
MP4056
The Converse of the Pythagorean Theorem
The Pythagorean Theorem can also be used to determine whether
any triangle is acute, right, or obtuse.
I. Find two shorter sides, square each, add the squares.
II. Square the longest side.
III. Compare the results:
longest side2 < short side2 + short side2 ⇒ ACUTE
longest side2 = short side2 + short side2 ⇒ RIGHT
longest side2 > short side2 + short side2 ⇒ OBTUSE
OB
TU
SE
LENGTHS
RI
GH
T
AC
UT
E
Determine whether the following lengths create an acute, right, or obtuse triangle.
Check the corresponding column in the chart and write its letter in the blanks below.
1.
3, 4, 5
A
T
D
U.S. Olympian Mike Conley won
2.
9, 9, 13
I
P
H
a silver medal in the 1984
3.
11, 11, 15
E
S
M
Summer Olympics and a gold medal
4.
7, 7, 7
T
A
C
for the same event in the 1992
5.
6, 8, 10
U
R
H
Summer Olympics. In which
6.
8, 10, 12
I
L
S
event did he compete?
7.
4, 6, 8
A
Z
P
8.
5, 12, 13
M
L
N
9.
8, 14, 17
T
I
E
10.
6, 7, 8
J
H
O
11.
9, 12, 15
E
U
L
12.
13, 14, 15
M
O
N
13.
7, 8, 11
S
U
P
© Milliken Publishing Company
____ ____ ____
____ ____ ____ ____ ____ ____
____ ____ ____ ____
12
MP4056
Congruent Figures
Two figures are congruent if they are the same size and shape. Congruent
parts of the figures (segments and angles) are called corresponding parts.
Name the corresponding parts for the congruent figures below. Remember that the order of the
letters is important.
1.
L
2.
P
T
K
M
N
R
a.
b.
c.
d.
e.
f.
g.
3.
ΔLMN ≅ Δ ______________________
ML ≅ __________________________
MN ≅ _________________________
∠L ≅ ∠ ________________________
PS ≅ __________________________
∠R ≅ ∠ ________________________
∠N ≅ ∠ ________________________
D
A
a.
b.
c.
d.
e.
f.
g.
L
D
I
R
S
C
B
U
N
TRUCK ≅ ______________________
KC ≅ _________________________
TR ≅ _________________________
CU ≅ _________________________
UR ≅ _________________________
TK ≅ _________________________
∠C ≅ ∠ ________________________
Q
P
T
R
a.
b.
c.
d.
DART ≅ ___________
TR ≅ ______________
QA ≅ _____________
∠T ≅ ∠ _______________
© Milliken Publishing Company
e.
f.
g.
h.
PQ ≅ ______________
RP ≅ ______________
∠DAR ≅ ∠ ____________
∠TRA ≅ ∠ ____________
13
i. PRAQ ≅ ____________
j. TDAR ≅ _____________
k. RPQA ≅ ____________
MP4056
Congruent Triangles
Three Methods of Proving Triangles Congruent
3 sides of one Δ ≅ to 3 sides of another Δ.
2 sides and the included angle of one Δ ≅
to 2 sides and the included angle of another Δ.
2 angles and the included side of one Δ ≅ to
2 angles and the included side of another Δ.
Side—Side—Side (SSS)
Side—Angle—Side (SAS)
Angle—Side—Angle (ASA)
Determine which of the above methods can prove the triangles congruent. If SSS, place an
X in the left letter box; if SAS, place an X in the middle letter box; if ASA, place an X in the
right letter box. The remaining letters will spell the name of a U.S. president who developed a
proof of the Pythagorean Theorem.
3.
2.
1.
RME
J AB
4.
S P A
5.
6.
S A M
B R E
7.
G L A
8.
R F L
9.
I E A
____ ____ ____ ____ ____
L T D
____ ____ ____ ____ ____
____ ____ ____ ____ ____ ____ ____ ____
© Milliken Publishing Company
14
MP4056
Congruent Triangles
Two Additional Triangle Congruence Methods
Angle—Angle—Side (AAS)
Hypotenuse—Leg (HL)
Use the following box as a guideline.
Two angles and a nonincluded side of one Δ ≅ to
two angles and a nonincluded side of another Δ.
In a right Δ, the hypotenuse and one leg ≅ to the
hypotenuse and leg of another right Δ.
SSS SAS ASA AAS HL
Place an X in all the boxes that can be applied to prove the triangles congruent.The remaining
letters will reveal the name of a process used by Archimedes to determine the volume of a
sphere using equal weights.
1.
3.
2.
M P E T H
T H A M E
4.
5.
T H E Q U
7.
O D O F R
6.
S T I L I
B R U S I
8.
M T H O D
U H E A P
____ ____ ____
____ ____ ____ ____ ____ ____
____ ____
____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____
© Milliken Publishing Company
15
MP4056
Angle and Triangle Word Search
Find the words listed below in the puzzle.
S
H
G
D
O
P
T
K
V
H
D
J
D
D
X
S
M
A
R
R
O
T
C
E
S
I
B
K
B
R
E
U
E
O
P
P
O
S
I
T
E
G
G
E
M
L
A
N
R
U
R
A
L
U
G
N
A
I
U
Q
E
S
B
E
S
N
J
S
T
R
A
I
G
H
T
C
A
M
I
N
E
H
M
A
T
S
K
L
A
N
S
G
A
R
O
X
O
D
R
S
T
E
U
A
S
O
S
I
E
K
T
E
O
S
T
H
H
S
B
C
S
R
H
V
D
A
L
T
I
P
X
W
G
O
L
I
O
S
B
E
Y
E
L
R
O
E
Q
U
I
L
A
T
E
R
A
L
V
T
S
E
E
R
D
P
R
Z
D
R
A
F
S
J
L
E
U
V
M
P
Y
V
E
S
J
E
S
U
E
F
S
V
P
I
U
E
E
T
U
C
A
V
D
W
R
U
B
C
P
C
C
N
J
H
W
A
C
N
M
A
N
R
K
B
L
R
U
D
E
G
G
L
E
E
M
E
A
N
D
A
E
V
E
I
H
W
A
E
N
I
T
N
V
E
U
K
M
X
F
C
E
G
K
N
T
O
B
T
U
S
E
Y
E
A
D
U
E
K
G
E
P
N
M
K
R
A
N
E
N
S
N
L
A
T
H
Y
A
N
S
M
A
I
L
A
T
N
D
A
E
R
H
U
S
A
R
S
H
U
L
L
A
C
Y
R
A
T
N
E
M
E
L
P
M
O
C
H
R
D
W
I
V
K
T
S
A
I
W
A
L
K
E
R
Y
A
obtuse
acute
scalene
right
equilateral
equiangular
isosceles
bisector
© Milliken Publishing Company
straight
sides
complementary
supplementary
16
vertical
hypotenuse
leg
adjacent
opposite
perpendicular
base
vertex
MP4056
Types of Quadrilaterals
Quadrilateral
Trapezoid
(only one pair
parallel sides)
Parallelogram
(two pairs of
parallel sides)
Rectangle
(four right angles)
Rhombus
(all sides equal)
Square
Check all terms that apply for the shapes below. Remember, more than one column may be
checked for a shape. Write the remaining letters in the blanks below to answer the question.
4
3
8
9
1.
2.
3.
4.
5
5
7
5
8
5. 3
6.
7.
6
2
9.
5
4
7
7
5
8
7
5
3
W
I
N
T
E
N
W
O
E
4
8.
5
5
3
quadrilateral
4
9
5
1.
2.
3.
4.
5.
6.
7.
8.
9.
6
6
10
10
4
6
12
10
trapezoid parallelogram
A
S
L
R
A
W
A
R
G
rectangle
rhombus
H
A
I
N
O
I
A
H
T
L
C
F
R
H
I
I
I
O
S
K
G
O
G
A
W
S
G
square
A
A
O
I
A
W
I
N
N
Which four U.S. states have active volcanoes?
____ ____ ____ ____ ____ ____ , ____ ____ ____ ____ ____ ____ ____ ____ ____ ____
____ ____ ____ ____ ____ ____ , and ____ ____ ____ ____ ____ ____ ____ ____ ____ ___
© Milliken Publishing Company
17
MP4056
Properties of Parallelograms
Parallelograms have all of the following properties:
—both pairs of opposite sides parallel
—both pairs of opposite sides congruent
—both pairs of opposite angles congruent
—diagonals bisect each other
Refer to the diagram to determine the following measures. Place the answers in the
cross–number puzzle.
I
K
2
3 4
1
S
78
E
ACROSS
5
6
T
1. If KI = 12, then ET = ______________________.
2. If KE = 23, then IT = ______________________.
3. If ∠KET = 150° and ∠3 = 70°, then ∠4 = ______°.
4. If ES = 7, then EI = ______________________.
DOWN
5. If ∠3 = 60° and ∠4 = 70°, then ∠ITE _________°.
1. If ∠IKE = 55°, then ∠KET = ________°.
6. If KT = 36, then ST = _____________________.
2. If KT = 48, then KS = ______________.
8. If ∠1 = 30° and ∠2 = 40°, then ∠KET = _______°.
3. If ∠EKI = 85 °, then ∠ETI = ________°.
9. If ∠EKI = 45°, then ∠KIT = ________________°.
6. If SI = 5, then EI = ________________.
10. If KS = 11.5, then KT = ___________________.
....
....
....
....
3
2
1
....
....
....
....
....
....
....
....
....
....
....
....
4
....
....
....
............................
....
....
....
....
5
6
7
....
....
....
....
....
....
....
....
....
....
....
....
....
....
8
....
....
....
....
........
....
........
....
....
....
....
....
9
10
....
....
....
....
....
....
....
....
....
....
....
........
........
....
....
....
....
....
....
........
........
....
© Milliken Publishing Company
18
7. If ∠1 = 30° and ∠2 = 40°, then
∠5=
__________°.
8. If ∠1 = 20° and ∠7 = 10°, then
∠KSE
= __________°.
9. If KS = 5, then KT = _______________.
10. If ∠EKI = 25°, then ∠ETI = _________°.
MP4056
Properties of Rectangles, Rhombuses, and Squares
Rectangles
all properties of parallelograms
plus
—diagonals are congruent
—all angles measure 90°
Rhombuses
all properties of parallelograms
plus
—all sides are congruent
—diagonals are perpendicular
—diagonals bisect opposite angles
Squares
—all properties of parallelograms
—all properties of rectangles
—all properties of rhombuses
Use the properties to find measures of segments and angles in the diagrams.
1. ABCD is a rectangle. If AB = 24, BC = 10, and ∠1 = 50°, find the following:
A
1 2
3
B
4
d. BD = ______ g. ∠DAB = ______
e. AX = ______ h. ∠3 = ________
f. BX = ______ i. ∠AXB = ______
a. CD = ______
b. AD = ______
c. AC = ______
X
C
D
2. ABCD is a rhombus. If AB = 6, XC = 3, and ∠DAB = 120°, find the following:
a.
b.
c.
j.
BC = _________
d. ∠AXB = _____
g. ∠3 = _____
= ______
e. ∠1 = _______
h. ∠4 = _____
= ______
f. ∠2 = _______
i. AX = _____
ΔABC is an _____________________________triangle.
∠ADC
∠DCB
A
3
1 2
B
4
X
D
C
3. ABCD is a square. If AB = 16 and AC = 16 2 , find the following:
A
B
1
2
3
4
a.
b.
c.
d.
BC = ________
BD = ________
AD = ________
∠1 = _________
e.
f.
g.
h.
∠2 = ___________
∠AXB = _________
∠BXC = _________
∠4 = ___________
X
D
© Milliken Publishing Company
C
19
MP4056
Trapezoids
All trapezoids have exactly one pair of parallel sides (called bases).
An isosceles trapezoid has congruent legs, base angles, and diagonals.
A right trapezoid has two right angles.
Any trapezoid can be divided into a rectangle and triangle(s) to help
find measurements of the sides and angles. To divide the trapezoid,
draw altitudes between the bases.
right trapezoid
isosceles trapezoid
general trapezoid
Use the properties of trapezoids, rectangles, and right triangles to find the missing measures in
the diagrams below. Remember to use the Pythagorean Theorem in right triangles.
6
1.
7
2.
5
10
6
4
4
z
10
5
6
6
y
x
20
b
4.
4
g
5.
f
c
3
d
a
3.
9
6
6.
e
12
10
5
7.
25
8
5
j
4
13
12
12
13
k
m
© Milliken Publishing Company
20
MP4056
Ratio and Proportion
An equation that sets two ratios equal to one another is a proportion.
Use the means–extremes property to solve a proportion.
The product of the means is equal to the product of the extremes.
a:b=c:d
a = c
means
OR
b
d
extremes
b and c are means
a and d are extremes
bc = ad
bc = ad
Solve the proportions below. Use the decoder to find the name of a special ratio and
where it is found in ancient times.
1. 7 = 21
2. 3x = 12
3. x + 5 = 1
4
x
6
4
4
2
4. x + 2 = 4
x+3
5
5. 18 = 9
x
4
6. 3 = 18
4
x
7. 36 = 18
x
8
8. x = 84
9 108
9. x – 3 = 4
x
5
10. x = 6
6
4
11. 7x – 5 = x + 9
4
3
12.
13. x + 6 = x
16
10
14. x + 9 = 5x
2
1
15. 4x + 2 = 18
6x + 13 17
16. x = 9
12
4
17. 9 = 3
4x 1
18. 2x + 1 = 4x – 3
18
21
19. x – 1 = 6
x
8
20. 2x + 4 = 13
3x – 3
15
21. 5 = x
8
40
A B C D
15 24 16 12
E
2
____ ____ ____
11
13
4
____ ____ ____
3
9
8
____ ____ ____
11
13
4
____ ____ ____ ____
9
2
8
10
F G H I K
1 34 10 –5 4
M
5
2
N O P
11 9 27
____ ____ ____ ____ ____ ____
17
10
2
1
4
20
____ ____ ____ ____
21
8
4
1
____ ____
11
10
R
5
____ ____ ____ ____ ____
17
12
4
4
19
S
7
T U W
3 25 –3
Y
8
____ ____ ____ ____ ____
12
9
11
15
10
____ ____ ____ ____ ____ ____
1
4
8
15 17
20
____ ____ ____ ____ ____ ____ ____ ____
16
5
12
9
18
15
1
8
____ ____ ____
9
20
1
© Milliken Publishing Company
L
6
5 = 15
x–2
x+4
____ ____ ____
9
20
1
____ ____ ____ ____ ____ ____ ____ ____ ____
6
21
15
2
1
15 20
17
8
____ ____ ____ ____ ____ ____ ____ ____ ____.
9
12
11 15 14
9
7
11
8
21
MP4056
Similar Figures
Figures are called similar when their corresponding sides are all in
proportion and their corresponding angles are congruent.
A
D
7
7
14
B
7
14
C
E
ΔABC ~ ΔDEF
F
14
ratio of sides is 7 : 14 or 1 : 2.
Use proportions to solve for the missing sides in the similar figures below.
1.
2.
4
a
y
8
8
4
8
x
3.
5
b
4.
8
6
12
3
6
6
c
e
8
f
8
g
10
i
18
h
d
5.
4
r
4
6
6.
s
12
6
5
13
t
v
w
u
7
39
14
© Milliken Publishing Company
22
MP4056
Trigonometric Ratios
Chief SOHCAHTOA says, “Use my name to remember the three basic
trigonometric ratios.”
Opposite
Sine =
Hypotenuse
Adjacent
Cosine =
Hypotenuse
Opposite
Tangent =
Adjacent
Use the triangles below to match the trigonometric ratios. Your answers will help you to find the
name of the first woman in American history to receive a patent.
A
V
8
D
E
12
S
R
X
25
4
5
B
17
C
3
5
15
1. sin ∠D
6. tan ∠C
11. sin ∠A
16. cos ∠W
cos ∠Y
tan ∠F
tan ∠Z
tan ∠D
2.
7.
12.
17.
13
9
W
T
F
24
tan ∠A
sin ∠C
tan ∠V
sin ∠Y
3.
8.
13.
18.
U
7
4.
9.
14.
19.
Y
sin ∠R
cos ∠R
cos ∠D
tan ∠W
5. cos ∠V
10. tan ∠T
15. tan ∠R
20. tan ∠Y
9
41
7
25
12
13
9
40
24
25
3
4
4
3
5
12
15
17
12
5
7
24
5
13
15
8
40
9
4
5
A
C
D
E
F
G
H
I
K
M
N
O
P
R
S
T
1
7
17
____ ____ ____ ____ ____
18
16
6
14
13
____ ____ ____ ____ ____ ____ ____ ____
17
9
2
9
6
____ ____ ____
12
13
19
9
____ ____ ____
17
3
9
© Milliken Publishing Company
7
2
3
5
V
W
8
17
40
41
X
Y
6
9
20
____ ____ ____ ____ ____ ____
4
7
8
9
10
8
____ ____ ____ ____ ____ ____ ____
17
3
15
7
11
9
7
____ ____ ____ ____ ____ ____ ____
1
24
7
____ ____ ____ ____
10
____
16
Z
41
8
15
____ ____ ____ ____
40
6
10
23
9
19
6
____ ____
6
10
5
1809.
10
MP4056
Perimeter
Perimeter is the distance around a figure. Simply add the lengths of
all sides that outline the figure to find its perimeter. Perimeter is
measured in linear units, such as inches, centimeters, yards, meters,
and so on.
Find the perimeter of each figure below. Shade the answers below to reveal the number
of bones in the human body.
2.
3.
1.
15 cm
8 in
6m
8 in
4 cm
8m
7 in
4.
5.
10 ft
6.
9m
15 cm
13 cm
9m
5 ft
5 cm
5 cm
3m
3 ft
8.
3 in
4 in
9 ft
8 cm
6 cm
1 cm
10. An equilateral triangle with sides of 8 cm.
11. A square with sides of 18 inches.
12. A regular octagon with sides of 5 cm.
© Milliken Publishing Company
9.
15 ft
10 cm
10 in
4 cm
2 cm
7.
13. A regular hexagon with sides of 10 ft.
14. A regular pentagon with sides of 6 yards.
15. A rhombus with sides of 9 mm.
24
MP4056
Area of Rectangles and Triangles
The area of a figure is the number of square units needed to fill its
interior. Area of some shapes can be found by using formulas.
Area of Rectangle
Area of Triangle
3 units
3 units
8 units
8 units
Area = 24 square units
OR
Area = base x height
8 units x 3 units = 24 units2
Counting square units is more difficult
than multiplication, but this triangle is
really half of the rectangle to the left.
1
Area = 2 (base x height)
Area = 12 (8 units x 3 units)
= 1 (24 units2)
2
= 12 units2
Remember: Base and height must be perpendicular.
Find the area of the figures below. Write the answers in the cross–number puzzle.
ACROSS
12
1.
3.
4.
5.
7
13
5
16
16
14
10
8
8.
3
2
1
9x
8
4
x
5
6
perimeter = 60
7
8
DOWN
4
1.
2.
37
3.
13
5
4.
15
50
8
9
6
6.
3
3
7.
5
5
15
8
6
6
© Milliken Publishing Company
25
MP4056
Area of Trapezoids
All trapezoids can be divided into a rectangle and one or two triangles.
These areas can be found separately and then added together, or the
formula for the area of a trapezoid can be used.
Area Addition Method
Formula Method
10
10
5
I
4
4
3
II
5
5
3
3
10
5
4
4
3
10
1
Rectangle = 10 x 4 = 40
1
ΔI = 2 (3 x 4) = 6
1
ΔII = 2 (3 x 4) = 6
total area = 52 units2
Area = 2 (height) (base 1 + base 2)
1
= 2 (4) (10 + 16)
= 12 (4) (26)
= 52 units2
Remember: Height is always perpendicular to the bases.
Find the area of the following trapezoids. Follow the answers through the maze.
1.
2.
10
5
4
13
4.
7
5.
6
10
5
3
13
6.
14
13
2
8.
12
13
7.
7
5
4
3
5
3.
7
8
9
8.
10
4
10
8
9.
6
10
13
13
6
3
14
6
20
© Milliken Publishing Company
26
MP4056
10.
11.
12.
10
12
12
5
5
5
5
13
12
16
20
14.
13.
15.
12
3
5
15
12
7
12
15
3
5
9
© Milliken Publishing Company
27
MP4056
Area of Parallelograms
A parallelogram can be divided into two congruent triangles and
one rectangle. Then the area formulas can be applied:
II
5
4
Notice what happens if we use the
formula for parallelograms.
5
I
3
6
area of rectangle = 4 x 6 = 24
1
area of ΔI = 2 (3 x 4) = 6
1
area of Δ II = 2 (3 x 4) = 6
total area = 36 square units
A = base x height
A = 9x4
A = 36 square units
Remember: Base and height must be perpendicular.
Find the area of the parallelograms. Match answers with the words below to learn
about Rear Admiral Grace Hopper.
1.
3
2.
7
3.
12
10
5
8
15
3
lady
first
COBOL
24
4.
5.
6.
7
10
2
4
5
1
computer
3
also
of
7.
8.
9.
5
2
17
13
4
7
known
© Milliken Publishing Company
5
3
coined
28
8
the
MP4056
10.
11.
12.
3
9
3
8
10
5
4
10
5
phrase
13.
was
14.
15
since
15.
18
8
5
12
12
3
2
she
13
bug
inventors
16.
17.
18.
6
5
8
10
4
4
1
2
5
software
________
__________
60
________
72
_________
60
_____________________
_______________
165
© Milliken Publishing Company
18
80
_________
44
________________.
180
________________
52
29
________
32
28
_________
18
______________________”
_______
216
______
20
28
______________
120
______
144
_______________
40
____________
________________
44
“___________
as
one
_________
28
__________
60
“______________
240
18
___________
8
_________”
156
MP4056
Word Search
As you find the words in the list below, shade in the squares containing the letters.
When finished, the remaining letters (read horizontally from top to bottom) will
explain what is so special about the number 1991 and the phrase, “some men interpret
nine memos.”
C
O
N
C
E
N
T
R
I
C
A
I
N
S
C
R
I
B
E
D
P
E
R
A
L
S
E
C
A
N
T
T
L
C
I
N
D
T
R
R
O
M
A
G
U
E
I
S
N
A
C
N
U
N
N
M
M
B
E
E
R
O
R
W
G
A
S
E
M
I
C
I
R
C
L
E
L
C
O
S
R
D
T
H
A
T
N
A
R
I
S
U
T
H
D
E
S
T
R
I
A
M
E
I
F
O
R
R
W
T
B
A
R
D
S
D
A
N
O
D
N
E
B
A
C
K
W
A
A
R
H
E
D
I
A
M
E
T
E
R
D
S
C
Word List:
____
RADIUS
DIAMETER
CHORD
ARC
SEMICIRCLE
CENTRAL ANGLE
CENTER
SECANT
CONCENTRIC
INSCRIBED
CIRCUMSCRIBED
TANGENT
____ ____ ____ ____ ____ ____ ____ ____ ____ ____
____ ____ ____ ____ ____ ____
____ ____ ____ ____
____ ____
____ ____
____ ____ ____
____ ____ ____ ____ ____ ____ ____ ____
____ ____
____
____ ____ ____ ____
____ ____ ____ ____
____ ____ ____
____ ____ ____ ____ ____ ____ ____ ____ ____ .
© Milliken Publishing Company
30
MP4056
Circumference
Circumference is the distance around a circle. Think of circumference
as the circle’s perimeter. Find circumference using the formula.
Circumference = π (diameter)
OR
C = πd
Remember: The diameter is twice the length of a radius.
Find the circumference for the circle. Leave your answers in terms of π. Match your
answers in the decoder to answer the riddle.
1. radius = 4 cm
5. radius = 11 ft
2. diameter = 18 m
6. diameter = 4 m
10. radius = 18 m
3. diameter = 10 cm
7. radius = 10 ft
11. radius = 9 ft
4. radius = 3 cm
8. diameter = 11 m
18π m
A
4π m
11π m
3π in
10π cm
18π ft
D
E
I
K
L
9. diameter = 3 in
8π cm 20π ft
M
6π cm
36π m
O
P
N
22π ft
U
What do mathematicians like to eat on Thanksgiving?
____ ____ ____ ____ ____ ____ ____
10
5
____
2
© Milliken Publishing Company
1
10
____ ____
11
3
9
____ ____
7
10
9
____ ____ ____ ____ .
2
1
31
4
6
8
MP4056
Area of a Circle
The area of a circle is found by using the formula
Area = π (radius)2
OR
A = πr2
Remember: Use square units when finding area.
Find the area of the circles described. Shade your answers below to reveal the name
of the mathematician who invented the roulette wheel.
1. radius = 4 cm
11. diameter = 4 cm
2. diameter = 18 cm
12. radius = 10 cm
3. diameter = 10 m
13. radius =13 cm
4. radius = 7 in
14. diameter = 28 in
5. radius = 3 m
15. diameter = 40 cm
6. diameter = 12 ft
16. radius = 3 ft
7. radius = 12 in
17. diameter = 32 in
8. diameter = 30m
18. radius = 9 ft
9. diameter = 22 cm
19. radius = 18 cm
10. radius = 8 m
© Milliken Publishing Company
20. diameter = 34 m
32
MP4056
Area of Irregular Shapes
Dividing irregular shapes helps in finding their area.
Try to form shapes that are listed below, then use the area formulas
to solve for the total area.
1
1
Triangle: A = 2 bh
Rectangle: A = bh
Parallelogram: A = bh
Trapezoid: A = 2 h(b1 + b2 )
Circle: A = πr 2
Use the decoder to find the animal with the highest blood pressure by solving for the area of each
shape.
1.
2.
3.
5
5
5
10
10
10
8
10
12
10
3
2
10
10
2
8
15
4.
5.
6.
10
5
10
5
10
10
10
7.
13
13
6
12
8.
14
8
5
5
4
10
5
16
5
8
96
224
A
E
9π
12 + 2
F
____ ____ ____
5
© Milliken Publishing Company
8
7
136
88
G
H
240
I
92
100 + 50π
R
T
____ ____ ____ ____ ____ ____ ____
2
6
33
1
3
4
4
7
MP4056
Area of a Shaded Region
To find the area of a shaded region, add or subtract areas of basic
figures (rectangle, triangle, circle, and so on).
Area of Large Circle = 72π = 49π
Example:
5
2
2
— Area of Small Circle = 5 π = 25π
Area of Shaded Ring = 49π – 25π = 24π units2
Find the area of the shaded regions below. Use your answers to reveal the game
James Naismith invented.
1.
2.
3.
4
3
8
6
4
10
10
4.
5.
6.
10
2
10
7
10
6
12
6
7
3 5
4
4
8
10
7.
10
13
12
12
13
25π – 24
84
60
400 – 100π
33π
51
18
A
B
E
K
L
S
T
____ ____ ____ ____ ____ ____ ____ ____ ____ ____
2
© Milliken Publishing Company
3
5
4
7
6
34
2
3
1
1
MP4056
Three–Dimensional Figures
Unscramble the words below. All are terms related to three dimensional figures. Then
place the circled letters in the blanks to reveal the diameter of a golf hole.
1.
AEFC
____ ____ ____ ____
2.
OCNE
____ ____ ____ ____
3.
LOVEUM
____ ____ ____ ____ ____ ____
4.
HGIRT
____ ____ ____ ____ ____
5.
L N A ST
EGIHTH
____ ____ ____ ____ ____
____ ____ ____ ____ ____ ____
6.
YAIPRDM
____ ____ ____ ____ ____ ____ ____
7.
TTADLIUE
____ ____ ____ ____ ____ ____ ____ ____
8.
BQEULIO
____ ____ ____ ____ ____ ____ ____
9.
LRTLAEA DGEE
____ ____ ____ ____ ____ ____ ____
____ ____ ____ ____
10.
ERXTVE
____ ____ ____ ____ ____ ____
11.
SDRAUI
____ ____ ____ ____ ____ ____
12.
YLCDREIN
____ ____ ____ ____ ____ ____ ____ ____
13.
UBCE
____ ____ ____ ____
14.
HEESPR
____ ____ ____ ____ ____ ____
15.
SPMRI
____ ____ ____ ____ ____
____ ____ ____ ____
____ ____ ____
____ ____ ____ ____ ____ ____ ____
© Milliken Publishing Company
35
____
____ ____ ____ ____ ____ ____
MP4056
Surface Area of Right Prisms
Right prisms have rectangles for lateral faces. Their bases can be any
polygon. To find total surface area, add the area of all lateral faces and
bases.
5
1
Example:
area of base = 2 (3 x 4) = 6
4
3 area of lateral faces = (4 x 10) + (5 x 10) + (3 x 10)
= 40 + 50 + 30
10
= 120
total area = 2 bases + lateral faces
= 6 + 6 + 120
= 132 units2
Find the total area of the right prisms.
1.
2.
8 cm
5 cm
12 in
13 in
12 cm
20 in
5 in
4.
10 m
3.
4m
10 m
5m
10 mm
15 mm
3 mm
25 m
16 m
5.
6.
15 cm
12 in
5 in
4 in
10 in
25 cm
12 cm
6 in
9 cm
7. A cube with base edge of 5 cm.
© Milliken Publishing Company
8. A right triangular prism whose
bases are right triangles with sides
6 cm, 8 cm, and 10 cm, and whose
height is 12 cm.
36
MP4056
Volume of Rectangular Prisms
The volume of a rectangular prism is found by multiplying length, width,
and height. Another way to think of it is finding the area of the base
(length x width) and multiplying by height. Volume is measured in
cubic units (units3).
OR
Volume = length x width x height
Volume = (area of base) x height
Find the missing measures in the chart below.
length
width
height
1.
10 cm
5 cm
8 cm
2.
20 m
10 m
15 m
3.
14 in
6 in
9 in
4.
3 ft
4 ft
5 ft
5.
25 mm
10 mm
5 mm
6.
7 cm
8 cm
4 cm
7.
13 m
6m
3m
8.
24 ft
9 ft
16 ft
9.
8 in
12 in
15 in
10.
30 mm
50 mm
60 mm
11.
15 cm
8 cm
12.
10 m
13.
6 mm
volume
480 cm3
15 m
750 m3
18 mm
2592 mm3
14.
100 ft
20 ft
100,000 ft3
15.
9m
12 m
1080 m3
© Milliken Publishing Company
37
MP4056
Volume of Right Prisms
The volume of right prisms is found by multiplying the area of
the base by the height of the prism.
Remember: The bases are the parallel faces of the prism.
Find the volume of the prisms described. Use your answers to reveal what
Archimedes said upon discovering the principle of buoyancy in his bath.
1.
2.
12 m
4m
15 m
3m
10 m
10 m
5m
6m
8m
3.
4.
6 cm
25 cm
8m
12 cm
13 cm
13 cm
5 cm
18 cm
16 cm
12 cm
5.
5m
25 m
5m
10 m
14 m
10 m
6m
720 m3
3300 cm3
72 m3
E
K
A
1800 m3
R
540 cm3
U
____ ____ ____ ____ ____ ____
3
© Milliken Publishing Company
4
5
38
3
1
2
MP4056
Surface Area of Regular Pyramids
A regular pyramid has a regular polygon for its base and congruent
isosceles triangles for its lateral faces. To find the total surface area,
add the area of the base and the area of all lateral faces.
Example:
base edge = 5 cm
lateral edge = 6.5 cm
slant height = 6 cm
The orange line is the height of the triangular
face — also known as the slant height.
6.5 cm
6 cm
5 cm
5 cm
Area of base = 5 x 5 = 25 cm2
1
Area of lateral face = 2 (6)(5) = 15 cm2 x 4 faces = 60 cm2
Total area = 25 + 60 = 85 cm2
Find the total surface area for the regular pyramids below. Use your answers to determine
what the E represents in the formula E = mc2.
84 cm2
336 m2
112 cm2
756 m2
340 cm2
736 m2
N
E
G
E
Y
R
____ ____ ____ ____ ____ ____
4
3
6
m
10
13
m
5c
12 m
5.
6.
m
cm
15 m
17
5
cm
18 m
© Milliken Publishing Company
16 m
3
12 m
8m
m
m
10 cm
15 m
2
3.
12 c
4c
6 cm
4.
5
2.
cm
1.
1
8 cm
39
MP4056
Volume of Regular Pyramids
The volume of any regular pyramid is found by finding the area of the
base, multiplying it by the height (or altitude) of the pyramid, and then
dividing the result by 3 (which is the same as multiplying by 13 ).
Volume is measured in cubic units.
Volume =
1
3
(area of base)(altitude of pyramid)
The altitude of the pyramid is the distance from the center of the base to
the tip (or vertex) of the pyramid. Notice the right triangle formed by the
altitude, the slant height, and half the length of the base in a regular square
pyramid:
altitude
slant height
1
base
2
Use this triangle and the Pythagorean Theorem to find needed
measurements when they are not given.
Example: Find the volume of a regular square pyramid with base edge of 6 cm and slant
height of 5 cm.
32 + x2 = 52
Volume:
1
9 + x2 = 25
= 3 (area of base) (altitude)
slant height
5 cm
altitude
x2 = 16
= 13 (6 x 6) (4)
x = 4 = altitude
= 1 (36) (4)
1
base
3
2
= 48 cm3
3 cm
Find the volume of each of the following regular square pyramids using the information given.
1. base edge of 8 cm and altitude of 6 cm.
2. base edge of 10 cm and altitude of 3 cm.
3. base edge of 6 cm and altitude of 5 cm.
4. base edge of 5 cm and altitude of 3 cm.
5. base edge of 7 cm and altitude of 9 cm.
6. base edge of 8 cm and slant height of 5 cm.
7. base edge of 10 cm and slant height of 13 cm.
8. base edge of 24 cm and slant height of 15 cm.
9. slant height of 10 cm and altitude of 6 cm.
10. slant height of 17 cm and altitude of 15 cm.
© Milliken Publishing Company
40
MP4056
Surface Area of Cylinders
A cylinder’s surface area is made up of two circles (the top and bottom) and a
rectangle (the lateral surface; picture the lateral surface as the label on a can).
The rectangle’s dimensions include the circumference of the
circle and the height of the cylinder.
height
circumferemce
Total surface area: = 2 (area of circular base) + (circumference) (height)
= 2πr2 + 2πrh
r
= 2πr (r+h)
h
Example:
radius = 5 cm
height = 12 cm
Area of base = πr2 = π (5)2 = 25π
Area of rectangle = 2πrh = 2π (5) (12) = 120π
OR
2πr (r+h) = 2π (5) (5+12)
= 2π (5) (17)
= 170π cm2
2 bases = 50π cm2
+ rectangle = 120π cm2
total = 170π cm2
Find the total surface area for the cylinders described below. Then use your answers
to find the name of the wheel Blaise Pascal invented in a search for perpetual motion.
1. radius = 5 cm; height = 10 cm
2. radius = 4 cm; height = 10 cm
3. radius = 8 cm; height = 3 cm
4. radius = 10 cm; height = 7 cm
5. diameter = 24 m; height = 4 m
6. diameter = 30 m; height = 9 m
7. diameter = 10 m; height = 8 m
8. diameter = 18 m; height = 11 m
340π cm2
E
150π cm2
130π m2
H
L
____ ____ ____
8
1
176π cm2
384π m2
O
360π m2
R
112π cm2
T
720π m2
U
W
____ ____ ____ ____ ____ ____ ____ ____
4
5
3
2
7
4
8
8
4
____ ____ ____ ____ ____
6
© Milliken Publishing Company
1
4
41
4
7
MP4056
Surface Area of Cones
A cone’s surface area is made up of one circular base and a
triangular–shaped sector.
Remember: A cone has a slant height, similar to that in pyramids.
A CONE’S SURFACE:
slant height
slant height
r
r
circle
circumference
circle area:
A = πr2
triangular sector area
A = 12 base x height
A = 12 circumference x slant height
1
2
total surface area = πr + 2 (2πr) (slant height) A = 1 (2πr) (slant height)
2
OR
= πr2 + πr (slant height)
Find the total surface area of the cones described below. Use the decoder to find the answer to
the question.
1.
2.
3.
4.
m
m
4 cm
cm
5c
m
12
7c
10
9 cm
3 cm
7m
5.
6.
7.
8.
30 m
15
12
m
20
m
8c
cm
22 m
m
18 cm
12 cm
Who was the author of “Conic Sections,” a work containing propositions about curves that are
created by slicing a cone with a plane?
24π cm2
A
84π cm2
44π cm2
171π cm2
I
L
N
286π m2
133π m2
525π m2
P
S
O
189π cm2
U
____ ____ ____ ____ ____ ____ ____ ____ ____ ____
4
© Milliken Publishing Company
2
7
1
1
7
42
3
6
5
8
MP4056
Volume of Cylinders and Cones
The volume of a cylinder is found
by multiplying the area of the circular
base by the height of the cylinder.
V = (area of base) (height)
2
V = πr h
The volume of a cone is found by taking 1 of the
3
product of the area of the base and the height
(altitude)
1
V = 3 (area of base) (height)
V = 13 πr 2h
h
r
radius
h
Recall the right triangle created by the radius,
altitude, and slant height.
altitude
slant height
radius
Use the Pythagorean Theorem to find the
needed measurements when they are not given.
Find the volume of the figure described. Follow your answers through the maze.
1. cylinder with radius 8 m; height 12 m
2. cone with radius 6 m; altitude 10 m
3. cylinder with radius 5 m; height 10 m
4. cone with radius 12 cm; altitude 16 cm
5. cylinder with radius 18 cm; height 20 cm
6. cone with radius 9 m; altitude 7 m
7. cylinder with radius 3 cm; height 2 cm
8. cone with radius 5 m; altitude 12 m
9. cylinder with radius 11 cm; height 7 cm
10. cone with radius 15 cm; altitude 10 cm
© Milliken Publishing Company
11. cylinder with diameter 10 m; height 8 m
12. cylinder with diameter 20 m; height 20 m
13. cylinder with diameter 18 m; height 5 m
14. cone with diameter 6 cm; altitude 4 cm
15. cone with diameter 10 cm; altitude 9 cm
16. cone with diameter 16 m; altitude 6 m
17. cylinder with diameter 4 m; height 2 m
18. cone with radius 4 cm; slant height 5 cm
19. cylinder with diameter 8 m; height 12 m
20. cone with radius 12 cm; slant height 13 cm
43
MP4056
Crossword Puzzle
Across
Down
2
2
2
1. longest side of a right triangle
2. a + b = c
3. polygon with equal sides
4. four–sided polygon
6. larger than 90°
5. parallelogram with four equal sides
8. base of a cone
7. no equal sides
9. quadrilateral with two pairs of parallel sides
8. lateral face is a rectangle, bases
12. angles adding to 180°
are circles
13. ratio of opposite side to hypotenuse
10. smaller than 90°
14. ratio of opposite side to adjacent side
11. equal
15. polygon with equal angles
16. quadrilateral with four right angles
19. distance around a circle
17. six–sided polygon
21. nonparallel, nonintersecting lines
18. lines that form right angles
22. ten–sided polygon
20. 90° angle
23. half of a diameter
Word List
acute
circle
circumference congruent
cylinder
decagon
equiangular
equilateral
hexagon
hypotenuse obtuse
parallelogram
perpendicular Pythagorean quadrilateral
radius
rectangle
rhombus
right
scalene
sine
skew
supplementary tangent
1
2
3
4
5
6
8
7
10
9
11
12
13
14
15
16
18
17
19
20
21
22
23
© Milliken Publishing Company
44
MP4056
Answer Key
Page 1:
ONE
Page 2:
EUCLID’S ELEMENTS
Page 3:
1.
2.
3.
4.
5.
Page 4:
All in Philadelphia
Page 6:
congruent: GOTTFRIED LEIBNIZ
Page 7:
The acute angles of a right triangle are complementary
Page 8:
1. x = 98° , y = 82°
2. x = 75° , y = 135°
z = 105°
3. x = 160° , y = 50°
Page 9:
CARL FRIEDRICH GAUSS
Page 12:
THE TRIPLE JUMP
Page 13:
Page 14:
right
obtuse
acute
acute
straight
1a.
b.
c.
d.
e.
f.
g.
ΔPRS
RP
RS
∠P
LN
∠M
∠S
6.
7.
8.
9.
10.
obtuse
straight
obtuse
right
acute
11.
12.
13.
14.
15.
Page 5:
straight
right
obtuse
acute
straight
16.
17.
18.
19.
20.
NIKOLAI LOBACHEVSKY
supplementary: ISAAC NEWTON
4. x = 30° , y = 60°
5. w = 110° , x = 150°
y = 100° , z = 70°
2a.
b.
c.
d.
e.
f.
g.
obtuse
obtuse
acute
obtuse
straight
6. w = 85° , x = 45°
y = 90° , z = 50°
7. x = 115° , y = 65°
z = 75°
Page 11: PYTHAGOREAN TRIPLES
NDBLI
IL
ND
LB
BD
NI
∠L
JAMES ABRAM GARFIELD
3a.
b.
c.
d.
e.
f.
g.
QARP
PR
DA
∠P
TD
RT
∠QAR
h.
i.
j.
k.
∠PRA
TRAD
PQAR
RTDA
Page 15: THE METHOD OF EQUILIBRIUM
Answer Key
Page 16:
Page 17:
S
M
E
R
S
E
O
O
I
O
E
M
U
C
U
E
F
D
N
D
Y
W
H
A
O
U
N
H
D
S
P
E
R
P
E
N
D
I
C
U
L
A
R
I
G
R
P
R
J
M
R
T
X
Q
D
Y
E
J
E
H
E
E
A
E
A
V
D
R
P
A
S
A
S
H
W
U
P
V
T
H
G
W
G
K
T
R
T
K
O
O
O
L
T
T
T
H
G
I
R
E
U
W
G
A
K
G
H
H
N
T
P
T
S
U
R
S
E
S
O
L
Z
S
C
A
L
E
N
E
Y
U
E
S
T
C
I
G
A
K
U
B
L
A
D
J
A
C
E
N
T
P
A
S
M
A
K
E
T
N
I
L
A
C
I
T
R
E
V
N
E
I
O
N
N
A
E
I
V
S
E
A
G
A
S
S
O
E
A
S
D
M
M
T
B
M
S
R
L
W
H
I
G
I
H
N
O
R
S
R
F
U
W
A
E
N
T
K
M
S
P
A
D
B
G
U
T
S
S
H
B
A
S
E
R
N
A
V
U
R
A
H
M
L
J
K
E
Q
C
G
I
V
E
L
J
F
U
R
N
E
S
A
I
U
O
K
D
B
M
E
A
A
E
D
Y
V
L
S
B
K
D
U
E
N
L
L
C
E
D
R
L
S
M
R
K
A
E
T
E
V
C
B
A
K
Y
E
A
L
H
R
X
E
A
B
I
O
T
L
L
S
U
P
P
L
E
M
E
N
Y
A
R
Y
S
U
N
E
N
X
E
T
R
E
V
I
C
R
V
X
A
S
N
C
D
A
ALASKA, CALIFORNIA, HAWAII, and WASHINGTON
....
....
....
....
2
....
....3 8 0
1 ....
2 .... 2 ....
3 ....
....
....
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....
....
4
....
....
1
4
5
2 ....
............................
....
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5
6
7
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1
8
0
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Page 18:
Page 19:
1a.
b.
c.
d.
e.
f.
g.
h.
i.
24
10
26
26
13
13
90°
40°
100°
2a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
6
60°
120°
90°
60°
60°
30°
30°
3
equilateral
3a.
b.
c.
d.
e.
f.
g.
h.
16
16 2
16
45°
45°
90°
90°
45°
Answer Key
Page 20:
1. x = 14
2. y = 16
3. z = 10 , a = 4
Page 21:
The Golden Ratio was used to design the pyramids and also Greek buildings
and artifacts.
Page 22:
1. x = 8, y = 8 2
2. a = 12, b = 7.5
3. c = 16, d = 8
e = 16
Page 23:
Mary Dixon Kies received a patent for her weaving machine in 1809.
Page 24:
206
Page 25:
4. b = 6 , c = 3 , d = 5
5. e = 13 , f = 15, g = 10
4. f = 13.5, g = 13.5
h = 22.5, i = 18
5. r = 8, s = 8, t = 12
u = 12
19
4
2
5
3
3
6
5
3
7
6
88
1
0
6. v = 36, w = 15
0
4
0
2
6
area = 46
area = 40
area = 50
area = 54
area = 138
2
8
6. j = 5 , k = 14
7. m = 15
6 4
3
4
6.
7.
8.
9.
10.
1
Page 26–27:
1.
2.
3.
4.
5.
area = 72
area = 180
area = 72
area = 36
area = 48
11.
12.
13.
14.
15.
area = 174
area = 52
area = 42
area = 252
area = 27
Page 29:
She was known as the “first lady of software” since she was one of the
inventors of COBOL. She also coined the phrase “computer bug.”
Page 30:
A palindrome is a number or word that is the same forwards and backwards.
Page 31:
PUMPKIN PI A LA MODE
Page 32:
PASCAL
Page 33:
THE GIRAFFE
Page 34:
BASKETBALL
Page 35: FOUR AND A QUARTER INCHES
Answer Key
4. 450 mm2
5. 1008 cm2
6. 352 in2
Page 36
1. 392 cm2
2. 660 in2
3. 1004 m2
Page 37:
1.
2.
3.
4.
5.
Page 38:
EUREKA
Page 40:
1. 128 cm3
2. 100 cm3
3. 60 cm3
Page 41:
THE ROULETTE WHEEL
Page 43:
1.
2.
3.
4.
5.
Page 44:
Across
© Milliken Publishing Company
400 cm3
3000 m3
756 in3
60 ft3
1250 mm3
6.
7.
8.
9.
10.
7. 150 cm2
8. 336 cm2
224 cm3
234 m3
3456 ft3
1440 in3
90,000 mm3
11.
12.
13.
14.
15.
Page 39:
ENERGY
4. 25 cm3
5. 147 cm3
6. 64 cm3
768π m3
120π m3
250π m3
768π cm3
6480π cm3
2.
3.
6.
8.
9.
12.
13.
14.
15.
19.
21.
22.
23.
6.
7.
8.
9.
10.
7.
8.
9.
10.
Page 42:
189π m3
18π cm3
100π m3
847π cm3
750π cm3
Pythagorean
equilateral
obtuse
circle
parallelogram
supplementary
sine
tangent
equiangular
circumference
skew
decagon
radius
48
4 cm
5m
24 mm
50 ft
10 m
11.
12.
13.
14.
15.
APOLLONIUS
200π m3
2000π m3
405π m3
12π cm3
75π cm3
Down
40 cm3
1728 cm3
512 cm3
1280 cm3
1.
4.
5.
7.
8.
10.
11.
16.
17.
18.
20.
16.
17.
18.
19.
20.
128π m3
8π m3
16π cm3
192π m3
240π cm3
hypotenuse
quadrilateral
rhombus
scalene
cylinder
acute
congruent
rectangle
hexagon
perpendicular
right
MP4056