Download POLYGON INTERIOR ANGLES THEOREM

Name______________________________________ Period ______
7.1 Angle Measures in Polygons
POLYGON INTERIOR ANGLES THEOREM
Sum of the measures of the interior angles of any polygon
n = _____________
and
_____________
Classify
polygons by #
of sides
3
4
5
Triangle
Quadrilateral
Pentagon
6
7
8
Hexagon
Heptagon
Octagon
Example 1: Use the Interior Angles Theorem
1. Find the sum of the measures of the interior angles of a
convex 22-gon.
Nonagon
Decagon
Dodecagon
n
n-gon
2. The sum of the measures of the interior angles of a
convex polygon is 2700°. Classify the polygon by the
number of sides.
Example 2: Apply the Interior Angles Theorem
1.
a. n = _____
b. Classify the polygon
c. Solve for x.
2.
9
10
12
a. n = _____
b. Classify the polygon
c. Find the
1
– , and
3. The measures of the interior angles of a pentagon are
measure of the largest angle?
What is the
POLYGON EXTERIOR ANGLES THEOREM
The sum of the measures of the exterior angles of a convex polygon, one angle at each
vertex, is ________.
Example 3: Find missing angles.
1. Find the value of x.
2. What is the sum of the exterior angles of a 14-gon?
3. Two interior angles of a triangle have measures 42° and 56°. Which of the following could not be a measure of an
exterior angle of the triangle?
A. 98°
B. 82°
C. 124°
D. 138°
REGULAR POLYGON: A polygon with all sides and angles _________________.
Example: Regular Pentagon
1) What is the measure of each
exterior angle?
2) What is the measure of each
interior angle?
Example 4: With Regular Polygons
1. What is the measure of each exterior angle of a regular
nonagon?
2. Each interior angle of the regular n-gon measures of
165°. Classify the polygon by the number of sides.
2
3. What is the measure of an interior angle of a regular
hexagon?
4. If an interior angle of a regular polygon is
the measure of an exterior angle?
, what is
5. Given a regular decagon, find the following information: (Don’t forget your diagram).
A. Sum of the exterior angles
B. Each exterior angle
C. Each interior angle
D. Sum of the interior angles
RECAP (note: n = ______________ and _____________)
ANY convex polygon
REGULAR convex polygon
SE =
EE =
SI =
EI =
At a vertex, an interior and
exterior angle are _____________
______________
1. If the interior angle sum of a regular polygon is 2160 ,
then find the measure of one exterior angle.
2. If an interior angle of a regular polygon is 165 , then
find…
(a) the measure of an exterior angle
(b) the interior angle sum of the polygon
3. If each exterior angle of a regular polygon is
classify the polygon by the number of sides.
4. Given a regular nonagon, what is the sum of the measures
of the exterior angles?
, then
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7.2 Circumference and Area of Circles
CIRCUMFERENCE
AREA OF A CIRCLE
A=
_____
C=
_____
“…in terms of
-
”
leave π in your answer
NO DECIMAL answer!
**NOTE**
Circumference is measured in single units (not
square units like area)
Example 1: Find the circumference or area.
1. Find the circumference and area of a circle with a
diameter of 26 meters. Leave your answers in terms of π.
2. Find the diameter of a circle with an area of 361π m2.
3. Find the area of the shaded region. Round your answer
the nearest tenth.
4. Find the area of the shaded region. Leave your answer in
terms of .
2𝑚
6𝑚
•
5. If the area of a circle is 121 cm2, what is the
circumference?
6. If the circumference of a circle is 48 inches, then what
is its area?
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7. The circle below is inscribed in the square. If the
circumference of the circle is
, then find the area of
the square.
8. The square is circumscribed about the circle. If the area
of the square is 64 square meters, then find the
circumference of the circle.
If a tire has a circumference of 13 inches, how far would it travel if it rotated twice?
Distance
traveled
Example 2: Finding distance traveled
1. How far does a tire with a diameter of 32 inches travel in
30 revolutions. Round your answer to the nearest tenth.
= _____ x _____________
2. The dimensions of the skateboard wheel shown at the
right are in millimeters. To the nearest millimeter, how
far does the wheel travel when it makes 35 revolutions?
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3. The tires of an automobile have a diameter of 24 inches.
Find the number of revolutions the tire will make if it
travelled 3120 inches. (Round answer to the tenths
place).
4. The tire of a tricycle travels 216 feet in 12 revolutions.
What is the circumference of the tricycle tire? What is
its radius?
MINI REVIEW
1. Four of the interiors angles of a pentagon are 90°, 143°,
77°, and 103°. Find the measure of the missing angle.
2. Solve for x.
70°
2x°
x°
5
80°
60°
7.3 Area of Basic Polygons
AREA OF A PARALLELOGRAM
(includes rectangles, rhombuses, and squares)
b=
A=
_____
h=
Example 1: Name the figure then use a formula to find area of the shaded region
1.
2.
ft
3.
m
ft
AREA OF A TRAPEZOID b
b
h
A=
**NOTE** The height of a
trapezoid is a perpendicular
segment between the bases.
(BASES⟶ parallel sides)
b
h
b
Example 2: Find the area of the shaded region
1.
2.
15 in
11 in
12 in
3.
4.
20 m
28 m
25 m
18 m
m
35 m
15 m
5.
is an isosceles trapezoid with legs ̅̅̅̅ and ̅̅̅̅ . If
6
°, find
and
.
AREA OF A RHOMBUS
AREA OF A KITE
A=
_____
A=
_____
[You can still use A = bh (parallelogram)]
Example 3: Name the figure then use a formula to find its area.
1.
2.
13 cm
5 cm
Example 4: Find the area of the equilateral triangle
* Remember!
Each angle in
an equilateral
∆ is 60˚
60˚
30˚
60˚
60˚
˚
60˚
1. Find the area of the figure below.
Notice a Pattern?
So when the altitude
cuts the triangle in half,
you get two 30˚-60˚-90˚
triangles
60˚
2. Find the area of an equilateral triangle with a side length
of 18.
EQUILATERAL ∆
Area Formula
1. Use the formula to find the area of an equilateral triangle
with a side length of 16 cm.
2. Find the area of an equilateral triangle with a side length
of 30 in.
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Example 5: Solve for unknown measures
1. A triangle has an area of 126 square feet and a height of
14 feet. What is the length of the base?
2. The parallelogram shown at the right has an area of 70
square feet. Find the value of x.
SHADED AREA: find the area of the shaded region
1.
2.
12
4
8
7
3. Find the area of the figure comprised of a kite and a
4. Find the area of the shaded region.
trapezoid.
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7.4
Perimeter/Area of Similar Figures + Review of Angles in Polygons
PERIMETER OF SIMILAR
POLYGONS
AREA OF SIMILAR
POLYGONS
If two polygons are similar with the lengths of
If two polygons are similar with the lengths of
corresponding sides in the ratio of a:b, then the
corresponding sides in the ratio of a:b, then the
ratio of their perimeters is______:______
ratio of their areas is______:______
( )
( )
SIDE2 : SIDE2 = AREA : AREA
SIDE : SIDE = PERIMETER : PERIMETER
Example 1: Use the similar polygons to find the unknown perimeters and areas.
1. ∆
∆
∆
2.
I
I
II
II
PI = 15 in
PII = _____
PI = _____
PII = _____
AI = _____
AII = 27 in2
AI = 72 m2
AII = _____
3. JANE is similar to BURT where the ratio of similarity is
5:12. If BURT is the smaller quadrilateral with a
perimeter of 32.5, then what is the perimeter of JANE?
4. Two similar trapezoids have a similarity ratio of 5:3. If
A. 78
B. 13.5
C. 93
D. 162
A. 270
B. 90
C. 19.4
D. 150
the area of the smaller trapezoid is 54 square feet, then
find the area of the larger trapezoid.
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5. If the ratios of the areas of two similar
hexagons is 225:64. What is the ratio
of the lengths of corresponding sides?
6. The hexagons below are similar. The perimeter of the larger hexagon is 64
inches and the perimeter of the smaller hexagon is 24 inches. (a) Find the
ratio of the corresponding side lengths. (b) Find the ratio of the areas of
the larger hexagon to the smaller hexagon.
7. Your mother has taken her favorite baby picture of you
and wants to enlarge the size of the photo to place over
the fireplace. She wants to place a nice ribbon along the
edges of the enlarged photo. What total length of ribbon
would she need to cover the border?
cm
8. Jill built a smaller version of a door that she liked for her
dollhouse. She would like to paint the door on the doll
house red. If the orginial door needed enough paint to
cover 80 square inches, then how many square inches
would need to covered by paint in the doll house?
cm
i
cm
cm
i
i
i
cm
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Review So Far…
1. What is the measure of one exterior angle of a regular
dodecagon?
A.
B.
C.
D.
2. Find the area of the shaded region.
A. 36
B.
C.
D.
36°
360°
30°
150°
3. If an interior angle of a regular polygon is
classify the polygon.
4. The diagram below shows the size of the tire that Billy
uses on his unicycle. The area of the tire is
ft2. If
Billy rode his unicycle
feet, then how many
revolutions did the tire make?
, then
10
12
5. Find the area of the shaded region.
6. If an interior angle of a regular polygon is
, then what is the measure of an exterior
angle?
7. Find the area of the figure below.
A.
B.
C.
D.
3000
980
2320
1160
A.
B.
C.
D.
9. If the circumference of a circle is
the area of the circle.
A.
B.
C.
D.
8. The ratio of similarity between to pentagons is 3:7. If the
area of the larger pentagon is 147 square units, then what
is the area of the smaller pentagon?
10. Find the area of the shaded region. Leave your answer in
terms of .
meteres, then find
A. 144
B.
C.
D.
225π
225
15
30π
11. Find the area of the parallelogram.
A.
B.
C.
D.
63
27
142
38
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12. If the ratio of the perimeters of two similar figures is 3:5,
then what is the ratio of the areas?
30
28
24
16
A. 3:5
B. 9:25
C. 25:9
D. 5:3
13. Daisy made a reduced model of her backyard for a school project. She wants to build a white picket fence enclosing the
model of the yard. What length of fencing would Daisy need in order to enclose the model of the yard?
14ft
8ft
Original
yard
6ft
5ft
13ft
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7.5 Spiral Questions
I. SOH-CAH TOA
1) Use ΔJAM to solve for x below.
2) Use the triangle below to find the length of ̅̅̅̅̅ .
P
A.
A
x
A.
J
B.
58°
22
9
C.
D.
T
i
23°
i
B.
C.
W
M
D.
3) Find the area of the given shape.
4) Find the area of the kite.
19 m
m
m
II. CONGRUENT TRIANGLES
1) If you are given two triangles, ΔMNZ and ΔUTH where
̅̅̅̅̅ ̅̅̅̅ and ̅̅̅̅ ̅̅̅̅ What additional information
would be sufficient to show that ΔMNZ ΔUTH?
A.
B. ̅̅̅̅̅ ̅̅̅̅
C. ̅̅̅̅̅ ̅̅̅̅
D.
and
is a right angle
2) Given ̅̅̅ ̅̅̅̅ , ̅̅̅
triangles congruent?
̅̅̅, and ̅̅̅
A. ASA
B. AAS
C. SAS
D. HL
̅̅̅̅. Why are the
K
I
M
E
Y
T
4) If you are given two triangles, ΔPLH and ΔRTV where
̅̅̅̅ ̅̅̅̅ and ̅̅̅̅ ̅̅̅̅ What additional information
would not be sufficient to show that ΔPLH ΔRTV?
I
A.
B. ̅̅̅̅ ̅̅̅̅
C.
and
are right angles
D. ̅̅̅̅ ̅̅̅̅
3) Given the diagram below, determine whether the
triangles are congruent are not. If so, why?
A. ASA
B. SAS
C. AAS
D.NONE
P
R
T
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III. Quadrilaterals
1) A quadrilateral has interior angles
,
, and
What is the measure of the missing interior angle?
3) Given
̅̅̅. If
2) Given JANE is an isosceles trapezoid with legs ̅̅̅ and
̅̅̅̅ . If
, then find
. Explain.
.
is an isosceles trapezoid with bases ̅̅̅̅ and
, then find
. Explain.
4) Three exterior angles of a quadrilateral are
,
, and
. Which of the following could not be the measure
of an interior angle?
A.
B.
C.
D.
A.
B.
C.
D.
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7.6 Review Day 
1. Find the area of the equilateral triangle below.
2. Find the area of the shaded region.
10 cm
6
cm
13 cm
3. If
4. What is the measure of an interior angle of a
regular nonagon?
, find the perimeter of each polygon.
5. Find the area of the shaded region.
6. What is the sum of the exterior
angles of the regular octagon?
5. A tire travels 324 feet in 9 revolutions. What is the
length of the radius of the tire?
7. The measures of the interior angles of a pentagon are
,
, and
. What is the value of
14
,
,
8. Two similar shapes have a similarity ratio of
If the
smaller shape has an area of 32 inches, what is the area of
the larger shape?
9. Find the area of the kite below.
10. Solve for x.
11. Find the area of the figure below.
(
𝑥
)
15 m
39 m
𝑥
12. Find the area of the figure below.
13. Each interior angle of a regular polygon has a measure
of 156°. Classify the polygon by the number of sides.
14. Four interior angles of a pentagon are 90°, 143°, 77°,
and 103°. Find the measure of the missing angle.
15. If the radius of a tire measures 14 inches, how many
inches will the tire roll in 7 revolutions. Leave your
answer in terms of .
16. Find the area of the shaded region.
17. If the area of a circle is 144 in2, find the circumference
of the circle.
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7.7 Area of a Regular Polygon
Vocabulary
REGULAR POLYGON
• Center
• Radius
• Apothem
• Central angle
MEASURE OF A CENTRAL ANGLE
** notice that the
# of central angles
is the same as the
# of sides.
Central =
Angle
Example 1: Find the measure of the indicated angle
1. m ONH
2. m EGH
3. m ARS and m ERA
T
Q
U
A
P
R
N
E
G
S
E
A
O
H
Finding the APOTHEM or SIDE LENGTH of a regular polygon
* In a regular polygon, the apothem, radius, and ½ side length make a right triangle
Use the right triangle to find the missing lengths
• GIVEN: 2 side lengths
1) Use _______________
a
• GIVEN: 1 side length
1) Find ½ the central angle
r
2) Use _________________
½s
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Example 2: Leave your answer as a simp. radical or round to the nearest tenth if necessary.
1. Find the length of the apothem of the regular polygon
2. Find the length of the apothem of the regular polygon.
with a side length of 14 m and a radius of 10 m.
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What do we need to find the Apothem and Side length for?
To find the area of the __________ and eventually the area of the entire ________!
• Find the area of the shaded region.
∆
* notice that the base of the triangle is the same as the _________
* notice that the height of the triangle is that same as the __________
SO:
a
∆
s
Example 3
1. Find the area of ∆
∆
2. Find the area of ∆
.
.
A
M
C
7
Finding the area of a regular polygon
Ex: Find the area of the regular hexagon.
1) Find the area of one triangle
A∆ =
=
9
9
8
=
8
=
2) Multiply the area of the triangle by the # of triangles
**NOTICE** # of triangles = # of sides (n)
Apolygon = A∆  n
=
=
17
AREA OF A REGULAR POLYGON
OR
A=
Area of ∆
# of sides (∆’ )
A=
• Height (h) of the ∆ ⇔ Apothem (a) of the polygon
• Base (b) of the ∆ ⇔ Side length (s) of the polygon
• Number of ∆’ ⇔ Number of sides (n)
Example 4: Find the area of the regular polygon
1.
a=
n=
s=
a=
n=
s=
Apolygon=
2.
a=
n=
s=
Apolygon=
3.
a=
n=
s=
Apolygon=
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7.8 Review
ANGLE MEASURES IN POLYGONS
SUM I =
EACH I =
1. Given a regular decagon, what is the
SUM E =
EACH E =
2. The measures of the interior angles of a hexagon are
measure of each exterior angle.
,
,
, and
3. What is the measure of one interior angle of a regular
,
,
. What is the value of
4. If an exterior angle of a regular polygon is 20°, what is
14-gon?
the measure of one interior angle?
CIRCLES
AREA of a circle
CIRCUMFERENCE
1. If the area of a circle is 121π meters2, what is the
Revolution Problems
2. If the radius of a bicycle wheel measures 12 inches, how
circumference of the circle.
many inches will the wheel roll in 20 revolutions?
AREA OF POLYGONS
Parallelogram
Triangle
Trapezoid
1. Find the area of the parallelogram.
Equilateral
Triangle
Rhombus
Kite
(2nd formula)
2. Find the area of the rhombus.
3. Find the area of the shaded region created by the
parallelogram and trapezoid.
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4. Find the area of the shaded region created by
the rectangle and 2 circles.
SIMILAR POLYGONS
PERIMETER of similar polygons
AREA of similar polygons
1. Two similar shapes have a similarity ratio of .
If the larger shape has an area of 256 mm, find
the area of the smaller shape.
2. The ratio of the areas of two similar figures is
196:64. What is the ratio of the length of their
corresponding sides?
AREA OF REGULAR POLYGONS
To find the area of a regular polygon you will need 3 things:
1)
2)
3)
To find the apothem (a) or side length (s), use a right ∆ created by a radius and apothem.
• If two sides are given, • If only one side is given, use a
use the Pyth. TH
*Must find 1 central first!
2
2
2
(a + b = c )
r
a
Central
r
a
1
s
=
* SOH-CAH-TOA or special
right triangle
s
(Notice that the right ∆ has
only half the side length of
the polygon)
• To find the AREA of the polygon
Use the formula
1. Find the area of the regular polygon.
2. Find the area of a regular dodecagon with a side length
of 30 inches.
a=
a=
n=
n=
s=
s=
Apolygon=
Apolygon=
20
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