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Geometry Review
1. If the perimeter of a rectangle is 24 inches
and the length of the rectangle is 3 inches,
what is the width of the rectangle?
April 23, 2012
6. Find the perimeter of the figure below. All
angles are right angles and all lengths are in
feet.
2. Suppose you have two different pieces of
wood and both are right rectangular prisms.
One piece of wood measures 3 cm by 4 cm by
5 cm. The other piece is similar to the first
piece, and its shortest edge measures 9 cm.
What is the total surface area of the larger
piece of wood?
3. A 26 ft long ladder is leaning up against a
house with its base 10 ft away from the house
on the ground. If the side of the house is
perfectly vertical, how far up the house would
you be if you climbed to the top of the
ladder?
7. The two squares shown side by side in the
figure below have a total area of 74 in2. The
side length of each square is a whole number
of inches. How many inches are in the
perimeter of the figure? (Note: The line
where the two squares touch is not to be
counted in the perimeter of the figure.)
4. Lines TP, BG, and DM intersect at point O.
∠BOT = 47 degrees and ∠MOG = 29 degrees.
How many degrees are in the measure of
angle DOP?
8. In the figure below, ABCD is a rectangle. The
points E and F trisect side DC, meaning they
cut it in three equal pieces. G is a point on
side AB. The length of side AB is 3 inches and
the length of BC is 1.2 inches. Find the area of
the shaded portion of the figure.
5. Lines m and n are parallel. ∠HIJ =148° and
∠QRS =133°. How many degrees are in the
measure of angle IJK?
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Geometry Review
April 23, 2012
BONUS PROBLEMS
9. What is the length of a side of a cube whose volume and surface area are numerically the same?
10. Suppose you have two blocks of wood and both are right rectangular prisms. The first measures
18 cm high and has a volume of 330 cm3. The second is half as tall, but twice as long and twice
as wide as the first block. How much more volume does the second block of wood have, in
cubic centimeters?
11. A pie baker likes to make pies that are neatly sliced into pieces that each have an area of π
square inches. How many slices would he have to cut his 14-inch diameter pie into in order to
continue his tradition?
12. The sum of the supplement of angle A and the complement of angle A measures sixteen degrees
more than a straight angle. How many degrees are in the measure of angle A?
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Geometry Review
April 23, 2012
SOLUTIONS:
1. 9 inches
Given that the length of the rectangle is 3 inches, we know that the perimeter is 3 + 3 + w + w = 24.
6 + 2w = 24. 2w = 18, so w = 9.
2. 846 cm2
Knowing that the prisms are similar, we know that their sides will be proportional. Therefore, the
shortest sides are in the ratio 3:9, which means the other two sides will be 4:12 and 5:15. So the
larger piece of wood has the dimensions 9 x 12 x 15. The surface area is then:
2 (9 x 12) + 2 (9 x 15) + 2 (12 x 15) = 2 (108) + 2 (135) + 2 (180) =
2 (108 + 135 + 180) =
2 x 423 =
846 cm2
3. 24 feet
Using the Pythagorean theorem, we’ll solve for a, how far up the house the ladder goes.
a2 + (10)2 = (26)2
a2 + 100 = 676
a2 = 576
a = 24
4. 104˚
The measures of angles BOT, TOM, and MOG must add up to 180 degrees since O is a point on line
BG. Thus the measure of angle TOM must be 180 – 47 – 29 = 104 degrees. Angles TOM and DOP
are opposite angles and therefore have the same measure. The measure of angle DOP is 104
degrees.
5. 101˚
Angle HIJ measures 148 degrees, so angle JIK must measure 180 – 148 = 32 degrees. Angle QRS and
angle IKR are corresponding angles, so they have the same measure. This means angle JKI must
measure 180 – 133 = 47. The total angle sum of triangle IJK has to be 180 degrees, so angle IJK must
measure 180 – 32 – 47 = 101 degrees.
6. 62 feet
The unmarked height on the left side of the figure is the sum of the three vertical lengths on the
right, namely 6 + 3 + 3 = 12 feet. Similarly, the other unmarked length must be 8 + 11 – 7 = 12 feet
also. The perimeter of the figure is thus: 8 + 6 + 11 + 3 + 12 + 3 + 7 + 12 = 62 feet. Alternatively, one
could reason that there must be two vertical totals of 12 feet and two horizontal totals of 19 feet,
for a perimeter of: 2 ´12 + 2 ´19 = 24 + 38 = 62 feet.
7. 38 inches
The two perfect squares with a sum of 74 must be 49 and 25. This means the side length of the
smaller square is 5 inches and the side length of the larger square must be 7 inches. Although there
are two unknown lengths where the smaller square meets the larger square (call one of them x and
the other y), the total height x + 5 + y must equal 7. Thus the perimeter of the figure is (4 x 7) + (2 x
5) = 28 + 10 = 38.
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Geometry Review
April 23, 2012
8. 3 in2
The area of rectangle ABCD is 1.2 x 3 = 3.6 square inches. Since E and F trisect side DC, the length of
segment EF must be 3 ÷ 3 = 1 inch. Also, regardless of where point G is on AB, the height of triangle
EFG is 1.2 inches. Thus the area of triangle EFG is ½ x 1 x 1.2 = 0.6 square inches. Subtracting the
area of triangle EFG from the area of rectangle ABCD, we get 3.6 – 0.6 = 3 square inches, which is
the area of the shaded regions.
9. 6
Using the formulas for volume and surface area:
V=s·s·s
SA = 6 · s · s
Therefore, each side must be 6 units long.
10. 330 cm3
Let x = the length of the first prism and y = the width of the first prism. Therefore, the first prism’s
volume = 18xy = 330
The second prism’s volume = 9 (2x) (2y) = 36xy. If 18xy = 330, then 36xy equals twice that, or 660.
Therefore, the second block has 330 cm3 more volume.
11. 49
Since the pie’s total area is πr2, and the pie’s radius is 7 inches, the pie’s total area is 49π square
inches. If each piece is π square inches in area, there must be 49 pieces.
12. 37˚
The supplement of angle A measures 180 – A. The complement of angle A measures 90 – A. Their
sum is (180 - A)+ (90 - A)= 270 - 2A. If this amount is sixteen degrees more than a straight angle,
then we can write the equation 270 - 2A =180 +16 and solve for A.
270 - 2A =180 +16
270 - 2A =196
270 =196 + 2A
270 -196 = 2A
74 = 2A
A = 37
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