Volume of Solids Key
Volume of Solids Key
Volume of Solids Key
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<strong>Volume</strong> <strong>of</strong> <strong>Solids</strong> KEY<br />
Geometry<br />
HS Mathematics<br />
Unit: 10 Lesson: 01<br />
<strong>Volume</strong> or capacity <strong>of</strong> a three dimensional figure (solid) is the amount <strong>of</strong> space it encloses. <strong>Volume</strong> is<br />
measured in cubic units.<br />
STAAR Geometry Reference Materials<br />
Examples:<br />
1. Find the volume <strong>of</strong> a right triangular prism with a right triangular base with legs 5 inches and 12<br />
inches and a height <strong>of</strong> 15 inches.<br />
450 in 3<br />
2. Anna’s room is in the shape <strong>of</strong> a regular hexagonal prism with the width <strong>of</strong> each wall being 16<br />
feet along the baseboard. The height <strong>of</strong> the room is 10 feet. If Anna wants to purchase an air<br />
purifier for the room, what is the minimum number cubic feet <strong>of</strong> air it must be able to handle?<br />
what is the minimum number cubic meters <strong>of</strong> air it must be able to handle?<br />
(3.28 feet = 1 meter)<br />
6651 ft 3<br />
≈ 188.5 m 3<br />
©2012, TESCCC 11/05/12 page 1 <strong>of</strong> 6
<strong>Volume</strong> <strong>of</strong> <strong>Solids</strong> KEY<br />
Geometry<br />
HS Mathematics<br />
Unit: 10 Lesson: 01<br />
3. A cylindrical grain silo is being built for Mr. Greenjeans. The silo is 13 meters tall with a<br />
diameter <strong>of</strong> 6 meters. How many cubic meters <strong>of</strong> grain can it hold? How many cubic feet <strong>of</strong><br />
grain can it hold? (3.28 feet = 1 meter)<br />
≈ 367.6 m 3<br />
≈ 12,970.5 ft 3<br />
4. A three-dimensional figure is formed by placing a square pyramid on top <strong>of</strong> a prism as shown<br />
below. Determine an equation to represent the volume in cubic inches <strong>of</strong> the threedimensional<br />
figure.<br />
2a<br />
s<br />
a<br />
<strong>Volume</strong> <strong>of</strong> Square Pyramid<br />
1 2 2<br />
( s s)(2 a)<br />
s a<br />
3 3<br />
<strong>Volume</strong> <strong>of</strong> Prism<br />
2<br />
s s a s a<br />
<strong>Volume</strong> <strong>of</strong> 3-D Figure<br />
2 2 2 5 2<br />
s a s a s a<br />
3 3<br />
©2012, TESCCC 11/05/12 page 2 <strong>of</strong> 6
<strong>Volume</strong> <strong>of</strong> <strong>Solids</strong> KEY<br />
Geometry<br />
HS Mathematics<br />
Unit: 10 Lesson: 01<br />
Guided Practice<br />
1. A model <strong>of</strong> right pyramid has an altitude <strong>of</strong> 15 cm. If the base is a regular pentagon with sides<br />
<strong>of</strong> 5.8 cm, what is the volume <strong>of</strong> the pyramid in cubic centimeters?<br />
≈ 289.4 cm 3<br />
2. The Luxor Hotel in Las Vegas, Nevada, is shaped like a square pyramid whose surface is<br />
covered in dark glass. The sides <strong>of</strong> the building are 606 feet in length. The height <strong>of</strong> the<br />
building at the apex <strong>of</strong> the pyramid is 350 feet. If the entire building were to be air conditioned,<br />
how many cubic feet capacity are in the building?<br />
42,844,200 ft 3<br />
3. Find the volume <strong>of</strong> a right cone, if the slant height is 14 meters and the diameter <strong>of</strong> the base is<br />
8 meters.<br />
≈ 224.8 m 3<br />
4. Find the air capacity <strong>of</strong> the following balls used in sports in cubic inches and cubic centimeters.<br />
(2.54 inches = 1 centimeter)<br />
a. Basketball with radius <strong>of</strong> 4.75 inches<br />
448.9 in. 3<br />
7356.5 cm 3<br />
b. Volleyball with circumference <strong>of</strong> 27 inches<br />
332.4 in. 3<br />
5446.8 cm 3<br />
©2012, TESCCC 11/05/12 page 3 <strong>of</strong> 6
<strong>Volume</strong> <strong>of</strong> <strong>Solids</strong> KEY<br />
Geometry<br />
HS Mathematics<br />
Unit: 10 Lesson: 01<br />
Practice Problems<br />
1. Find the volume <strong>of</strong> each right solid. Assume bases are regular.<br />
a. b.<br />
8,137.3 ft 3 203,374 mm 3<br />
56 mm<br />
35 ft<br />
34 mm<br />
8 ft<br />
c. d.<br />
12 cm<br />
4 cm<br />
18 cm<br />
248.1 cm 3<br />
400 cm 3<br />
10 cm<br />
e. f.<br />
8 yd<br />
2,513 ft 3 6 yd<br />
24 ft<br />
1,093 yd 3 7 yd<br />
10 ft<br />
©2012, TESCCC 11/05/12 page 4 <strong>of</strong> 6
<strong>Volume</strong> <strong>of</strong> <strong>Solids</strong> KEY<br />
Geometry<br />
HS Mathematics<br />
Unit: 10 Lesson: 01<br />
2. Orange cylindrical containers are used to set <strong>of</strong>f construction sites along the road. Each<br />
container is filled with sand. If the containers have a diameter <strong>of</strong> 28 inches and a height <strong>of</strong> 4<br />
feet, how many cubic feet <strong>of</strong> sand does each container hold, if filled to the top? How many<br />
cubic meters <strong>of</strong> sand does each container hold, if filled to the top? (3.28 feet = 1 meter)<br />
17.1 ft 3 ; 0.5 m 3<br />
3. In order to hang a block <strong>of</strong> gold from a chain necklace, a jeweler drilled a hole in the center, 3<br />
mm in diameter. If the original block <strong>of</strong> gold was 26 mm long, 7mm wide, and 18 mm high, find<br />
the volume <strong>of</strong> the remaining piece <strong>of</strong> gold.<br />
3092.2 mm 3<br />
3 mm<br />
18 mm<br />
7 mm<br />
26 mm<br />
4. The drill team will be selling ice cream cones at the baseball play<strong>of</strong>f. Each cone will be 12 cm<br />
high with a diameter <strong>of</strong> 6 cm. Ice cream will be packed into the cone and also form a<br />
hemisphere above the cone. Due to health regulations each cone must be covered in a piece<br />
<strong>of</strong> paper before served.<br />
a. What is the minimum amount <strong>of</strong> paper needed to cover the outside <strong>of</strong> the cone?<br />
116.6 cm 2<br />
b. What volume <strong>of</strong> ice cream will be needed to fill each cone?<br />
169.6 cm 3<br />
c. If the top ice cream dome is to be dipped in chocolate, what is the surface area <strong>of</strong> ice<br />
cream to be covered?<br />
56.5 cm 2<br />
d. If the layer <strong>of</strong> dipped chocolate averages .2 mm in thickness, what volume <strong>of</strong> chocolate<br />
is needed for each cone?<br />
1.13 cm 3<br />
5. In chemistry class Sue Ann poured pure water into a cylindrical vessel with a diameter <strong>of</strong> 3<br />
inches. She poured the water to a height <strong>of</strong> 5 inches. How many milliliters <strong>of</strong> water did Sue<br />
Ann pour into the cylindrical container?<br />
(1 inch = 2.54 centimeters, 1 milliliter = 1 cubic centimeter)<br />
V<br />
V<br />
V<br />
r h<br />
2 2<br />
3<br />
( ) (5) 35.343 cubic inches<br />
2<br />
3<br />
35.343(2.54) 579.17 cubic centimeters<br />
579.17 milliliters<br />
©2012, TESCCC 11/05/12 page 5 <strong>of</strong> 6
<strong>Volume</strong> <strong>of</strong> <strong>Solids</strong> KEY<br />
Geometry<br />
HS Mathematics<br />
Unit: 10 Lesson: 01<br />
6. The three-dimensional figure below is formed by placing a cone on top <strong>of</strong> a cylinder.<br />
Determine an equation that could be used to represent the volume in cubic inches <strong>of</strong> the threedimensional<br />
figure.<br />
<strong>Volume</strong> <strong>of</strong> Cone<br />
1 2<br />
3 rh<br />
<strong>Volume</strong> <strong>of</strong> the Cylinder<br />
2 2<br />
r (2 h) 2 r h<br />
<strong>Volume</strong> <strong>of</strong> 3-D Figure<br />
1 2 2 7 2<br />
r h 2 r h r h<br />
3 3<br />
©2012, TESCCC 11/05/12 page 6 <strong>of</strong> 6
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