Volume of Composite Shapes

Show your work for each problem.

1.
A solid is a cylinder with a cone attached to its top, so the cone's circular base sits exactly on the top of the cylinder. How do you find the volume of the whole solid?
1. a)Subtract the volume of the cone from the volume of the cylinder
2. b)Use only the cylinder, because the cone is too small to count
3. c)Multiply the volume of the cylinder by the volume of the cone
4. d)Add the volume of the cylinder and the volume of the cone
Answer: Add the volume of the cylinder and the volume of the cone
The cone is attached to the cylinder, so the solid takes up the space of both parts: $V = V_{\text{cylinder}} + V_{\text{cone}}$ . You subtract only when a part is removed.
2.
A cube with edges of $4$ cm is glued on top of a bigger cube with edges of $6$ cm. What is the total volume of the solid in cm³?
Answer: 280
Small cube: $4^3 = 64$ cm³. Big cube: $6^3 = 216$ cm³. Total: $64 + 216 = 280$ cm³.
3.
A step block is made of two boxes. The bottom box is $8$ cm long, $4$ cm wide and $2$ cm high. A second box, $4$ cm long, $4$ cm wide and $2$ cm high, sits on top of it. Find the total volume in cm³.
Answer: 96
1. What is the volume of the bottom box in cm³? 64
2. What is the volume of the top box in cm³? 32
3. What is the total volume in cm³? 96
4.
Which expression gives the volume of a hemisphere (half a sphere) with radius $r$ ?
1. a) $\frac{1}{3}\pi r^3$
2. b) $2\pi r^2$
3. c) $\frac{4}{3}\pi r^3$
4. d) $\frac{2}{3}\pi r^3$
Answer: $\frac{2}{3}\pi r^3$
A hemisphere is half a sphere: $\frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$ . $\frac{4}{3}\pi r^3$ is the whole sphere and $2\pi r^2$ is an area, not a volume.
5.
A small shed is a cube with edges of $2$ m. Its roof is a pyramid with a square base of $2$ m by $2$ m (the top of the cube) and a height of $1.5$ m. What is the total volume of the shed in m³?
Answer: 10
Cube: $2^3 = 8$ m³. Pyramid: $\frac{1}{3} \times 4 \times 1.5 = 2$ m³. Total: $8 + 2 = 10$ m³.
6.
A model house is a box $6$ m long, $4$ m wide and $3$ m high, with a pyramid roof on top. The roof has the same $6$ m by $4$ m rectangular base and a height of $2$ m. Find the total volume in m³.
Answer: 88
1. What is the volume of the box in m³? 72
2. What is the base area $B$ of the roof in m²? 24
3. What is the volume of the roof in m³? 16
4. What is the total volume in m³? 88
7.
A wooden cube has edges of $10$ cm. A small cube with edges of $4$ cm is cut out of one corner. What volume of wood is left, in cm³?
Answer: 936
1. What is the volume of the whole cube in cm³? 1000
2. What is the volume of the piece cut out in cm³? 64
3. How much wood is left in cm³? 936
8.
A rocket model is $15$ cm tall: a cylinder with radius $2$ cm and height $9$ cm, with a cone of radius $2$ cm and height $6$ cm on top. Which expression gives its volume?
1. a) $\pi \cdot 2^2 \cdot 9 + \frac{1}{3}\pi \cdot 2^2 \cdot 6$
2. b) $\pi \cdot 2^2 \cdot 15$
3. c) $\pi \cdot 2^2 \cdot 9 + \pi \cdot 2^2 \cdot 6$
4. d) $\pi \cdot 2^2 \cdot 15 + \frac{1}{3}\pi \cdot 2^2 \cdot 6$
Answer: $\pi \cdot 2^2 \cdot 9 + \frac{1}{3}\pi \cdot 2^2 \cdot 6$
Cylinder with its own height $9$ plus cone with its own height $6$ and the factor $\frac{1}{3}$ : $\pi \cdot 2^2 \cdot 9 + \frac{1}{3}\pi \cdot 2^2 \cdot 6 = 36\pi + 8\pi = 44\pi$ . The total height $15$ is not used in either formula.
9.
A crayon is a cylinder with radius $1$ cm and length $9$ cm, with a cone-shaped tip of radius $1$ cm and height $3$ cm. Use $\pi \approx 3.14$ to find its volume in cm³.
Answer: 31.4
1. What is the volume of the cylinder in cm³? 28.26
2. What is the volume of the cone in cm³? 3.14
3. What is the total volume in cm³? 31.4
10.
An ice cream cone has radius $3$ cm and height $8$ cm and is completely filled. A hemisphere-shaped scoop of radius $3$ cm sits on top. The total volume of ice cream is $k\pi$ cm³. What is $k$ ?
Answer: 42
Cone: $\frac{1}{3}\pi \times 9 \times 8 = 24\pi$ . Hemisphere: $\frac{2}{3}\pi \times 27 = 18\pi$ . Total: $24\pi + 18\pi = 42\pi$ , so $k = 42$ .
11.
A capsule-shaped container is a cylinder with radius $3$ cm and length $12$ cm, with a hemisphere of radius $3$ cm on each end. Use $\pi \approx 3.14$ and round the total volume to the nearest whole cm³.
Answer: 452
1. What is the volume of the cylinder in cm³? 339.12
2. What is the volume of the two hemispheres together in cm³? 113.04
3. What is the exact total with $\pi \approx 3.14$ , in cm³? 452.16
4. Round to the nearest whole cm³. 452
12.
A wooden block is $10$ cm long, $10$ cm wide and $6$ cm high. A cylindrical hole with radius $2$ cm is drilled straight down through its whole $6$ cm height. Using $\pi \approx 3.14$ , what volume of wood is left, rounded to the nearest tenth?
1. a)675.4 cm³
2. b)524.6 cm³
3. c)600 cm³
4. d)298.6 cm³
Answer: 524.6 cm³
Block: $600$ cm³. Hole: $3.14 \times 4 \times 6 = 75.36$ cm³. Wood left: $600 - 75.36 = 524.64 \approx 524.6$ cm³. Adding gives $675.4$ , and using the diameter $4$ as the radius gives $298.6$ .
13.
A silo is a cylinder with radius $2$ m and height $8$ m, with a hemisphere-shaped roof of radius $2$ m. Use $\pi \approx 3.14$ . What is the volume of the silo in m³, rounded to the nearest whole number?
Answer: 117
Cylinder: $\pi \times 2^2 \times 8 = 32\pi$ . Hemisphere: $\frac{2}{3}\pi \times 2^3 = \frac{16}{3}\pi$ . Total: $\frac{112}{3}\pi \approx \frac{112 \times 3.14}{3} \approx 117.23$ , so about $117$ m³.
14.
A solid cylinder has radius $6$ cm and height $10$ cm. A cone with the same base (radius $6$ cm) and the same height ( $10$ cm) is hollowed out of it, with its tip pointing down inside the cylinder. Use $\pi \approx 3.14$ to find the volume of material that is left, in cm³.
Answer: 753.6
1. What is the volume of the cylinder in cm³? 1130.4
2. What is the volume of the cone in cm³? 376.8
3. What volume of material is left in cm³? 753.6
15.
A metal pipe is $50$ cm long. Its outer radius is $6$ cm and its inner radius (the hole) is $5$ cm. Which expression gives the volume of metal in the pipe?
1. a) $\pi (6 - 5)^2 \cdot 50$
2. b) $\pi \cdot 6^2 \cdot 50$
3. c) $\pi \cdot 6^2 \cdot 50 + \pi \cdot 5^2 \cdot 50$
4. d) $\pi (6^2 - 5^2) \cdot 50$
Answer: $\pi (6^2 - 5^2) \cdot 50$
Metal = outer cylinder minus hole: $\pi \cdot 6^2 \cdot 50 - \pi \cdot 5^2 \cdot 50 = \pi (6^2 - 5^2) \cdot 50 = 550\pi$ . The choice $\pi (6 - 5)^2 \cdot 50 = 50\pi$ subtracts the radii before squaring, which is wrong.
16.
A concrete drain pipe is $200$ cm long. Its outer radius is $10$ cm and its inner radius is $8$ cm. Use $\pi \approx 3.14$ to find how many liters of concrete it contains ( $1$ L $= 1000$ cm³).
Answer: 22.608
1. What is the volume of the outer cylinder in cm³? 62800
2. What is the volume of the hole in cm³? 40192
3. What is the volume of concrete in cm³? 22608
4. How many liters is that? 22.608
17.
A fish tank is a box $30$ cm long, $20$ cm wide and $20$ cm high. A solid decoration shaped like a cylinder, with radius $5$ cm and height $20$ cm, stands upright inside it. Using $\pi \approx 3.14$ , how many liters of water are needed to fill the rest of the tank? ( $1$ L $= 1000$ cm³)
Answer: 10.43
Tank: $30 \times 20 \times 20 = 12000$ cm³. Decoration: $3.14 \times 5^2 \times 20 = 1570$ cm³. Water: $12000 - 1570 = 10430$ cm³ $= 10.43$ L.
18.
A solid is a cylinder with a hemisphere on top, both with radius $3$ cm. Its total volume is $90\pi$ cm³. How tall is the cylinder part? (Work in terms of $\pi$ .)
Answer: 8
1. The hemisphere has volume $k\pi$ cm³. What is $k$ ? 18
2. The cylinder part has volume $m\pi$ cm³. What is $m$ ? 72
3. Solve $\pi \times 3^2 \times h = 72\pi$ . What is $h$ in cm? 8