Volume of Cubes

Show your work for each problem.

1.
Which formula gives the volume of a cube with edge $s$ ?
1. a) $V = s^2$
2. b) $V = s^3$
3. c) $V = 3s$
4. d) $V = 6s^2$
Answer: $V = s^3$
$V = s \times s \times s = s^3$ . $3s$ is three edges added up, $s^2$ is the area of one face and $6s^2$ is the area of all six faces.
2.
What is the volume of a cube with an edge of $4$ cm?
1. a)48 cm³
2. b)12 cm³
3. c)16 cm³
4. d)64 cm³
Answer: 64 cm³
$4 \times 4 = 16$ and $16 \times 4 = 64$ , so $V = 64$ cm³. $12$ is $4 \times 3$ and $16$ is only one face.
3.
The edge of a cube is doubled. By how much is the volume multiplied?
1. a) $\times 2$
2. b) $\times 8$
3. c) $\times 4$
4. d) $\times 6$
Answer: $\times 8$
All three edges double, so the volume is multiplied by $2 \times 2 \times 2 = 8$ . Example: $1^3 = 1$ and $2^3 = 8$ .
4.
Which of these numbers is a perfect cube?
1. a)16
2. b)36
3. c)125
4. d)100
Answer: 125
$125 = 5 \times 5 \times 5 = 5^3$ . The others are perfect squares ( $16 = 4^2$ , $36 = 6^2$ , $100 = 10^2$ ), not perfect cubes.
5.
Cube A has an edge of $2$ cm and cube B has an edge of $6$ cm. How many times larger is the volume of cube B?
1. a)27
2. b)3
3. c)216
4. d)9
Answer: 27
$V_A = 2^3 = 8$ cm³ and $V_B = 6^3 = 216$ cm³. $216 \div 8 = 27$ . Shortcut: the edge is $3$ times longer, so the volume is $3^3 = 27$ times larger.
6.
A cube has an edge of $3$ m. What is its volume in m³? Write only the number.
Answer: 27
$V = 3^3 = 3 \times 3 \times 3 = 9 \times 3 = 27$ m³.
7.
What is the volume in cm³ of a cube with an edge of $10$ cm? Write only the number.
Answer: 1000
$10^3 = 10 \times 10 \times 10 = 1000$ cm³. That is exactly $1$ liter!
8.
A small cube has an edge of $1.5$ cm. What is its volume in cm³? Write only the number.
Answer: 3.375
$1.5 \times 1.5 = 2.25$ and $2.25 \times 1.5 = 3.375$ , so $V = 3.375$ cm³. Check: it is between $1^3 = 1$ and $2^3 = 8$ .
9.
A cube has a volume of $216$ cm³. How long is its edge in cm? Write only the number.
Answer: 6
$5^3 = 125$ is too small, $6^3 = 6 \times 6 \times 6 = 216$ . So $\sqrt[3]{216} = 6$ and the edge is $6$ cm.
10.
A cube-shaped water tank has an inside edge of $50$ cm. How many liters of water does it hold? ( $1$ L $= 1000$ cm³) Write only the number.
Answer: 125
$50 \times 50 = 2500$ and $2500 \times 50 = 125000$ cm³. $125000 \div 1000 = 125$ L. Shortcut: $50$ cm is $5$ times $10$ cm, so $5^3 = 125$ liter cubes fit.
11.
Find the volume of a cube with an edge of $7$ cm.
Answer: 343
1. What is $7 \times 7$ (the area of one face in cm²)? 49
2. Multiply by the third edge: what is $49 \times 7$ (the volume in cm³)? 343
12.
A big cube is built from small $1$ cm cubes, with $4$ small cubes along each edge. How many small cubes are used?
Answer: 64
1. How many small cubes are in the bottom layer ( $4$ by $4$ )? 16
2. How many layers are there? 4
3. How many small cubes are there in total? 64
13.
A cube-shaped box has an edge of $0.4$ m. Find its volume in m³.
Answer: 0.064
1. What is $0.4 \times 0.4$ ? 0.16
2. What is $0.16 \times 0.4$ (the volume in m³)? 0.064
14.
A cube has a volume of $512$ cm³. Find the length of its edge.
Answer: 8
1. Try $7$ : what is $7^3$ ? 343
2. $343$ is too small. Try $8$ : what is $8^3$ ? 512
3. So how long is the edge in cm? 8
15.
A cube has an edge of $5$ cm. Its edge is doubled. How many times larger does the volume get?
Answer: 8
1. What is the original volume in cm³? 125
2. What is the new edge in cm? 10
3. What is the new volume in cm³? 1000
4. How many times larger is the new volume? 8
16.
A cube-shaped aquarium has an inside edge of $60$ cm. How many liters of water does it hold when full? ( $1$ L $= 1000$ cm³)
Answer: 216
1. What is the area of the bottom, $60 \times 60$ , in cm²? 3600
2. What is the volume in cm³? 216000
3. How many liters is that? 216
17.
A cube has a volume of $27$ m³. Each edge is made $1$ m longer. What is the new volume in m³?
Answer: 64
1. What is the original edge in m? 3
2. What is the new edge in m? 4
3. What is the new volume in m³? 64
18.
A cube-shaped box has an inside edge of $12$ cm. How many small cubes with a $2$ cm edge fit inside it?
Answer: 216
1. How many small cubes fit along one edge? 6
2. How many small cubes fit in one layer? 36
3. How many small cubes fit in the whole box? 216