Volume of Pyramids

Show your work for each problem.

1.
Which formula gives the volume of any pyramid? ( $B$ = base area, $h$ = height)
1. a) $V = \frac{1}{3}Bh$
2. b) $V = \frac{1}{2}Bh$
3. c) $V = 3Bh$
4. d) $V = Bh$
Answer: $V = \frac{1}{3}Bh$
$V = \frac{1}{3}Bh$ . A pyramid is exactly one third of the prism with the same base and height, whose volume is $Bh$ .
2.
A pyramid has a square base with sides of $3$ m and a height of $5$ m. What is its volume in m³?
Answer: 15
$B = 3^2 = 9$ m², so $V = \frac{1}{3} \times 9 \times 5 = \frac{45}{3} = 15$ m³.
3.
Find the volume of a square pyramid with base sides of $5$ cm and a height of $12$ cm.
Answer: 100
1. What is the base area in cm²? 25
2. Multiply the base area by the height. What do you get? 300
3. Divide by 3. What is the volume in cm³? 100
4.
A prism and a pyramid have the same base and the same height. The prism has a volume of $90$ cm³. What is the volume of the pyramid?
1. a)45 cm³
2. b)90 cm³
3. c)30 cm³
4. d)270 cm³
Answer: 30 cm³
The pyramid is one third of the prism: $90 \div 3 = 30$ cm³.
5.
A pyramid has a base area of $24$ cm² and a height of $7$ cm. What is its volume in cm³?
Answer: 56
$V = \frac{1}{3} \times 24 \times 7 = 8 \times 7 = 56$ cm³.
6.
A pyramid has a rectangular base of $9$ m by $4$ m and a height of $5$ m. Find its volume.
Answer: 60
1. What is the base area in m²? 36
2. Multiply by the height. What do you get? 180
3. Divide by 3. What is the volume in m³? 60
7.
In $V = \frac{1}{3}Bh$ , which length is $h$ ?
1. a)The perpendicular distance from the apex to the base
2. b)The length of a base edge
3. c)The slant height of a triangular face
4. d)The length of an edge from the apex to a corner
Answer: The perpendicular distance from the apex to the base
$h$ is the perpendicular height: the straight distance from the apex down to the base at a right angle. The slant height runs along a face and is longer.
8.
A pyramid has a rectangular base of $7$ cm by $6$ cm and a height of $10$ cm. What is its volume in cm³?
Answer: 140
$B = 7 \times 6 = 42$ cm², so $V = \frac{1}{3} \times 42 \times 10 = 14 \times 10 = 140$ cm³.
9.
The base of a pyramid is a right triangle with legs of $6$ cm and $8$ cm. The pyramid is $15$ cm tall. Find its volume.
Answer: 120
1. What is the area of the base triangle in cm²? 24
2. Multiply by the height of the pyramid. What do you get? 360
3. Divide by 3. What is the volume in cm³? 120
10.
A square pyramid has base sides of $4$ cm and a height of $6$ cm. Which calculation gives its volume?
1. a) $\frac{1}{2} \times 4^2 \times 6$
2. b) $\frac{1}{3} \times 4^2 \times 6$
3. c) $\frac{1}{3} \times 4 \times 6$
4. d) $4^2 \times 6$
Answer: $\frac{1}{3} \times 4^2 \times 6$
$B = 4^2 = 16$ , so $V = \frac{1}{3} \times 4^2 \times 6 = \frac{96}{3} = 32$ cm³. The other choices forget the square, the $\frac{1}{3}$ , or use $\frac{1}{2}$ .
11.
The base of a pyramid is a triangle with a base of $10$ cm and a height of $6$ cm. The pyramid is $8$ cm tall. What is its volume in cm³?
Answer: 80
$B = \frac{1}{2} \times 10 \times 6 = 30$ cm², so $V = \frac{1}{3} \times 30 \times 8 = 10 \times 8 = 80$ cm³.
12.
A pyramid with a rectangular base of $8$ cm by $5$ cm has a volume of $120$ cm³. How tall is it?
Answer: 9
1. What is the base area in cm²? 40
2. What is $3V$ , three times the volume? 360
3. Divide by the base area. What is the height in cm? 9
13.
A box (rectangular prism) has a square base of $6$ cm by $6$ cm and is $10$ cm tall. A hollow pyramid has the same base and height. How many full pyramids of sand fill the box?
Answer: 3
1. What is the volume of the box in cm³? 360
2. What is the volume of the pyramid in cm³? 120
3. How many pyramids fill the box? 3
14.
You double the side of a square pyramid's base and keep the height the same. By how much is the volume multiplied?
1. a)×8
2. b)×3
3. c)×4
4. d)×2
Answer: ×4
The base area is $s^2$ , so doubling $s$ gives $(2s)^2 = 4s^2$ : four times the base area. The height is unchanged, so the volume is multiplied by $4$ .
15.
A square pyramid has a volume of $200$ cm³ and a height of $6$ cm. How long is one side of its base, in cm?
Answer: 10
$B = \frac{3V}{h} = \frac{600}{6} = 100$ cm², and $s = \sqrt{100} = 10$ cm.
16.
A square pyramid has base sides of $16$ cm and a slant height of $17$ cm. Find its volume.
Answer: 1280
1. What is half of the base side, in cm? 8
2. Use Pythagoras. What is the height $h$ in cm? 15
3. What is the base area in cm²? 256
4. What is the volume in cm³? 1280
17.
A large group tent is a square pyramid. Its floor has sides of $6$ m and the slant height of each face is $5$ m. How much air is inside?
Answer: 48
1. What is half of the floor side, in m? 3
2. Use Pythagoras. What is the height of the tent in m? 4
3. What is the floor area in m²? 36
4. What is the volume in m³? 48
18.
A glass paperweight is a square pyramid with a volume of $384$ cm³ and a height of $8$ cm. How long is one side of its base, in cm?
Answer: 12
1. What is $3V$ ? 1152
2. Divide by the height. What is the base area in cm²? 144
3. What is the side of the square base in cm? 12