Volume of Cones

Show your work for each problem.

1.
Which formula gives the volume of a cone with radius $r$ and height $h$ ?
1. a) $V = \frac{4}{3}\pi r^3$
2. b) $V = \frac{1}{3}\pi r^2 h$
3. c) $V = 2\pi r h$
4. d) $V = \pi r^2 h$
Answer: $V = \frac{1}{3}\pi r^2 h$
The cone formula is $V = \frac{1}{3}\pi r^2 h$ . $\pi r^2 h$ is the cylinder, $\frac{4}{3}\pi r^3$ is the sphere and $2\pi r h$ is the curved surface of a cylinder, not a volume.
2.
A cylinder has a volume of 90 cm³. A cone has the same base and the same height. What is the volume of the cone in cm³?
Answer: 30
The cone holds one third of the cylinder: $90 \div 3 = 30$ cm³.
3.
A cone has radius 3 cm and height 5 cm. Its volume is $\square\,\pi$ cm³. What number goes in the box?
Answer: 15
$V = \frac{1}{3}\pi \times 3^2 \times 5 = \frac{45}{3}\pi = 15\pi$ cm³.
4.
A traffic cone has a base with a diameter of 10 cm. Which value do you put in for $r$ in the volume formula?
1. a)20 cm
2. b)2.5 cm
3. c)5 cm
4. d)10 cm
Answer: 5 cm
$r = 10 \div 2 = 5$ cm. Using 10 cm would make $r^2$ four times too big.
5.
Find the volume of a cone with radius 2 cm and height 6 cm. Use $\pi \approx 3.14$ and give the answer in cm³ (no rounding needed).
Answer: 25.12
1. What is $r^2$ ? 4
2. What is $r^2 \times h$ ? 24
3. Divide by 3. What do you get? 8
4. Multiply by 3.14. What is the volume in cm³? 25.12
6.
A party hat is a cone with radius 3 cm and height 4 cm. Its volume is $\square\,\pi$ cm³. Find the number in the box.
Answer: 12
1. What is $r^2$ ? 9
2. What is $r^2 \times h$ ? 36
3. Divide by 3. Which number goes in front of $\pi$ ? 12
7.
A cone has a radius of 6 cm and a slant height of 10 cm. What is its perpendicular height?
1. a)8 cm
2. b)4 cm
3. c)10 cm
4. d)16 cm
Answer: 8 cm
$h^2 = 10^2 - 6^2 = 100 - 36 = 64$ , so $h = 8$ cm. The slant height (10 cm) is the hypotenuse, so the height must be shorter.
8.
A pile of gravel is a cone with a diameter of 4 m and a height of 3 m. How many m³ of gravel are in the pile? Use $\pi \approx 3.14$ and round to the nearest hundredth.
Answer: 12.56
$r = 2$ m. $V = \frac{1}{3} \times 3.14 \times 2^2 \times 3 = 3.14 \times 4 = 12.56$ m³.
9.
An ice cream cone has a diameter of 6 cm and a height of 11 cm. Find its volume with $\pi \approx 3.14$ , rounded to the nearest tenth of a cm³.
Answer: 103.6
1. What is the radius in cm? 3
2. What is $r^2 \times h$ ? 99
3. Divide by 3. What do you get? 33
4. Multiply by 3.14 (do not round yet). 103.62
5. Round to the nearest tenth. 103.6
10.
You double the height of a cone and keep the radius the same. What happens to its volume?
1. a)It triples
2. b)It stays the same
3. c)It doubles
4. d)It becomes 4 times as large
Answer: It doubles
The volume is proportional to $h$ , so doubling $h$ doubles $V$ . Example: $r = 3$ , $h = 4$ gives $12\pi$ ; $h = 8$ gives $24\pi$ .
11.
A cone has a volume of $12\pi$ cm³ and a radius of 3 cm. What is its height in cm?
Answer: 4
$3\pi h = 12\pi$ , so $h = 12 \div 3 = 4$ cm. Check: $\frac{1}{3}\pi \times 9 \times 4 = 12\pi$ .
12.
A kitchen funnel is a cone with radius 5 cm and height 9 cm. How many cm³ does it hold when full? Use $\pi \approx 3.14$ (no rounding needed).
Answer: 235.5
1. What is $r^2$ ? 25
2. What is $r^2 \times h$ ? 225
3. Divide by 3. What do you get? 75
4. Multiply by 3.14. How many cm³? 235.5
13.
A cylinder and a cone both have radius 4 cm and height 6 cm. The cylinder holds $\square\,\pi$ cm³ more than the cone. Find the number in the box.
Answer: 64
1. The cylinder is $\square\,\pi$ cm³. What number? 96
2. The cone is $\square\,\pi$ cm³. What number? 32
3. Subtract. What number goes in front of $\pi$ ? 64
14.
You double the radius of a cone and keep the height the same. How does the new volume compare with the old one?
1. a)8 times as large
2. b)4 times as large
3. c)2 times as large
4. d)3 times as large
Answer: 4 times as large
$(2r)^2 = 4r^2$ , so the volume becomes 4 times as large. Example: $r = 3$ , $h = 4$ gives $12\pi$ ; $r = 6$ , $h = 4$ gives $48\pi$ .
15.
A cone has radius 5 cm and slant height 13 cm. Its volume is $\square\,\pi$ cm³. What number goes in the box?
Answer: 100
$h^2 = 169 - 25 = 144$ , so $h = 12$ . $V = \frac{1}{3}\pi \times 25 \times 12 = 100\pi$ cm³.
16.
A cone has a volume of 301.44 cm³ and a height of 8 cm. Find its radius in cm. Use $\pi \approx 3.14$ .
Answer: 6
1. What is $3 \times V$ ? 904.32
2. What is $\pi \times h$ with $\pi \approx 3.14$ ? 25.12
3. Divide to get $r^2$ . 36
4. What is $r$ ? 6
17.
A cone-shaped tent has a radius of 9 dm and a slant height of 15 dm. Find its volume in dm³. Use $\pi \approx 3.14$ (no rounding needed).
Answer: 1017.36
1. What is $h^2 = l^2 - r^2$ ? 144
2. What is the height $h$ in dm? 12
3. What is $\frac{1}{3} \times r^2 \times h$ ? 324
4. Multiply by 3.14. What is the volume in dm³? 1017.36
18.
Cone-shaped paper cups have radius 3 cm and height 10 cm. How many full cups do you need to fill a 1 liter (1000 cm³) jug completely? Use $\pi \approx 3.14$ .
Answer: 11
1. What is $\frac{1}{3} \times r^2 \times h$ ? 30
2. What is the volume of one cup in cm³? 94.2
3. $1000 \div 94.2 \approx 10.6$ . How many full cups do you need? 11