Volume of Cylinders

Show your work for each problem.

1.
Which formula gives the volume of a cylinder with radius $r$ and height $h$ ?
1. a) $V = 2\pi r h$
2. b) $V = \pi r h$
3. c) $V = \pi r^2 h$
4. d) $V = \pi d^2 h$
Answer: $V = \pi r^2 h$
The base is a circle with area $\pi r^2$ , and volume is base area times height, so $V = \pi r^2 h$ . The formula $2\pi r h$ belongs to the curved surface, not to the volume.
2.
A cylinder has radius $2$ cm and height $5$ cm. Its volume is $V = \square\pi$ cm³. What number goes in the box?
Answer: 20
$V = \pi \times 2^2 \times 5 = \pi \times 4 \times 5 = 20\pi$ cm³.
3.
A cylinder has a diameter of $10$ cm. Which value should you use for $r$ in $V = \pi r^2 h$ ?
1. a)2.5 cm
2. b)10 cm
3. c)5 cm
4. d)20 cm
Answer: 5 cm
The radius is half the diameter: $r = 10 \div 2 = 5$ cm. Using $10$ would make the volume 4 times too big.
4.
A cylinder has radius $10$ cm and height $10$ cm. Find its volume in cm³. Use $\pi \approx 3.14$ .
Answer: 3140
$V \approx 3.14 \times 10^2 \times 10 = 3.14 \times 1000 = 3140$ cm³.
5.
Find the volume of a cylinder with radius $3$ cm and height $4$ cm. Use $\pi \approx 3.14$ and do not round.
Answer: 113.04
1. What is $r^2$ ? 9
2. What is $r^2 \times h$ ? 36
3. What is the volume in cm³? 113.04
6.
A can has a diameter of $6$ cm and a height of $10$ cm. What is its volume in terms of $\pi$ ?
1. a) $90\pi$ cm³
2. b) $360\pi$ cm³
3. c) $30\pi$ cm³
4. d) $60\pi$ cm³
Answer: $90\pi$ cm³
$r = 6 \div 2 = 3$ , so $V = \pi \times 3^2 \times 10 = 90\pi$ cm³. $360\pi$ comes from squaring the diameter.
7.
A cylinder has a diameter of $8$ cm and a height of $5$ cm. Find its volume using $\pi \approx 3.14$ . Round to the nearest whole cm³.
Answer: 251
$r = 4$ , $V \approx 3.14 \times 16 \times 5 = 3.14 \times 80 = 251.2$ , which rounds to $251$ cm³.
8.
A cylindrical water tank has radius $50$ cm and height $100$ cm. How many liters does it hold? Use $\pi \approx 3.14$ .
Answer: 785
1. What is $r^2$ ? 2500
2. What is $r^2 \times h$ ? 250000
3. What is the volume in cm³? 785000
4. How many liters is that? 785
9.
A cylinder has a volume of $942$ cm³ and a radius of $5$ cm. Find its height. Use $\pi \approx 3.14$ .
Answer: 12
1. What is $r^2$ ? 25
2. What is the base area $B$ in cm²? 78.5
3. What is the height in cm? 12
10.
You double the radius of a cylinder and keep the height the same. What happens to the volume?
1. a)It becomes 4 times as big
2. b)It doubles
3. c)It becomes 8 times as big
4. d)It stays the same
Answer: It becomes 4 times as big
The radius is squared, so doubling it multiplies the volume by $2^2 = 4$ : $\pi (2r)^2 h = 4\pi r^2 h$ .
11.
A cylinder has a volume of $50\pi$ cm³ and a radius of $5$ cm. What is its height in cm?
Answer: 2
$h = V \div (\pi r^2) = 50\pi \div 25\pi = 2$ cm. No need for $3.14$ here, because $\pi$ cancels.
12.
A soup can has a diameter of $10$ cm and a height of $12$ cm. How many liters does it hold? Use $\pi \approx 3.14$ .
Answer: 0.942
1. What is the radius in cm? 5
2. What is $r^2 \times h$ ? 300
3. What is the volume in cm³? 942
4. How many liters is that? 0.942
13.
A round tank is $2$ m wide (diameter) and $1.5$ m tall. How many liters does it hold? Use $\pi \approx 3.14$ and $1$ m³ $= 1000$ L.
Answer: 4710
1. What is the radius in meters? 1
2. What is the volume in m³? 4.71
3. How many liters is that? 4710
14.
A cylinder is $20$ cm tall and has a volume of $1570$ cm³. Find its radius. Use $\pi \approx 3.14$ .
Answer: 5
1. What is $3.14 \times h$ ? 62.8
2. What is $r^2$ ? 25
3. What is the radius in cm? 5
15.
Cylinder A has radius $4$ cm and height $3$ cm. Cylinder B has radius $3$ cm and height $4$ cm. Which one holds more?
1. a)Cylinder B
2. b)You cannot tell without a calculator
3. c)Cylinder A
4. d)They hold the same amount
Answer: Cylinder A
$V_A = \pi \times 4^2 \times 3 = 48\pi$ and $V_B = \pi \times 3^2 \times 4 = 36\pi$ . A holds more, because the radius is squared and counts more than the height.
16.
A cylinder has radius $2.5$ cm and height $7$ cm. Find its volume using $\pi \approx 3.14$ . Round to the nearest tenth of a cm³.
Answer: 137.4
$6.25 \times 7 = 43.75$ and $43.75 \times 3.14 = 137.375$ , which rounds to $137.4$ cm³.
17.
A full can (radius $4$ cm, height $15$ cm) is poured into glasses. Each glass is a cylinder with radius $3$ cm and height $8$ cm. How many glasses can you fill completely?
Answer: 3
1. The can holds $V = \square\pi$ cm³. What number goes in the box? 240
2. One glass holds $V = \square\pi$ cm³. What number goes in the box? 72
3. How many glasses can you fill completely? 3
18.
A cylindrical aquarium has a diameter of $40$ cm and a height of $50$ cm. How many liters does it hold? Use $\pi \approx 3.14$ and round to the nearest liter.
Answer: 63
1. What is the radius in cm? 20
2. What is $r^2 \times h$ ? 20000
3. What is the volume in cm³? 62800
4. How many liters is that, rounded to the nearest liter? 63