Volume of Cylinders
A cylinder is a stack of identical circles. Find the area of one circle, multiply by how tall the stack is, and you have its volume.
Know these first
- Master List skill 7.G.4
- Master List skill 7.G.6
This unlocks
- Volume of Cones 8.G.9
- Volume of Spheres 8.G.9
What you'll be able to do
- Find the volume of a cylinder with V = πr²h, in terms of π or rounded.
- Work backward to a missing radius or height.
- Find the volume of a hollow cylinder, like a pipe.
- Predict how volume changes when the radius or height changes.
The big idea
Picture a stack of coins. Every coin is the same circle, and the stack's volume is the area of one coin times the height of the stack.
The base of a cylinder is a circle with area πr². So V = πr²h: base area times height.
Volume is measured in cubic units, like cm³ or cubic feet, because it counts how many unit cubes fit inside.
Because the radius is squared, it matters more than the height. Double the height and the volume doubles; double the radius and the volume becomes 4 times as big.
Because the radius is squared, it matters more than the height. Double the height and the volume doubles; double the radius and the volume becomes 4 times as big.
Words you need
- Radius (r)
- The distance from the center of the circle to its edge.
- Diameter
- All the way across the circle: twice the radius.
- Height (h)
- The distance between the two circular bases.
- In terms of π
- Leaving π as a symbol, like 45π, for an exact answer.
Worked examples
Example 1
A cylinder has radius 3 and height 5. Find its volume in terms of π.
- Base area: π × 3² = 9π.
- Multiply by the height: 9π × 5 = 45π.
Answer: 45π cubic units
Example 2
A cylinder has radius 4 cm and height 10 cm. Find its volume to the nearest tenth.
- V = π × 4² × 10 = 160π.
- 160 × 3.14159... ≈ 502.65, so V ≈ 502.7 cm³.
Answer: about 502.7 cubic cm
Example 3
A cylinder has radius 5 and volume 200π. Find its height.
- Start from V = πr²h: 200π = π × 5² × h.
- So 200π = 25πh.
- Divide both sides by 25π: h = 8.
Answer: h = 8 units
Common mistakes
Using the diameter as the radius. If you're given the distance across, halve it first.
Forgetting to square the radius. It's r² - the radius times itself - not 2r.
Using square units. Volume is in cubic units, like cm³.
"Doubling the radius doubles the volume." The radius is squared, so doubling it multiplies the volume by 2² = 4.
"Doubling the radius doubles the volume." The radius is squared, so doubling it multiplies the volume by 2² = 4.
Practice: 20 questions
14 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Find the volume in terms of π.
- Find the volume in terms of π.
- A soup can has a radius of 4 cm and a height of 11 cm. How much can it hold, to the nearest tenth of a cubic centimeter? Word problem
- A candle is a cylinder with radius 2 inches and height 6 inches. What is its volume in terms of π? Word problem
- A water tank has radius 3 feet and height 8 feet. What is its volume, to the nearest tenth of a cubic foot? Word problem
Tier 2 Intermediate One more step to think through
- Find the volume to the nearest tenth.
- The volume is 36π and the radius is 2. Find the height.
- The volume is 64π and the height is 4. Find the radius.
- A round cake pan is 10 inches across and 2 inches deep. What is its volume in terms of π? Word problem
- A grain silo with radius 4 meters holds 320π cubic meters. How tall is it? Word problem
- A glass 10 cm tall holds 90π cubic centimeters. What is its radius? Word problem
- A round pool has a radius of 6 meters and is 1.5 meters deep. How much water does it hold, to the nearest tenth of a cubic meter? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- A cylinder's radius is doubled and its height stays the same. How many times as large is the new volume?
- A metal pipe is 100 cm long. Its outer radius is 5 cm and the hole through it has radius 4 cm. How much metal is in the pipe, in terms of π? Word problem
- Can B has twice the radius of can A but half the height. How many times as much does can B hold? Word problem
- A cylindrical tank with radius 3 feet and height 4 feet is filled at 2 cubic feet per minute. How many minutes does it take to fill, to the nearest minute? Word problem
- A water bottle is 8 cm across and holds 400π cubic centimeters. How tall is it? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- A roll of paper towels has radius 6 cm and height 28 cm. The cardboard tube in the middle has radius 2 cm. What volume of paper is on the roll, to the nearest tenth? Word problem
- A stone with volume 50π cubic centimeters is dropped into a cylindrical glass of radius 5 cm, and it sinks under the water. How many centimeters does the water level rise? Word problem
- Can A has radius 3 inches and height 8 inches. Can B has radius 4 inches and height 5 inches. Which holds more, and by how much (in terms of π)? Word problem
Answers
Tier 1 Foundations
- 28π cubic units. V = πr²h = 28π.
- 100π cubic units. V = πr²h = 100π.
- about 552.9 cubic cm. V = πr²h = 176π ≈ 552.9.
- 24π cubic inches. V = πr²h = 24π.
- about 226.2 cubic feet. V = πr²h = 72π ≈ 226.2.
Tier 2 Intermediate
- about 339.3 cubic units. V = πr²h = 108π ≈ 339.3.
- h = 9 units. Start from V = πr²h, put in what you know, and solve for h.
- r = 4 units. Start from V = πr²h, put in what you know, and solve for r.
- 50π cubic inches. V = πr²h = 50π.
- h = 20 meters. Start from V = πr²h, put in what you know, and solve for h.
- r = 3 cm. Start from V = πr²h, put in what you know, and solve for r.
- about 169.6 cubic meters. V = πr²h = 54π ≈ 169.6.
Tier 3 Advanced
- 4 times as much. In V = πr²h, r is squared and h is not: 2² × 1 = 4.
- 900π cubic cm. Outer cylinder minus the hole: π × 5² × 100 − π × 4² × 100 = 900π.
- 2 times as much. In V = πr²h, r is squared and h is not: 2² × 0.5 = 2.
- about 57 minutes. V = π × 3² × 4 = 36π ≈ 113.1 cubic feet. At 2 cubic feet per minute that's 113.1 ÷ 2 ≈ 57 minutes.
- h = 25 cm. Start from V = πr²h, put in what you know, and solve for h.
Tier 4 Challenge
- about 2814.9 cubic cm. Outer cylinder minus the hole: π × 6² × 28 − π × 2² × 28 = 896π ≈ 2814.9.
- 2 cm. The water rises by a cylinder of volume 50π with radius 5: π × 5² × rise = 50π, so the rise is 50 ÷ 25 = 2 cm.
- 8 π cubic inches more (can B). Can A holds π × 3² × 8 = 72π and can B holds π × 4² × 5 = 80π, so can B holds 8π cubic inches more.
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