Key takeaway: The volume of a cylinder is the amount of space inside it, found by multiplying the area of the circular base by the height, which gives V equals pi times r squared times h.
Grade 88.G.C.9Geometry and Measurement
Definition of Volume of a Cylinder
The volume of a cylinder tells how much a can, pipe, silo or drum holds. A cylinder has two identical circular bases joined by a curved side, so it is just a stack of thin circles piled up to a certain height.
Each circle has area \(\pi r^{2}\). Stacking them to height \(h\) multiplies that base area by \(h\). Volume is measured in cubic units, for instance ft³ or in³, because it counts unit cubes that fill the solid. A tall skinny can and a short wide one can hold the same amount.
Formula and notation
To get the volume of a cylinder, use the base radius \(r\), height \(h\) and the constant \(\pi \approx 3.14159\):
\[V = \pi r^{2} h\]
Here \(\pi r^{2}\) is the base area. Use the radius, which is half of the diameter. Leave the answer in terms of \(\pi\) for an exact value, or round for a decimal.
Key facts
- Doubling the radius makes the volume four times as large; doubling the height only doubles it.
- Both measurements must use the same unit before you multiply.
- A cone sharing its base and height holds only a third as much.
- The volume of a cylinder in gallons needs a conversion from cubic units.
Volume of a Cylinder example
A farm water tank is shaped like an upright can: 4.5 feet from the center to the wall, and 12 feet tall. Find its volume to the nearest tenth of a cubic foot.
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Base area: \(\pi r^{2} = \pi (4.5)^{2} = 20.25\pi\).
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Multiply by the height: \(V = 20.25\pi \times 12 = 243\pi\).
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Approximate: \(243 \times 3.14159 \approx 763.4\).
The tank holds about 763.4 cubic feet, exactly \(243\pi\) cubic feet.
Common mistake
Watch out
A soup can is 9 inches tall and measures 6 inches straight across its top. Rae writes \(V = \pi (6)^{2}(9) = 324\pi\) cubic inches.
Why it is wrong
She squared the diameter. The radius is half of 6, which is 3, so \(V = \pi (3)^{2}(9) = 81\pi \approx 254.5\) cubic inches. Her answer was four times too large. Halve the diameter before you square.
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Geometry and Measurement