Volume of a Cylinder
In short: Volume of a cylinder measures how much a round can or tube can hold: multiply the area of the circular end by the height, V = πr²h.
8th GradeGeometry8.G.C.9
What is Volume of a Cylinder?
Piling identical coins into a tower builds a cylinder layer by layer, and the volume of a cylinder follows the same idea: every layer is a circle of area \(\pi r^2\), and the height \(h\) says how many layers there are.
The result is \(V = \pi r^2 h\), reported in cubic units such as cubic feet. Take care that \(r\) is the radius, half the diameter, and that squaring it comes before the multiplication.
How to spot it
- The problem describes a can, tank, pipe, silo, drum or candle with a round base.
- You are asked how much a container holds, how much liquid fills it or how much material it contains.
- Two measurements are given: something across the circle and something along the height.
- The answer is wanted in cubic units, sometimes in terms of π.
The method
- Find the radius; if the diameter is given, divide it by 2.
- Compute the base area \(\pi r^2\) by squaring the radius first.
- Multiply the base area by the height \(h\).
- Attach cubic units, and round only at the very end if a decimal is requested.
Volume of a Cylinder: solved example
Problem
A grain bin on a Nebraska farm is a cylinder 10 feet across and 24 feet tall. Compute its capacity in cubic feet, using 3.14 for π.
Solution
Half of 10 is the radius, 5 ft, so the floor area is \(\pi \cdot 5^2 = 25\pi\).
Times the height: \(V = 25\pi \cdot 24 = 600\pi\). With \(\pi \approx 3.14\), \(600 \times 3.14 = 1{,}884\). The bin holds about 1,884 cubic feet.
Try it
Question 1
A candle mold has radius 2.5 in. and height 14 in. State its volume in terms of π.
Show the answer
\(\pi \cdot 2.5^2 \cdot 14 = 87.5\pi\) cubic inches, about 274.9.
Question 2
A cylinder with radius 5 cm has volume \(150\pi\) cm³. How tall is it?
Show the answer
\(25\pi h = 150\pi\), so \(h = 6\) cm.
Volume of a Cylinder vs. Surface Area
The volume of a cylinder counts the cubic units that fill the inside, so it uses \(\pi r^2 h\). Surface area counts the square units of material needed to cover the outside: two circles plus the wrapped side. Fill a can with water for the first; label it with paper for the second.
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