Volume of a Cylinder

The space inside a cylinder equals the area of its circular base times its height.

V = π r² h

The circular base has radius r. Multiply its area by the height h to fill the cylinder.

What each symbol means

A cylinder is a can shape: a circle on the bottom, the same circle on top, and straight sides joining them. To find how much fits inside, first find the area of the circular base, then multiply by how tall it is.

The formula is V = π r² h. The π r² part is just the area of the circular base. The h stacks that circle all the way up.

It builds on the circle: You already know the area of a circle is π r². Volume of a cylinder is simply that base area carried upward through the height.

A worked example

A can has radius 3 cm and height 10 cm

  1. Write the formula: V = π r² h.
  2. Put in the numbers: V = π × 3² × 10.
  3. Square the radius: 3² = 9.
  4. Multiply: π × 9 × 10 = π × 90.
  5. With π ≈ 3.14: V ≈ 3.14 × 90 ≈ 282.6 cubic cm.

Cubic units: Volume is measured in cubic units (like cubic cm), because you are filling a 3D space, counting little 1-by-1-by-1 cubes.

Why base area times height works

  1. Think of the cylinder as a stack of many identical paper circles, one on top of another.
  2. Each circle has area π r². That is one thin layer.
  3. Stacking them to a height h multiplies that layer by the height.
  4. So the total space is base area times height: π r² × h = π r² h. This same "base area times height" idea works for any prism.

Practice questions

Question 1

Find the volume of a cylinder with radius 2 cm and height 5 cm. Use π ≈ 3.14.

  1. Write the formula: V = π r² h.
  2. Square the radius first: 2² = 4.
  3. Put the numbers in with π ≈ 3.14: V = 3.14 × 4 × 5.
  4. Multiply across: 3.14 × 4 = 12.56, then 12.56 × 5 = 62.8.
  5. The volume is 62.8 cubic cm.

Answer: 62.8 cubic cm

Question 2

Find the volume of a cylinder with radius 5 m and height 2 m. Use π ≈ 3.14.

  1. Write the formula: V = π r² h.
  2. Square the radius first: 5² = 25.
  3. Put the numbers in with π ≈ 3.14: V = 3.14 × 25 × 2.
  4. Multiply across: 3.14 × 25 = 78.5, then 78.5 × 2 = 157.
  5. The volume is 157 cubic m.

Answer: 157 cubic m

Question 3

A cylinder has radius 7 cm and height 10 cm. Find its volume using π ≈ 227.

  1. Write the formula: V = π r² h.
  2. Square the radius first: 7² = 49.
  3. Put the numbers in with π ≈ 227: V = 227 × 49 × 10.
  4. Simplify the fraction: 49 ÷ 7 = 7, so this becomes 22 × 7 × 10.
  5. Multiply across: 22 × 7 = 154, then 154 × 10 = 1540, so the volume is 1540 cubic cm.

Answer: 1540 cubic cm

Question 4

A water tank is a cylinder with radius 1 m and height 3 m. Find its volume. Use π ≈ 3.14.

  1. The tank is a cylinder, so use V = π r² h.
  2. Square the radius first: 1² = 1.
  3. Put the numbers in with π ≈ 3.14: V = 3.14 × 1 × 3.
  4. Multiply across: 3.14 × 3 = 9.42.
  5. The tank holds 9.42 cubic m.

Answer: 9.42 cubic m

Question 5

If two cylinders have the same radius but one is twice as tall, how do their volumes compare?

  1. Look at the formula V = π r² h: with the radius fixed, the part π r² stays the same for both cylinders.
  2. That fixed base area is simply multiplied by the height, so volume grows in step with height.
  3. Replacing h with 2h gives π r² × 2h = 2 × (π r² h), which is twice the original volume.
  4. So the taller cylinder has twice the volume.

Answer: The taller one has twice the volume.