Volume of a Cylinder Formula

The Formula

V = \pi r^2 h

When to use: Imagine stacking hundreds of identical circular coins into a tall tower. Each coin is a thin circle with area \pi r^2, and stacking h units high gives you a cylinder. The volume is just the area of one coin times the height of the stack.

Quick Example

A cylinder with radius 3 and height 10: V = \pi(3)^2(10) = 90\pi \approx 282.74 \text{ cubic units}

Notation

V for volume, r for radius of the base, h for height

What This Formula Means

The amount of three-dimensional space inside a cylinder, found by multiplying the area of the circular base by the height.

Imagine stacking hundreds of identical circular coins into a tall tower. Each coin is a thin circle with area \pi r^2, and stacking h units high gives you a cylinder. The volume is just the area of one coin times the height of the stack.

Formal View

V = \pi r^2 h = \int_0^h \pi r^2\,dz (Cavalieri's principle: stacking circular cross-sections of constant area \pi r^2)

Worked Examples

Example 1

easy
A cylinder has a radius of 4 cm and a height of 10 cm. Find its volume. Use \pi \approx 3.14.

Solution

  1. 1
    Step 1: Write the formula for the volume of a cylinder: V = \pi r^2 h.
  2. 2
    Step 2: Substitute the values: V = \pi \times 4^2 \times 10 = \pi \times 16 \times 10 = 160\pi.
  3. 3
    Step 3: Approximate: V \approx 160 \times 3.14 = 502.4 cm³.

Answer

V = 160\pi \approx 502.4 cm³.
The volume of a cylinder is the area of the circular base (\pi r^2) multiplied by the height (h). This formula makes intuitive sense: you are stacking up infinitely many thin circular disks of area \pi r^2 to a total height h.

Example 2

medium
A cylindrical water tank holds 2000\pi liters. Its height is 20 m. Find the radius of the tank.

Common Mistakes

  • Using diameter instead of radius in the formula
  • Confusing height with slant height for tilted cylinders
  • Forgetting that volume uses cubic units, not square units

Why This Formula Matters

Cylinders are everywhere—cans, pipes, tanks, silos. Knowing the volume tells you how much they hold.

Frequently Asked Questions

What is the Volume of a Cylinder formula?

The amount of three-dimensional space inside a cylinder, found by multiplying the area of the circular base by the height.

How do you use the Volume of a Cylinder formula?

Imagine stacking hundreds of identical circular coins into a tall tower. Each coin is a thin circle with area \pi r^2, and stacking h units high gives you a cylinder. The volume is just the area of one coin times the height of the stack.

What do the symbols mean in the Volume of a Cylinder formula?

V for volume, r for radius of the base, h for height

Why is the Volume of a Cylinder formula important in Math?

Cylinders are everywhere—cans, pipes, tanks, silos. Knowing the volume tells you how much they hold.

What do students get wrong about Volume of a Cylinder?

Make sure h is the perpendicular height (straight up), not the slant height.

What should I learn before the Volume of a Cylinder formula?

Before studying the Volume of a Cylinder formula, you should understand: area of circle, volume.

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