Volume of a Cylinder Formula

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

The Formula

V=πr2hV=\pi r^2h

When to use: Imagine stacking hundreds of identical circular coins into a tall tower. Each coin is a thin circle with area πr2\pi r^2, and stacking hh 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 33 and height 1010: V=π(3)2(10)=90π≈282.74 cubic unitsV = \pi(3)^2(10) = 90\pi \approx 282.74 \text{ cubic units}

Notation

πr2\pi r^2 is the circular base area; hh is 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 πr2\pi r^2, and stacking hh units high gives you a cylinder. The volume is just the area of one coin times the height of the stack.

Formal View

V=πr2h=∫0hπr2 dzV = \pi r^2 h = \int_0^h \pi r^2\,dz (Cavalieri's principle: stacking circular cross-sections of constant area πr2\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 π≈3.14\pi \approx 3.14.

Answer

V=160π≈502.4V = 160\pi \approx 502.4 cm³.

First step

1
Step 1: Write the formula for the volume of a cylinder: V=πr2hV = \pi r^2 h.

Full solution

  1. 2
    Step 2: Substitute the values: V=π×42×10=π×16×10=160πV = \pi \times 4^2 \times 10 = \pi \times 16 \times 10 = 160\pi.
  2. 3
    Step 3: Approximate: V≈160×3.14=502.4V \approx 160 \times 3.14 = 502.4 cm³.
The volume of a cylinder is the area of the circular base (πr2\pi r^2) multiplied by the height (hh). This formula makes intuitive sense: you are stacking up infinitely many thin circular disks of area πr2\pi r^2 to a total height hh.

Example 2

medium
A cylindrical water tank holds 2000π2000\pi liters. Its height is 20 m. Find the radius of the tank.

Example 3

medium
A cylindrical mug has diameter 88 cm and height 1010 cm. How many cm³ does it hold (use π≈3.14\pi\approx 3.14)?

Common Mistakes

  • Using 2πr2\pi r instead of πr2\pi r^2 for the base — volume uses base area, not circumference.
  • Forgetting to square the radius — base area is circular area.
  • Using diameter as radius — halve the diameter before substituting for rr.

Why This Formula Matters

Cylinder volume extends prism volume to circular bases and prepares students for cones, spheres, and real-world capacity problems. Recognizing it by "Can I identify the circular base area and the height?" — rather than by familiar numbers — is what lets a student tell it apart from volume of a cone and surface area of cylinder in a mixed problem set.

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 πr2\pi r^2, and stacking hh 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?

πr2\pi r^2 is the circular base area; hh is height.

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

Cylinder volume extends prism volume to circular bases and prepares students for cones, spheres, and real-world capacity problems. Recognizing it by "Can I identify the circular base area and the height?" — rather than by familiar numbers — is what lets a student tell it apart from volume of a cone and surface area of cylinder in a mixed problem set.

What do students get wrong about Volume of a Cylinder?

The procedure for volume of a cylinder is the easy part; the trap is using 2πr2\pi r instead of πr2\pi r^2 for the base. Asking "Can I identify the circular base area and the height?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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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