Area of a Circle Formula

The Formula

A = \pi r^2

When to use: Imagine cutting a pizza into many thin slices and rearranging them into a shape that looks like a rectangle. The 'height' of that rectangle is the radius r, and the 'width' is half the circumference (\pi r). So the area is r \times \pi r = \pi r^2.

Quick Example

A circle with radius r = 5: A = \pi(5)^2 = 25\pi \approx 78.54 \text{ square units}

Notation

A for area, r for radius

What This Formula Means

The amount of space enclosed inside a circle, calculated as \pi times the square of the radius.

Imagine cutting a pizza into many thin slices and rearranging them into a shape that looks like a rectangle. The 'height' of that rectangle is the radius r, and the 'width' is half the circumference (\pi r). So the area is r \times \pi r = \pi r^2.

Formal View

A = \pi r^2 = \iint_{x^2+y^2 \leq r^2} dA; in polar coordinates: A = \int_0^{2\pi}\int_0^r \rho\,d\rho\,d\theta = \pi r^2

Worked Examples

Example 1

easy
Find the area of a circle with radius 6 cm. Leave your answer in terms of \pi.

Solution

  1. 1
    The area enclosed by a circle of radius r is A = \pi r^2. This can be understood by imagining the circle divided into many thin triangles from the centre; their combined area gives \frac{1}{2} \times (2\pi r) \times r = \pi r^2.
  2. 2
    Substitute r = 6 cm: A = \pi(6)^2 = \pi \times 36.
  3. 3
    Result: A = 36\pi cm² \approx 113.1 cm². Note that doubling the radius quadruples the area (since r is squared), a key scaling insight.

Answer

A = 36\pi \text{ cm}^2
The area of a circle depends on the square of the radius. Doubling the radius quadruples the area, which illustrates the quadratic relationship between radius and area.

Example 2

medium
Find the area of a semicircle with diameter 20 cm. Give your answer to one decimal place.

Common Mistakes

  • Using the diameter instead of the radius in \pi r^2
  • Confusing area (\pi r^2) with circumference (2\pi r)
  • Forgetting to square the radius—writing \pi r instead of \pi r^2

Why This Formula Matters

Essential for calculating the space covered by circular objects—pizzas, gardens, pipes, wheels, and lenses.

Frequently Asked Questions

What is the Area of a Circle formula?

The amount of space enclosed inside a circle, calculated as \pi times the square of the radius.

How do you use the Area of a Circle formula?

Imagine cutting a pizza into many thin slices and rearranging them into a shape that looks like a rectangle. The 'height' of that rectangle is the radius r, and the 'width' is half the circumference (\pi r). So the area is r \times \pi r = \pi r^2.

What do the symbols mean in the Area of a Circle formula?

A for area, r for radius

Why is the Area of a Circle formula important in Math?

Essential for calculating the space covered by circular objects—pizzas, gardens, pipes, wheels, and lenses.

What do students get wrong about Area of a Circle?

The formula uses the radius, not the diameter. If given the diameter, divide by 2 first.

What should I learn before the Area of a Circle formula?

Before studying the Area of a Circle formula, you should understand: circles, pi, area.

← Back to Area of a Circle See All Examples Practice Problems