Area⏱ 5 min read

Area of a Circle

A = πr²

What is the Area of a Circle?

The area of a circle scales with the square of its radius, which is why doubling the radius quadruples the area rather than just doubling it.

If you're given the diameter instead of the radius, divide it by two before squaring — a mix-up between the two is the single most common error on this formula.

What Each Variable Means

AreaThe amount of two-dimensional space enclosed by the circle.

RadiusThe distance from the circle's center to its edge.

PiA constant, approximately 3.14159, equal to a circle's circumference divided by its diameter.

When to Use It

Finding the surface area of a circular object (a table top, a pool cover, a pipe's cross-section)
As a building block for the volume and surface area formulas of spheres, cylinders, and cones
Comparing how area scales when a circular design is resized

Where This Formula Comes From

Cut the circle into equal sectors

Slice the circle into many thin, equal pie-shaped wedges — the more slices, the better the next step works.

Rearrange the sectors into a near-rectangle

Lay the wedges alternately point-up and point-down in a row. As the number of slices grows, this arrangement looks more and more like a rectangle.

Identify the rectangle's dimensions

Its height is approximately the circle's radius r, and its width is approximately half the circle's circumference, since the wedges' curved edges — half pointing each way — make up the top and bottom.

width ≈ ½ × 2πr = πr

Multiply width by height

The rectangle's area — and therefore the circle's area — is width times height.

A ≈ πr × r = πr²

Step-by-Step Examples

Example 1: Finding area from radius

Problem: Find the area of a circle with radius 7 cm.

Confirm you have the radius, not the diameter

The problem already gives the radius directly.

r = 7 cm

Square the radius

Multiply r by itself.

r² = 7 × 7 = 49

Multiply by π

Use π ≈ 3.14159.

A = π × 49 ≈ 153.94

✓

Answer: A ≈ 153.94 cm²

Example 2: Finding radius from area

Problem: A circle has an area of 78.54 cm². Find its radius.

Use the rearranged formula

Since area is known and radius isn't, solve for r instead of A.

r = √(A / π)

Substitute the known area

Plug in A = 78.54 and divide by π.

r = √(78.54 / π) = √(25.0)

Take the square root

Finish by taking the square root of 25.

r ≈ 5

✓

Answer: r ≈ 5 cm

Interactive Calculator

r (radius)

Result will appear here

Solving for Other Variables

r = √(A / π)Solve for the radius when the area is known.

Common Mistakes

Mistake: Using the diameter in place of the radius.
Fix: If you're given the diameter, divide it by 2 to get the radius first — A = πr² needs the radius, not the diameter.
Mistake: Multiplying by 2 instead of squaring.
Fix: A = πr² squares the radius. Multiplying by 2π instead gives you the circumference, not the area.

Practice Questions

Find the area of a circle with radius 10 m.
A circle has a diameter of 8 cm. What is its area?
Hint: First find the radius: 8 ÷ 2 = 4 cm.

Frequently Asked Questions

What if I only know the diameter?

Divide it by 2 to get the radius, then use A = πr² as usual. Equivalently, A = π(d/2)² = πd²/4.

Does this formula work for any size circle?

Yes — A = πr² holds for every circle, from a coin to a planet's cross-section, as long as r is measured consistently.

How is area different from circumference?

Circumference (C = 2πr) measures the distance around the circle's edge, in the same linear units as the radius. Area measures the two-dimensional space inside it, in squared units.