Area of a Circle

Q: Why is the area of a circle π r² and not 2πr?

The expression $2\pi r$ gives the circumference (the length around the circle), not the area. Area measures the two-dimensional space inside the circle, which grows with the square of the radius. One intuitive way to see this: if you cut a circle into many thin wedges and rearrange them, they form a shape close to a rectangle with height $r$ and width $\pi r$, giving area $\pi r^2$.

Area of a Circle

The formula is below.

Circle with radius r labeled. Formulas: Area = πr², Circumference = 2πr or πd (d = diameter).

See also

Circle, circumference

Key Formula

A = \pi r^2

Where:

$A$ = Area of the circle (in square units)
$\pi$ = Pi, approximately 3.14159
$r$ = Radius of the circle (distance from center to edge)

Worked Example

Problem: Find the area of a circle with a radius of 5 cm.

Step 1: Write the formula for the area of a circle.

A = \pi r^2

Step 2: Substitute the radius into the formula.

A = \pi (5)^2

Step 3: Square the radius.

A = \pi \times 25

Step 4: Multiply by π to get the exact area, or use 3.14159 for an approximation.

A = 25\pi \approx 78.54 \text{ cm}^2

Answer: The area of the circle is 25π cm², which is approximately 78.54 cm².

Another Example

Problem: A circular garden has a diameter of 12 meters. How much ground does it cover?

Step 1: The diameter is 12 m, so find the radius by dividing by 2.

r = \frac{12}{2} = 6 \text{ m}

Step 2: Apply the area formula.

A = \pi r^2 = \pi (6)^2

Step 3: Square the radius and multiply by π.

A = 36\pi \approx 113.10 \text{ m}^2

Answer: The garden covers 36π m², which is approximately 113.10 m².

Frequently Asked Questions

How do you find the area of a circle if you only know the diameter?

Divide the diameter by 2 to get the radius, then use the standard formula

A = \pi r^2

. Alternatively, you can substitute

r = \frac{d}{2}

directly into the formula to get

A = \frac{\pi d^2}{4}

, where

d

is the diameter.

Why is the area of a circle π r² and not 2πr?

The expression

2\pi r

gives the circumference (the length around the circle), not the area. Area measures the two-dimensional space inside the circle, which grows with the square of the radius. One intuitive way to see this: if you cut a circle into many thin wedges and rearrange them, they form a shape close to a rectangle with height

r

and width

\pi r

, giving area

\pi r^2

Area of a Circle vs. Circumference of a Circle

Area (

A = \pi r^2

) measures the space enclosed inside the circle in square units. Circumference (

C = 2\pi r

) measures the distance around the circle's edge in linear units. Area depends on

r^2

, so doubling the radius quadruples the area. Circumference depends on

r

directly, so doubling the radius only doubles the circumference.

Why It Matters

Knowing how to calculate the area of a circle comes up constantly in real life—sizing a pizza, planning a circular garden, or designing round components in engineering. It is also foundational in higher math: computing volumes of cylinders and cones, working with polar coordinates, and understanding integrals all build on this formula.

Common Mistakes

Mistake: Using the diameter instead of the radius in the formula.

Correction: The formula

A = \pi r^2

requires the radius. If you are given the diameter, divide it by 2 first. Plugging the diameter in directly gives an answer four times too large.

Mistake: Multiplying by 2 before squaring, writing

A = \pi (2r)

or confusing the area formula with the circumference formula.

Correction: Remember: area squares the radius (

\pi r^2

), while circumference multiplies the radius by

2\pi

(

2\pi r

). These are different operations that measure different things.

Related Terms

Circle — The shape whose area this formula calculates
Circumference — Perimeter of a circle, uses a different formula
Formula — General term for mathematical equations like πr²
Radius — Key measurement used in the area formula
Diameter — Twice the radius; often given in problems
Pi (π) — The constant approximately 3.14159 in the formula
Area — General concept of measuring enclosed 2D space