How to find the area of a circle

The area of a circle, the amount of two-dimensional space it covers, is found with the formula A = πr², where r is the radius (the distance from the center of the circle to any point on its edge) and π (pi) is a constant, approximately3.14159, that relates a circle's dimensions to each other no matter how big or small the circle is.

What π actually is

π is the ratio of a circle's circumference to its diameter. It's the same number, approximately 3.14159, for every circle that exists, from a coin to a planet's orbit, which is what makes it possible to write one formula that works for any circle. π is an irrational number, meaning its decimal digits never repeat or terminate, so in practice you either round it (commonly to 3.14) or leave it in the answer as an exact symbolic value, like25π.

Basic worked example

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

1. Square the radius: 5² = 25.
2. Multiply by π: 25 × π ≈ 25 × 3.14159 ≈ 78.5.
3. Attach the correct units: the area is approximately 78.5 cm².

Note the order of operations: square the radius first, then multiply by π. Multiplying by π before squaring, or squaring the whole expression πr instead of just r, both give the wrong answer.

Starting from the diameter instead

Problems often give the diameter, the distance all the way across the circle through its center, rather than the radius. The diameter is always exactly twice the radius, so divide it by 2 before applying the area formula.

Worked example

Find the area of a circle with a diameter of 10 cm.

1. Find the radius: 10 / 2 = 5 cm.
2. Square it: 5² = 25.
3. Multiply by π: 25π ≈ 78.5 cm².

This example uses the same numbers as the first one on purpose: a 10 cm diameter and a 5 cm radius describe the identical circle, and skipping the diameter-to-radius conversion is the single most common mistake in this topic. Plugging the diameter directly intoA = πr² without dividing by 2 first roughly quadruples the answer, since doubling r inside a squared term multiplies the result by 4, not 2.

Working with fractional and decimal radii

The same formula applies regardless of whether the radius is a whole number.

Worked example: a decimal radius

Find the area of a circle with a radius of 3.5 inches.

1. Square the radius: 3.5² = 12.25.
2. Multiply by π: 12.25 × π ≈ 38.48.
3. The area is approximately 38.48 in².

Leaving the answer in terms of π

Some problems, especially in geometry courses, ask for an exact answer rather than a decimal approximation. In that case, leave π as a symbol instead of substituting 3.14159.

Worked example

Find the exact area of a circle with a radius of 6 cm, in terms of π.

1. Square the radius: 6² = 36.
2. The exact area is 36π cm², without approximating π to a decimal.

If a decimal is needed afterward, 36π ≈ 113.1 cm², but the exact form36π is considered more precise, since it carries no rounding error at all.

Working backward: finding the radius from the area

Sometimes the area is known and the radius is the unknown. Reverse the formula by dividing by π first, then taking the square root.

Worked example

A circle has an area of 78.5 square centimeters. Find its radius.

1. Divide both sides by π: 78.5 / π ≈ 25.
2. Take the square root of both sides: √25 = 5.
3. The radius is 5 cm, which matches the original example above.

Area versus circumference

It's easy to mix up the two main circle formulas, especially since both use π:

Circumference scales directly with the radius: doubling the radius doubles the circumference. Area scales with the radius squared: doubling the radius quadruples the area. That difference is worth remembering on its own, since it's a fast way to sanity-check whether an answer seems reasonable, independent of redoing the arithmetic.

Common mistakes

• Using the diameter instead of the radius in A = πr². Always divide the diameter by 2 first if that's what's given.
• Squaring the wrong thing. It's r that gets squared, not π or the whole expression πr treated as one unit before squaring.
• Leaving off square units, or using linear units by mistake. Area is always reported in square units (cm², m², in²), never plain centimeters or inches.
• Rounding π too early or too aggressively. Using 3 instead of 3.14 introduces a noticeable error, since π rounds up meaningfully from 3.
• Confusing the area and circumference formulas. Mixing up πr²and 2πr is one of the most frequent errors on circle problems specifically because both formulas involve π and r.

Where this shows up outside of class

Circular area calculations come up when buying enough sod or mulch for a round garden bed, figuring out how much pizza you're actually getting for the price by comparing two different sizes, sizing a circular rug or tablecloth, or estimating material needed for a round pool cover. In each case, the same two-step process, radius squared, then multiplied by π, gives the answer.

Practice it yourself

Try the geometry calculator for a quick answer, or head togeometry practice for an unlimited set of area and circumference problems with step-by-step explanations.