How to Find the Area and Circumference of a Circle

The two formulas

Area: A = πr² — the space inside the circle, measured in square units.

Circumference: C = 2πr — the distance all the way around the edge, measured in ordinary units.

For a circle of radius 5: A = π × 25 ≈ 78.54 square units, and C = 2 × π × 5 ≈ 31.42 units.

How to remember which is which

Mixing these two up is easily the most common circle mistake, so it is worth having a way to tell them apart that does not rely on memory.

Use the units. Area is measured in square units, so its formula must contain something squared — r². Circumference is a length, measured in plain units, so it cannot have anything squared in it.

There is also a rhyme that many people are taught: "twice pi r is the circle round, pi r squared is the area found." Slightly clunky, but it sticks.

When you only know the diameter

Real objects are usually measured across rather than from the centre, so you will often have the diameter instead of the radius.

The radius is half the diameter. Halve it first, then use the formulas as normal.

A very common error is to square the diameter instead of the radius, which gives an answer four times too big. If your circle area seems suspiciously large, this is the first thing to check.

For circumference you can skip the halving entirely, since C = πd works directly from the diameter.

What value of π should you use?

For hand calculations, 3.14 is fine and 3.1416 is more than enough. The calculator on this page uses the full precision your device supports, which is why its answers may differ from yours in the fourth decimal place.

If a question asks you to leave your answer "in terms of π", it wants 25π rather than 78.54. That is not being awkward — it is exact, whereas any decimal is rounded.

The fraction 22/7 is a rough approximation that some courses still teach. It is accurate to about 0.04%, which is fine for arithmetic practice and not fine for anything you are going to build.

Semicircles, quarters and sectors

For a fraction of a circle, work out the whole circle first and then take that fraction of it. A semicircle is half the area, a quarter circle is a quarter.

Perimeter needs more care. The perimeter of a semicircle is not half the circumference — it is half the curved edge *plus* the straight diameter across the bottom, because that straight edge is now part of the outline.

Forgetting the straight edge is one of the most frequently lost marks in geometry papers, and it is worth drawing the shape and tracing round it with a finger before answering.