Area of a Circle — Formula, Derivation & Examples

To find the area of a circle, square the radius and multiply by $\pi$:

This formula uses the radius, not the diameter. The $r^2$ tells you the area grows with the square of the radius: if the radius doubles, the area becomes four times as large, not two times. That squared term is the main idea to remember, because it makes circle area change quickly.

When to use this, and what you need first

Use circle area when you need the size of a circular region on a flat surface: a pizza, a round tabletop, a circular garden bed, or a pipe cross-section. If a question asks for material covering a round surface, paint for a circular face, or space inside a round boundary, area is the right idea. Before you start, check whether the problem gives the radius or the diameter, because the formula only takes the radius. If a problem gives the diameter $d$, convert with $r = d/2$. The same relationship can be written as

The procedure, step by step

1. Check the given measure. Decide whether the problem gives the radius or the diameter.
2. Convert if needed. If the diameter is given, divide by $2$ to get the radius.
3. Apply the formula. Substitute into $A = \pi r^2$.
4. Square before multiplying. Compute $r^2$ first, then multiply by $\pi$.
5. Report square units. Write the answer in square units, and keep $\pi$ unless the problem asks for a decimal.

If the problem asks for an exact answer, leave the result in terms of $\pi$. If it asks for a decimal, use an approximation such as $\pi \approx 3.14$.

Why the formula is $A = \pi r^2$

One common derivation cuts a circle into many thin sectors and rearranges them in alternating directions. As the sectors get thinner, the rearranged shape gets closer to a rectangle. In that picture, the rectangle's height is about $r$, and its base is about half the circle's circumference:

So the area approaches

The more sectors you imagine, the closer the rearranged shape gets to a true rectangle, which gives a solid intuition for the formula without advanced geometry.

A full example, radius then diameter

Suppose a circle has radius $6$ cm. Apply the formula:

So the exact area is $36\pi\ \text{cm}^2$. If a decimal is required,

Use the exact form when the problem says "in terms of $\pi$," and the decimal form only when it asks for an estimate.

Now suppose the diameter is $12$ cm instead. Step 2 converts first:

Then the usual formula:

This is where many mistakes happen. Putting $12$ directly into $A = \pi r^2$ gives $144\pi$, four times too large.

Where each step trips people up

1. Using the diameter directly in place of the radius (skip Step 2 and the answer is four times too big).
2. Forgetting to square the radius.
3. Writing the result in plain units instead of square units.
4. Rounding too early when the problem wants an exact answer in terms of $\pi$.
5. Mixing up area and circumference. Area measures the space inside; circumference measures the distance around the edge.

One fast self-check at the end: ask whether the answer size makes sense. A circle with radius $10$ should have much more area than one with radius $5$, because doubling the radius multiplies the area by $4$. That quick check catches many radius-versus-diameter mistakes.

Run the steps yourself

Take a diameter of $18$ cm. Convert to radius first, then find the exact area, and only after that compute a decimal approximation if needed. To feel the squared term, compare the area when the radius changes from $4$ cm to $8$ cm and confirm the area changes by a factor of $4$, not $2$.