Area of a Circle — Formula, Derivation & Examples

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

$$A = \pi r^2$$

This formula uses the radius, not the diameter. If a problem gives the diameter $d$, convert first with $r = d/2$. The same relationship can be written as

$$A = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}$$

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 $\pi \approx 3.14$.

Area of a circle formula: what it means

$r^2$ 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 as large.

That is the main idea to remember. Circle area changes quickly because the radius is squared.

Why the area of a circle is $A = \pi r^2$

One common derivation is to cut a circle into many thin sectors and rearrange 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:

$$\frac{1}{2}(2\pi r) = \pi r$$

So the area approaches

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

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

Area of a circle example with radius $6$ cm

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

$$A = \pi r^2 = \pi(6)^2 = 36\pi$$

So the exact area is $36\pi\ \text{cm}^2$.

If a decimal approximation is required, then

$$A \approx 36(3.14) = 113.04\ \text{cm}^2$$

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

How to find the area of a circle from the diameter

If the diameter is $12$ cm, first convert to radius:

$$r = \frac{12}{2} = 6$$

Then use the usual formula:

$$A = \pi(6)^2 = 36\pi\ \text{cm}^2$$

This is where many mistakes happen. If you put $12$ directly into $A = \pi r^2$, you get $144\pi$ instead of $36\pi$, which is four times too large.

Common mistakes with circle area

1. Using the diameter directly in place of the radius.
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.

When to use the area of a circle

Use circle area when you need the size of a circular region on a flat surface. Common examples include a pizza, a round tabletop, a circular garden bed, or a pipe cross-section.

If the question asks for material covering a round surface, paint needed for a circular face, or space inside a round boundary, area is usually the right idea.

One fast check before you finish

Ask whether the answer size makes sense. A circle with radius $10$ should have much more area than a circle with radius $5$, because doubling the radius multiplies the area by $4$.

That quick check catches many radius-versus-diameter mistakes.

Try a similar problem

Try your own version with diameter $18$ cm. First convert to radius, then find the exact area, and only after that compute a decimal approximation if needed. If you want to solve a similar problem, compare the area when the radius changes from $4$ cm to $8$ cm and check why the area changes by a factor of $4$, not $2$.