Area of a Circle: Formula, Derivation & Examples

What Is the Area of a Circle?

The area of a circle is the amount of flat, two-dimensional space enclosed inside the circle's boundary. It is measured in square units, such as square centimetres ($\text{cm}^2$) or square metres ($\text{m}^2$), because area always counts how many unit squares fit inside a region.

A circle is the set of all points the same distance from a fixed centre. That fixed distance is the radius ($r$). The full distance across the circle through the centre is the diameter ($d$), and it is always twice the radius, so $d = 2r$. Every area question about a circle comes back to the radius, so finding $r$ is almost always the first move.

The Area of a Circle Formula

The area of a circle depends on one measurement, the radius, and one constant, $\pi$:

$$A = \pi r^2.$$

Here $r$ is the radius and $\pi$ (pi) is the constant ratio of any circle's circumference to its diameter, roughly $3.14159$. The $r^2$ is the radius multiplied by itself, not the radius doubled, which is the single most common place this formula goes wrong. Squaring the radius is what gives area its two-dimensional, square-unit character.

Where Does πr² Come From? Unrolling the Circle

A formula you can derive is a formula you never forget. Here is the classic argument, the one Archimedes reached for over two thousand years ago.

Slice the circle into many thin wedges, like a pizza, all meeting at the centre. Each wedge is almost a thin triangle. Now lay the wedges side by side, alternating point-up and point-down, so they interlock into a shape close to a parallelogram.

• The two long, wavy edges are made of the circle's outer rim. Together they total the full circumference, $2\pi r$, so one long edge of the parallelogram has length $\pi r$ (half the circumference).

• The slanted short side of each wedge is the radius, so the height of the parallelogram is $r$.

The more wedges you cut, the straighter those edges become, and the shape gets closer and closer to a true rectangle of width $\pi r$ and height $r$. The area of that rectangle is width times height:

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

The circle and the rearranged shape hold the same space, so the circle's area is $\pi r^2$.

How Do You Find the Area From the Diameter?

A common version of this question is, "What if I'm only given the diameter, not the radius?" Since the radius is half the diameter, $r = \tfrac{d}{2}$, substitute that straight into $A = \pi r^2$:

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

So a circle of diameter $d$ has area $\dfrac{\pi d^2}{4}$. You can either halve the diameter first and use $\pi r^2$, or use this diameter form directly. Both give the same answer; halving first is usually safer because it keeps you in the familiar $\pi r^2$ habit.

How Do You Find the Area From the Circumference?

If you are handed the circumference $C$ instead, recover the radius first. Since $C = 2\pi r$, rearranging gives $r = \dfrac{C}{2\pi}$. Substituting into the area formula:

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

This is the bridge a student needs when a problem gives the distance around a circular track and asks for the ground it encloses. Notice how each version, radius, diameter, or circumference, is the same single formula wearing a different measurement.

Area of a Sector: A Slice of the Whole

A sector is a pie-slice region cut from a circle by two radii. Its area is just a fraction of the full circle, set by the central angle $\theta$:

$$A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2 \quad (\theta \text{ in degrees}).$$

A quadrant, the slice from a $90^\circ$ angle, is therefore exactly one quarter of the circle: $A = \tfrac{1}{4}\pi r^2$. The sector formula sits one step beyond the main idea, so we keep it brief here; for the full treatment see the sibling article on the sector of a circle.

Examples of the Area of a Circle

With the formula derived and the diameter and circumference versions in hand, here is the area doing real work. The problems build from a clean radius up to working backward from a known area.

Example 1 - Find the area of a circle with radius $7$ cm. Use $\pi \approx \tfrac{22}{7}$.

$A = \pi r^2 = \tfrac{22}{7} \times 7^2 = \tfrac{22}{7} \times 49 = 22 \times 7 = 154$.

Final answer: $154 \ \text{cm}^2$.

Example 2 - A circle has diameter $10$ cm. Find its area, using $\pi \approx 3.14$.

A tempting first move is to drop the diameter straight into the formula: $A = \pi \times 10^2 = 314 \ \text{cm}^2$. Check that against the picture. The formula needs the radius, the distance from centre to edge, but $10$ cm is the full width across. Using the diameter squares a length twice as long, inflating the area by a factor of four.

Done correctly: the radius is half the diameter, $r = \tfrac{10}{2} = 5$ cm. Then $A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \ \text{cm}^2$.

Final answer: $78.5 \ \text{cm}^2$.

Example 3 - A circular garden has a radius of $14$ m. How much turf is needed to cover it? Use $\pi \approx \tfrac{22}{7}$.

$A = \tfrac{22}{7} \times 14^2 = \tfrac{22}{7} \times 196 = 22 \times 28 = 616$.

Final answer: $616 \ \text{m}^2$ of turf.

Example 4 - A circle has circumference $C = 31.4$ cm. Find its area, using $\pi \approx 3.14$.

First recover the radius: $r = \dfrac{C}{2\pi} = \dfrac{31.4}{2 \times 3.14} = \dfrac{31.4}{6.28} = 5$ cm.

Then $A = 3.14 \times 5^2 = 78.5 \ \text{cm}^2$.

Final answer: $78.5 \ \text{cm}^2$.

Example 5 - A circle has area $A = 50.24 \ \text{cm}^2$. Find its radius, using $\pi \approx 3.14$.

Working backward, solve $\pi r^2 = A$ for $r$: $r^2 = \dfrac{A}{\pi} = \dfrac{50.24}{3.14} = 16$, so $r = \sqrt{16} = 4$ cm.

Final answer: $r = 4 \ \text{cm}$.

Example 6 - Find the area of a quadrant (quarter circle) of radius $6$ cm. Use $\pi \approx 3.14$.

A quadrant is one quarter of the whole circle, so $A = \tfrac{1}{4}\pi r^2 = \tfrac{1}{4} \times 3.14 \times 6^2 = \tfrac{1}{4} \times 3.14 \times 36 = \tfrac{1}{4} \times 113.04 = 28.26$.

Final answer: $28.26 \ \text{cm}^2$.

Why Area of a Circle Matters Beyond the Page

Knowing the space inside a round boundary is one of the most reused calculations in the physical world, because so much of what we build and grow is round.

• Sizing what spreads outward. A lawn sprinkler, a radar dish, or a cell-tower signal each cover a circular region. The reach $r$ decides the covered area $\pi r^2$, and because area grows with $r^2$, doubling the reach quadruples the coverage. That square relationship is why a slightly stronger sprinkler covers far more lawn than you would guess.

• Material and cost. Cutting a circular tabletop, a manhole cover, or a pizza base all start with $\pi r^2$ to know how much wood, steel, or dough a single piece takes.

• Cross-sections that carry flow. The amount of water a round pipe can move, or air a duct can push, depends on the area of its circular cross-section, not its diameter alone, which is why a small change in pipe radius changes capacity sharply.

• The pull of pi. Every one of these uses drags in $\pi$, the constant the ancient world chased for centuries. The framework that ties a curved boundary to a clean algebraic formula is the same coordinate-and-measurement thinking you will lean on in every later geometry topic.

Where Students Trip Up on Area of a Circle

Mistake 1: Using the diameter in place of the radius

Where it slips in: The problem gives the diameter, and the student plugs it straight into $\pi r^2$.

Don't do this: Write $A = \pi d^2$ with the full width as $r$.

The correct way: Halve the diameter first, $r = \tfrac{d}{2}$, then use $A = \pi r^2$. The memorizer who learned "pi r squared" as a chant often forgets that $r$ has a specific meaning, centre to edge, and that the diameter is twice that.

Mistake 2: Doubling the radius instead of squaring it

Where it slips in: Computing $r^2$, the student writes $2r$.

Don't do this: Treat $r^2$ as $r \times 2$.

The correct way: $r^2$ means $r \times r$. For $r = 5$, that is $25$, not $10$. The rusher who races to a number skips the squaring step entirely; reading the exponent aloud, "r times r", catches it.

Mistake 3: Dropping or mismatching the units

Where it slips in: The answer is written with no unit, or with a length unit such as cm instead of $\text{cm}^2$.

Don't do this: Report an area as "$78.5$ cm".

The correct way: Area is always in square units. Because the formula multiplies a length by a length, $\text{cm} \times \text{cm} = \text{cm}^2$. Keep one unit convention through the whole problem.

Conclusion

• The area of a circle is the space inside its boundary, given by $A = \pi r^2$ in square units.

• The formula comes from unrolling the circle into a rectangle of width $\pi r$ and height $r$, not from memorisation.

• From the diameter, area is $\tfrac{\pi d^2}{4}$; from the circumference, it is $\tfrac{C^2}{4\pi}$; each is the same formula in disguise.

• The most common mistake is using the diameter where the radius belongs, which inflates the area fourfold.

• A sector is a fraction $\tfrac{\theta}{360^\circ}$ of the whole circle's area.

Practice These Problems to Solidify Your Understanding

1. Find the area of a circle with radius $10$ cm (use $\pi \approx 3.14$).

2. A circle has diameter $28$ cm. Find its area (use $\pi \approx \tfrac{22}{7}$).

3. A circle has area $113.04 \ \text{cm}^2$. Find its radius (use $\pi \approx 3.14$).

Answer to Question 1: $314 \ \text{cm}^2$. Answer to Question 2: $616 \ \text{cm}^2$. Answer to Question 3: $r = 6$ cm. If Question 2 gave a much larger number, check whether you used the radius ($14$ cm) rather than the diameter inside the formula.

Want a live Bhanzu trainer to walk your child through circle area and the wider mensuration chapter? Book a free demo class — online globally.

Also Read:

• Geometry — Concepts, Formulas, Shapes & Topics Guide

• Circumference of a Circle

• Radius of a Circle

• Sector of a Circle

• Diameter of a Circle

• Parts of a Circle

• Chord of a Circle