Area of a Circle: Formula, Derivation & Examples
Area of a Circle: Formula, Derivation & Examples
TL;DR
The area of a circle is the flat space enclosed inside its boundary, given by the formula A=πr², where r is the radius. This article defines the area, derives πr² by unrolling the circle into a triangle, covers area from the diameter and circumference, and works through six examples.
Last updated on June 9, 2026 9 min read
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 (cm²) or square metres (m²), 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, π:
A=πr².
Here r is the radius and π (pi) is the constant ratio of any circle's circumference to its diameter, roughly 3.14159. The r² 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.
| Symbol | Meaning | Units |
|---|---|---|
| A | Area enclosed by the circle | square units (cm², m²) |
| r | Radius, centre to edge | length units (cm, m) |
| π | Circumference ÷ diameter, a fixed constant ≈3.14159 | none (a pure ratio) |
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, 2πr, so one long edge of the parallelogram has length πr (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 πr and height r. The area of that rectangle is width times height:
A=(πr)×r=πr².
The circle and the rearranged shape hold the same space, so the circle's area is πr².
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=d/2, substitute that straight into A=πr²:
A=π(d/2)²=πd²/4.
So, a circle of diameter d has area πd²/4. You can either halve the diameter first and use πr², or use this diameter form directly. Both give the same answer; halving first is usually safer because it keeps you in the familiar πr² habit.
How Do You Find the Area From the Circumference?
If you are handed the circumference C instead, recover the radius first. Since C=2πr, rearranging gives r=C/2π. Substituting into the area formula:
A=π(C/2π)²=C²/4π.
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 θ:
Aₛₑ𝒸ₜₒ𝓇= θ/360°×πr² (θ in degrees).
A quadrant, the slice from a 90° angle, is therefore exactly one quarter of the circle: A=1/4πr².
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 π≈22/7.
A=πr²=22/7×7²=22/7×49=154.
Final answer: 154 cm².
Example 2 - A circle has diameter 10 cm. Find its area, using π≈3.14.
Correctly: the radius is half the diameter, r=10/2=5 cm. Then A=3.14×5²=3.14×25=78.5 cm².
Final answer: 78.5 cm².
Example 3 - A circular garden has a radius of 14 m. How much turf is needed to cover it? Use π≈22/7.
A=22/7×14²=22/7×196=616.
Final answer: 616 m² of turf.
Example 4 - A circle has circumference C=31.4 cm. Find its area, using π≈3.14.
First recover the radius: r=C/2π=31.4/(2×3.14)=5 cm.
Then A=3.14×5²=78.5 cm².
Final answer: 78.5 cm².
Example 5 - A circle has area A=50.24 cm². Find its radius, using π≈3.14.
r²=A/π=50.24/3.14=16, so r=4 cm.
Final answer: r=4 cm.
Example 6 - Find the area of a quadrant (quarter circle) of radius 6 cm. Use π≈3.14.
A=1/4πr²=1/4×3.14×6²=28.26.
Final answer: 28.26 cm².
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.
- Material and cost. Cutting a circular tabletop, a manhole cover, or a pizza base all start with πr² to know how much wood, steel, or dough.
- Cross-sections that carry flow. The amount of water a round pipe can move depends on the area of its circular cross-section.
Conclusion
- The area of a circle is the space inside its boundary, given by A=πr² in square units.
- The formula comes from unrolling the circle into a rectangle of width πr and height r.
- The most common mistake is using the diameter where the radius belongs, which inflates the area fourfold.