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Area of a Circle
If you doubled a pizza's radius, you wouldn't get twice the pizza — you'd get four times as much. But how?
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What is the Area of a Circle?
The area of a circle is the amount of space enclosed within the boundary of a circle. The region within the boundary of the circle is the area occupied by the circle — it may also be referred to as the total number of square units inside that circle.
Area of Circle = πr² or πd²/4 in square units, where:
- π (Pi) = 22/7 or 3.14 — the ratio of circumference to diameter of any circle, a special mathematical constant
- r = radius of the circle (distance from centre to edge)
- d = diameter of the circle (d = 2r)
Use the sliders to increase slices, unfold the sectors, then rearrange them into a rectangle. Watch how the circle's area becomes πr × r = πr².
💡 Try this: drag Slices to 32, then Unfold to 100%, then Rearrange to 100%. The circle becomes a rectangle — width πr, height r. Area = πr².
A circle's area is 100π cm². What is its radius? What is its circumference? Start by solving πr² = 100π for r, then use C = 2πr.
Area of a Circle — Three Formulas
The same area can be computed from the radius, the diameter, or the circumference — depending on what information is given.
The most common form. Square the radius, then multiply by π. Use π ≈ 3.14 for approximate answers or keep it as π for exact answers.
Use this when the diameter is given directly. A pizza described as "14 inches" means 14-inch diameter — r = 7.
Useful when you've measured the perimeter of a round object — for example, a wire bent into a circle.
Differences Between Area and Circumference of a Circle
Area and circumference are both fundamental properties of a circle, but they measure different things and behave differently when the radius changes.
| Aspect | Circumference (C) | Area (A) |
|---|---|---|
| Definition | The length of the circle's boundary | The amount of space enclosed within the circle |
| Formula | C = 2πr | A = πr² |
| Units | Linear units (cm, m, in) | Square units (cm², m², in²) |
| When radius doubles | Doubles (proportional to r) | Quadruples (proportional to r²) |
| Example (r = 7 cm) | C = 2 × (22/7) × 7 = 44 cm | A = (22/7) × 49 = 154 cm² |
Circumference tells you how far you'd walk around the edge of a circle. Area tells you how much paint you'd need to fill it. A pizza's circumference is the crust length; its area is the amount of topping.
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Practice
An 18-inch pizza has more than twice the area of a 12-inch — because area scales with r², not r.
Painting a 12-inch round clock? You need its exact area to calculate the paint required.
A regulation bullseye is 12.7 mm across. Its painted area is π × 6.35² ≈ 126.7 mm².
A circular sprinkler with radius 15 m covers π × 225 ≈ 707 m² of farmland per rotation.
Downloadable Worksheets
Key Takeaways
| Term | Definition |
|---|---|
| Area of a Circle | The region enclosed within the circle boundary. Measured in square units (cm², m², in²). |
| Radius | A = πr² — square the radius, multiply by π. |
| Diameter | A = πd²/4 — since r = d/2. |
| Circumference | A = C²/(4π) — useful when the perimeter is known. |
| π (pi) | ≈ 3.14159. The ratio of circumference to diameter. The same for every circle. |
| Radius doubles → Area ×4 | Area ∝ r². Doubling r makes r² grow by 4× — area quadruples, not doubles. |
| Derivation | Slice into wedges, rearrange into a rectangle of width πr and height r. Area = πr × r = πr². |
