The area circle formula is A = πr². Square the radius, then multiply by π. The radius r is the distance from the centre to the edge.
Cut a circle into slices and it becomes a rectangle.
A = πr × r = πr²
Hello, curious mind!
What is the area of a circle?
The area of a circle is the flat space inside its edge, A = πr². We measure it in square units, such as cm² or m².
Why does it work? Cut a circle into thin slices, like a cake. Lay them in a row, one up and one down. The row is almost a rectangle.
- Its long side is half the edge of the circle. The full edge (the circumference) is 2πr, so half is πr.
- Its short side is the length of one slice: the radius, r.
- Rectangle area = long side × short side = πr × r = πr².
Thinner slices make a straighter row. So π is in both formulas: the area comes from the circumference.
r² means r × r (see exponents). π ≈ 3.14159. Keep π until the last step.
Remember this
2πr goes round the circle. πr² fills it.
How to use the area circle formula: 3 worked examples
Example 1: area from the radius
A wall clock has a radius of 15 cm. What is the area of its face?
Square the radius: 15² = 225.
A = π × 225 = 225π.
A ≈ 706.9 cm².
Given the diameter? Halve it first. A round mirror 24 cm across has r = 12 cm, so A = 144π ≈ 452.4 cm².
Example 2: radius from the area
A round rug covers 2.5 m² of floor. What is its radius?
Turn the formula round: r² = A ÷ π.
r² = 2.5 ÷ π ≈ 0.7958.
Take the square root: r ≈ 0.892 m, about 89 cm. The rug is about 1.78 m across.
Example 3: area from the circumference
A tape round a tree trunk reads 157 cm. What is the area of its cross-section?
Find r from C = 2πr: r = 157 ÷ 2π ≈ 24.99 cm.
A = πr². Keep the full value of r.
A ≈ 1,961.5 cm², about 0.196 m².
Short cut: A = C² ÷ 4π = 24,649 ÷ 12.566 ≈ 1,961.5 cm².
Sectors and rings: one rule each
A sector is a slice of the circle. Its area is the same fraction of πr² as its angle is of 360°.
Sector area = (angle ÷ 360) × πr²
- A pizza slice. A pizza with radius 18 cm is cut into 8 equal slices. Each angle is 360 ÷ 8 = 45°. One slice = 324π ÷ 8 = 40.5π ≈ 127.2 cm².
- A garden sprinkler. It turns through 270° and sprays 10 m. It waters ¾ of 100π = 75π ≈ 235.6 m².
A ring is a big circle with a small circle taken out. Find both areas, then subtract.
A Dubai roundabout (made-up numbers)
A road ring has an outer radius of 40 m. The garden in the middle has a radius of 25 m. How much road is there?
Big circle: π × 40² = 1,600π. Small circle: π × 25² = 625π.
Road = 1,600π − 625π = 975π.
Road ≈ 3,063.1 m².
Do not subtract the radii first: π × (40 − 25)² = 225π is far too small.
Bonus: a square of side 10 cm fits exactly inside a circle. Its diagonal is the diameter, √(10² + 10²) ≈ 14.14 cm, by Pythagoras’ theorem. So r² = 50 and the area is 50π ≈ 157.1 cm².
Common area of a circle mistakes
- Squaring the diameter. For the 24 cm mirror, π × 24² = 576π. That is 4 times the true area.
- Using 2πr for the area. For the clock, 2π × 15 ≈ 94.2 cm. That is the length of the edge, not the space inside.
- Working out (πr)². (15π)² ≈ 2,220.7. Only r is squared, not π. This answer is about 3.14 times too big.
- Rounding π too early. With π = 3, the road is 975 × 3 = 2,925 m², about 138 m² short of 3,063.1 m².
- Writing cm, not cm². Area is always in square units.
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- Squares the width across (the diameter), not the radius: DRC, did not read carefully.
- Uses 2πr (the circumference) for the area: CNC, concept not clear.
- Works out (πr)² instead of πr²: CNC, concept not clear.
- Rounds π to 3 before the last step: CMT, calculation mistake.
- Writes cm, not cm²: BPT, bad presentation.
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Test prep corner
Circle area on the SAT
SAT Geometry and Trigonometry covers “Area and volume” and “Circles”, says College Board. Its digital SAT practice tests open the maths part with a reference sheet. It gives A = πr², C = 2πr and 360 degrees of arc. So the SAT tests how you use them. These are SAT-style questions, not real past questions.
SAT-style: A circle has a circumference of 26π. What is its area? (A) 13π (B) 26π (C) 169π (D) 676π
2πr = 26π, so r = 13. Area = π × 13² = 169π, answer (C). Trap: (D) squares 26, the diameter.
SAT-style: A sector of a circle with radius 12 has a central angle of 150°. Its area is kπ. What is the value of k?
The whole circle is 144π. The sector is 150/360 of it: (150 ÷ 360) × 144 = 60. Trap: using 2πr gives an arc length, not an area.
Want a longer set? Try our SAT-style maths practice questions, with full working.
Facts checked on 12 November 2026 on satsuite.collegeboard.org.
Quick check: 5 area of a circle questions
1. A circle has a radius of 20 cm. What is its area?
Not quite. 2πr = 40π is the circumference.
Yes! 20² = 400, so A = 400π.
Not quite. 1,600 is 40², the diameter squared.
2. A plate is 16 cm across. What is its area?
Yes! r = 8, and 8² = 64.
Not quite. 16 cm is the diameter. Halve it first.
Not quite. 8 × 2 = 16, but 8² is 8 × 8.
3. A circle has an area of 196π cm². What is its radius?
Not quite. Take the square root, do not halve.
Not quite. 196 is r². Take its square root.
Yes! r² = 196, so r = 14.
4. What is the area of a quarter circle with radius 8 cm?
Not quite. That is half the circle.
Yes! 64π ÷ 4 = 16π.
Not quite. That is the whole circle.
5. A ring has outer radius 10 cm and inner radius 6 cm. What is its area?
Not quite. Find both areas before you subtract.
Not quite. Take the hole away, do not add it.
Yes! 100π − 36π = 64π.
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Questions people ask about the area of a circle
What is the area circle formula?
A = πr², where r is the radius. The answer is in square units.
How do I find the area from the diameter?
Halve it to get the radius, then use A = πr². For 24 cm across, A = 144π ≈ 452.4 cm².
How do I find the radius from the area?
Divide the area by π, then take the square root. For 2.5 m², r = √(2.5 ÷ π) ≈ 0.892 m.
What is the area of a sector?
(angle ÷ 360) × πr². A 45° slice with radius 18 cm is 40.5π ≈ 127.2 cm².
Why is π in both the area and the circumference?
The slices make a row πr long (half the circumference) and r tall. So the area is πr × r.
Is the formula on the SAT sheet?
Yes. A = πr² and C = 2πr are both on the sheet in College Board’s digital SAT practice tests. The sector rule is not.
Keep exploring
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