One 30 cm round cake has more cake in it than two 20 cm cakes put together. That surprises most students. The reason is the area of a circle: area grows with the square of the radius, not the radius itself.

You probably met A = πr² in middle school. It comes back in IGCSE Extended Maths and IB Maths AA, where it hides inside sector, arc length, composite shape and radian questions. This guide covers all of it, from the basic formula to exam-style problems.

By the end you will be able to:

• find the area from the radius, the diameter or the circumference
• explain why the formula is πr²
• calculate sector area and arc length in degrees and in radians
• avoid the mistakes that cost marks in exams

Quick checklist: area of a circle formulas

• Area from the radius: A = πr²
• Area from the diameter: A = πd² ÷ 4
• Area from the circumference: A = C² ÷ 4π
• Radius from the area: r = √(A ÷ π)
• Area is always in square units (cm², m²)
• Double the radius and the area is four times bigger
• Sector area (degrees): (θ ÷ 360) × πr²
• Arc length (degrees): (θ ÷ 360) × 2πr
• Sector area (radians): ½r²θ
• Arc length (radians): rθ
• In IGCSE, use the π button or 3.142, and round to 3 significant figures unless told otherwise

Need every geometry formula in one place? See our Geometry Formulas guide for IB and IGCSE.

What is the area of a circle?

The area of a circle is the amount of flat space inside its boundary. Think of it as the number of unit squares needed to cover the circle exactly.

The area of a circle is sometimes called the area of a disc, the round area or the area of a circular plate. Its formula, A = πr², is read aloud as "pi r squared".

Three measurements describe a circle:

• Radius (r): the distance from the centre to the edge
• Diameter (d): the distance across the circle through the centre, so d = 2r
• Circumference (C): the distance around the edge, so C = 2πr

Pi (π) is the ratio of any circle's circumference to its diameter. It is about 3.14159. Cambridge IGCSE papers ask you to use your calculator's π key or 3.142. Some school textbooks use 22/7, which works neatly when the radius is a multiple of 7.

Area of a circle formula

You can find the area from the radius, the diameter or the circumference. Pick the version that matches what the question gives you.

1. Area using the radius

A = \pi r^2

Square the radius first, then multiply by π.

Example: a circle has radius 5 cm.

• A = π × 5² = 25π
• A ≈ 78.5 cm²

2. Area of a circle using the diameter

Since r = d ÷ 2, substitute it into A = πr²:

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

Example: a circle has diameter 12 cm.

• A = π × 12² ÷ 4 = 144π ÷ 4 = 36π
• A ≈ 113 cm² (3 s.f.)

A faster route is to halve the diameter first (r = 6 cm) and use πr². You get the same answer.

3. Area of a circle using the circumference

Rearrange C = 2πr to get r = C ÷ 2π, then substitute:

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

Example: a circle has circumference 31.4 cm.

• A = 31.4² ÷ (4π) = 985.96 ÷ 12.566
• A ≈ 78.5 cm² (this is the r = 5 cm circle again)

4. Finding the radius from the area

Work backwards: divide by π, then take the square root.

r = \sqrt{\frac{A}{\pi}}

This reverse step appears in a lot of IGCSE questions, often as the first part of a longer problem.

5. Radius of a circle formula

You can find the radius from any of the other measurements:

• From the diameter: r = d ÷ 2
• From the circumference: r = C ÷ 2π
• From the area: r = √(A ÷ π)

Why is the area of a circle πr²?

The formula isn't something to memorise blindly. Two short arguments show where it comes from.

The slice-and-rearrange proof

1. Cut a circle into many equal sectors, like slices of a pizza.
2. Lay the slices in a row, pointing up and down alternately.
3. The shape now looks like a wavy parallelogram. The more slices you cut, the closer it gets to a rectangle.
4. The rectangle's height is the radius, r.
5. Its length is half the circumference, because half the curved edges sit on top and half on the bottom. Half of 2πr is πr.
6. The area of the rectangle is length × height = πr × r = πr².

Image suggestion: a circle cut into 16 coloured sectors, then the same sectors rearranged into a near-rectangle with "πr" along the bottom and "r" up the side.

Area of a circle using integration: the rings proof (IB Maths AA)

Picture the circle as many thin rings, like the rings of a tree. A ring at distance x from the centre has length 2πx and a tiny thickness dx. Adding up all the rings from the centre to the edge is an integral:

A = \int_0^r 2\pi x \, dx = \left[\pi x^2\right]_0^r = \pi r^2

This is a good example of integration as "adding up thin strips". Our Integration Formulas guide covers the rules used here.

Area and perimeter (circumference) of a circle

Students often mix these two up. They measure different things.

Circumference

• It is the length of the boundary.
• Formula: C = 2πr or C = πd
• It is measured in linear units: cm, m
• It grows in proportion to r. Double the radius and the circumference doubles.

Area

• It is the space enclosed by the boundary.
• Formula: A = πr²
• It is measured in square units: cm², m²
• It grows in proportion to r². Double the radius and the area becomes four times bigger.

A quick way to remember: circumference is the length of fence around a round garden, and area is the amount of grass seed you need to cover it.

Why the cake example works: a 30 cm cake has radius 15 cm, so its area is 225π ≈ 707 cm². Two 20 cm cakes have a total area of 2 × 100π = 200π ≈ 628 cm². The single bigger cake wins.

Real-life uses of the area of a circle

• Sports pitches: a football centre circle has a radius of 9.15 m, so it covers π × 9.15² ≈ 263 m² of grass.
• Phone signal coverage: a mast that reaches 2 km in every direction covers about π × 2² ≈ 12.6 km².
• Pie charts: each slice is a sector, so its angle shows what fraction of the whole circle it represents.
• Car wipers: a wiper blade sweeps part of a ring, so the cleared area is a sector of the outer circle minus a sector of the inner one.
• Buying materials: the fabric for a round tablecloth or the cover for a round pool is worked out from πr², plus extra for the overhang.

Area of a sector and arc length

A sector is a "slice" of a circle between two radii. An arc is the curved part of its edge. Both are a fraction of the full circle, and that fraction is set by the angle at the centre.

In degrees (IGCSE and IB Maths AI)

The fraction of the circle is θ ÷ 360.

\text{Sector area} = \frac{\theta}{360} \times \pi r^2

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\text{Arc length} = \frac{\theta}{360} \times 2\pi r

• The perimeter of a sector = arc length + 2r. Don't forget the two straight edges.
• A minor sector has θ less than 180°. The major sector is the rest of the circle.

In radians (IB Maths AA)

IB Maths AA uses radians, where a full turn is 2π. The formulas become shorter:

l = r\theta \qquad A = \frac{1}{2} r^2 \theta

• To convert degrees to radians, multiply by π ÷ 180.
• To convert radians to degrees, multiply by 180 ÷ π.
• Set your calculator to radian mode before using sin, cos or tan with these formulas.

Image suggestion: a sector with radius r, angle θ at the centre and the arc length l labelled along the curve.

Area of a segment (IB)

A segment is the region between a chord and its arc. Take the sector and remove the triangle:

\text{Segment area} = \frac{1}{2} r^2 (\theta - \sin\theta) \quad (\theta \text{ in radians})

The formula above gives the minor segment. To find the area of the major segment, subtract the minor segment from the whole circle: major segment = πr² − minor segment.

Semicircle, quadrant and ring

• Semicircle: area = ½πr², perimeter = πr + 2r
• Quadrant (quarter circle): area = ¼πr², perimeter = ½πr + 2r
• Ring (annulus, also called a hollow circle or the area between concentric circles): area = πR² − πr² = π(R² − r²), where R is the outer radius and r is the inner radius

These show up as composite shapes in IGCSE papers: a window with a semicircular top, a running track, or a path around a pond.

Solved examples

The examples move from basic to exam level. Answers are given to 3 significant figures unless stated.

Example 1: area from the radius

Find the area of a circle with radius 7 cm. Give the exact answer and a decimal answer.

• A = π × 7² = 49π cm² (exact)
• A ≈ 154 cm²

Example 2: area from the diameter

A circular garden has a diameter of 18 m. Find its area.

• r = 18 ÷ 2 = 9 m
• A = π × 9² = 81π ≈ 254 m²

Example 3: area from the circumference

A circle has a circumference of 50 cm. Find its area.

• A = C² ÷ 4π = 2500 ÷ 4π
• A ≈ 199 cm²
• Check: r = 50 ÷ 2π ≈ 7.96 cm, and π × 7.96² ≈ 199 cm² ✓

Example 4: radius from the area

A circle has an area of 200 cm². Find its radius.

• r = √(200 ÷ π) = √63.66...
• r ≈ 7.98 cm

Example 5: sector in degrees (IGCSE)

A sector has radius 10 cm and angle 72°. Find its area, arc length and perimeter.

• Fraction of the circle = 72 ÷ 360 = 1/5
• Sector area = 1/5 × π × 10² = 20π ≈ 62.8 cm²
• Arc length = 1/5 × 2π × 10 = 4π ≈ 12.6 cm
• Perimeter = 12.57 + 10 + 10 ≈ 32.6 cm

Example 6: finding the angle (IGCSE)

A sector of radius 8 cm has an area of 30 cm². Find the angle at the centre.

• (θ ÷ 360) × π × 8² = 30
• θ = 30 × 360 ÷ 64π
• θ ≈ 53.7°

Example 7: sector in radians (IB Maths AA)

A sector has radius 6 cm and angle 1.2 radians. Find its arc length and area.

• l = rθ = 6 × 1.2 = 7.2 cm
• A = ½ × 6² × 1.2 = 21.6 cm²

Example 8: area of a ring

A path 2 m wide runs around a circular pond of radius 5 m. Find the area of the path.

• Outer radius R = 5 + 2 = 7 m, inner radius r = 5 m
• Area = π(7² − 5²) = 24π ≈ 75.4 m²

Example 9: area of a segment (IB Maths AA)

A chord cuts off an angle of π/3 radians at the centre of a circle of radius 5 cm. Find the area of the minor segment.

• Segment area = ½ × 5² × (π/3 − sin(π/3))
• = 12.5 × (1.0472 − 0.8660)
• ≈ 2.26 cm²

Example 10: a textbook classic (CBSE Class 10)

Find the area of a circle of diameter 8.4 m.

• r = 8.4 ÷ 2 = 4.2 m
• A = π × 4.2² = 17.64π
• A ≈ 55.4 m² using the π key (55.44 m² if you use 22/7)

Common mistakes

1. Using the diameter as the radius. If the question gives the diameter, halve it before you use πr². This is the most common lost mark.
2. Squaring πr instead of r. πr² means π × (r × r), not (π × r)². Type it into your calculator as π × r².
3. Mixing up area and circumference. 2πr gives a length, not an area. Check that your answer is in square units.
4. Using a degree formula with radians, or the other way round. ½r²θ only works when θ is in radians.
5. Leaving the calculator in the wrong mode. For segment questions, sin θ must be worked out in radian mode.
6. Forgetting the two radii in a sector's perimeter. Perimeter = arc + r + r.
7. Rounding too early. Keep full calculator values until the last line, then round to 3 s.f.
8. Using 22/7 when the paper says to use 3.142 or the π key. The answers differ slightly and may fall outside the mark scheme's accepted range.

Practice questions

Try these before checking the answers.

1. Find the area of a circle with radius 3 cm.
2. Find the area of a circle with diameter 20 cm.
3. A circle has a circumference of 44 cm. Find its area.
4. A circle has an area of 50π cm². Find its radius and circumference.
5. A sector has radius 12 cm and angle 150°. Find its area and arc length.
6. (IB) A sector has radius 9 cm and arc length 12 cm. Find the angle in radians and the sector's area.
7. The radius of a circle increases from 4 cm to 6 cm. By what factor does the area increase, and by how many cm²?
8. A semicircle has diameter 14 cm. Find its area and perimeter.

Answers

1. 9π ≈ 28.3 cm²
2. r = 10 cm, so the area is 100π ≈ 314 cm²
3. 44² ÷ 4π ≈ 154 cm²
4. r = √50 ≈ 7.07 cm, and the circumference is 2π√50 ≈ 44.4 cm
5. Area = 60π ≈ 188 cm², and the arc length = 10π ≈ 31.4 cm
6. θ = 12 ÷ 9 = 4/3 ≈ 1.33 rad, and the area = ½ × 81 × 4/3 = 54 cm²
7. Factor = (6 ÷ 4)² = 2.25. The increase is 36π − 16π = 20π ≈ 62.8 cm²
8. r = 7 cm. Area = 24.5π ≈ 77.0 cm², and the perimeter = 7π + 14 ≈ 36.0 cm

Frequently asked questions

What is the formula for the area of a circle?

The area of a circle is A = πr², where r is the radius. If you know the diameter, use A = πd² ÷ 4.

How do you find the area of a circle with the diameter?

Halve the diameter to get the radius, then use A = πr². For a 10 cm diameter, r = 5 cm and A = 25π ≈ 78.5 cm².

How do you find the area of a circle from the circumference?

Use A = C² ÷ 4π. Alternatively, find the radius first with r = C ÷ 2π, then use πr².

How do you find the radius if you know the area?

Divide the area by π, then take the square root: r = √(A ÷ π).

What happens to the area if the radius doubles?

The area becomes four times bigger, because area depends on r². Tripling the radius makes the area nine times bigger.

Why is the formula πr² and not 2πr?

2πr is the circumference, which is a length. Area needs square units, so the radius must be multiplied by itself. The slice-and-rearrange proof shows the circle becomes a rectangle measuring πr by r.

Should I use 3.14, 22/7 or the π button?

Follow the question. Cambridge IGCSE papers ask for the calculator's π key or 3.142. If a question asks for an exact answer, leave it in terms of π, for example 49π cm².

What is the difference between sector area and arc length?

Sector area is the space inside the slice, measured in cm². Arc length is the length of the curved edge, measured in cm.

Is the area of a circle tested in IB Maths?

Yes. IB Maths AA tests sector area, arc length and segment area in radians, often inside longer trigonometry questions. IB Maths AI applies circle and sector areas in modelling contexts. See our IB Maths AI tutoring page for support.

What is the area of a semicircle?

It is half the area of the full circle: ½πr². For a radius of 4 cm, that is 8π ≈ 25.1 cm².

Does a circle have volume?

No. A circle is a flat, two-dimensional shape, so it has area but no volume. A 3D shape built on a circle, such as a cylinder or a sphere, does have volume.

What units is the area of a circle measured in?

Area is always measured in square units, such as cm², m² or km². If the radius is in cm, the area is in cm².

What is the surface area, TSA or CSA of a circle?

A circle is flat, so its "surface area" is just its area, πr². Total surface area (TSA) and curved surface area (CSA) belong to 3D shapes. For example, a sphere has surface area 4πr², a hemisphere has curved surface area 2πr², and a cylinder has CSA 2πrh.

What is the cross-sectional area of a circle?

It is the area of the circular face you see when you cut a pipe, wire or rod straight across: πr². Engineers use it to work out how much water flows through a pipe or how much current a wire can carry.

What is the area of a 12-inch circle?

If 12 inches is the diameter, as with a pizza, then r = 6 in and A = 36π ≈ 113 in². If 12 inches is the radius, A = 144π ≈ 452 in².

Is area of a circle in CBSE Class 10?

Yes. The chapter "Areas Related to Circles" builds on A = πr² with sectors, segments and combined shapes, just as IGCSE Extended does.

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