Geometry & Measures
Grade 4-6Circumference & Area of Circles
Circle questions test whether you can pick the correct formula (circumference or area), whether you can work backwards from a given circumference or area, and whether you can adapt these formulae for a fraction of a circle, such as a semicircle or sector. This lesson covers the circumference and area formulae, working backwards to find a radius or diameter, and finding the area and arc length of sectors.
Written by Asad, Co-Founder of Teachably
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The written lesson below covers everything you need in the meantime.
Circumference of a circle
The circumference of a circle is π × diameter, or equivalently 2 × π × radius. Use whichever form matches the measurement given in the question.
A circle with radius 7 cm has a circumference of 2 × π × 7 = 14π cm. A circle with diameter 9 cm has a circumference of π × 9 = 9π cm.
Area of a circle
The area of a circle is π × radius². If the diameter is given instead of the radius, halve it first before squaring.
A circle with radius 6 cm has an area of π × 6² = 36π cm². A circle with diameter 10 cm has radius 5 cm, so its area is π × 5² = 25π cm².
Working backwards from circumference or area
If the circumference is known, divide by π (or by 2π) to find the diameter (or radius). If the area is known, divide by π and take the square root to find the radius.
A circle has a circumference of 20π cm, so 2 × π × r = 20π, giving r = 10 cm. A circle has an area of 49π cm², so r² = 49, giving r = 7 cm.
Semicircles and sectors
A semicircle is half a circle, so its area is ½ × π × radius², and its curved length (the arc) is half the circumference. Its perimeter also includes the straight diameter edge. A sector is a fraction of a circle defined by its angle at the centre, so its area is (angle ÷ 360) × π × radius², and its arc length is (angle ÷ 360) × 2 × π × radius.
A semicircle with diameter 14 cm has an arc length of half of π × 14 = 21.98 cm (using π = 3.14), so its perimeter is 21.98 + 14 = 35.98 cm. A sector with radius 9 cm and angle 120° has an area of (120 ÷ 360) × π × 9² = 27π cm².
Worked Examples
Three exam-style questions, fully solved.
The circle shown has radius 7 cm. Find its circumference, leaving your answer in terms of π.
Easy- 1.Use the circumference formula with the radius: 2 × π × 7
Answer: 14π cm
A circle has a circumference of 20π cm. Find the radius of the circle.
Medium- 1.Set up an equation using the circumference formula: 2 × π × r = 20π
- 2.Divide both sides by 2π: r = 20π ÷ 2π
Answer: r = 10 cm
The diagram shows a sector of a circle with radius 9 cm and angle 120°. Find the area of the sector and the length of the arc, both leaving your answer in terms of π.
Hard- 1.Find the fraction of the full circle: 120 ÷ 360 = ⅓
- 2.Find the sector area: ⅓ × π × 9² = 27π cm²
- 3.Find the arc length: ⅓ × 2 × π × 9
Answer: Sector area = 27π cm², arc length = 6π cm
Avoid These
The most common mistakes students make.
Using the diameter instead of the radius in the area formula without halving it first, since area = π × radius², not π × diameter².
Confusing the circumference and area formulae, for example squaring the radius when finding circumference.
When working backwards from a given circumference or area, forgetting to take the square root when solving for the radius from the area formula.
Forgetting to add the straight edge (the diameter) when finding the perimeter of a semicircle, and only calculating the curved arc length.
Using the full circle formula instead of multiplying by the correct fraction (angle ÷ 360) when finding the area or arc length of a sector.
FAQ
Questions parents and students ask.
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