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.
What you need to know
- Circumference is pi times diameter, which is the same as 2 times pi times radius.
- Area is pi times radius squared; always square the radius, not the diameter.
- The diameter is twice the radius.
- For an arc or sector, take the fraction angle over 360 of the whole circle.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
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
Practice
6 questions, marked instantly
Type an answer and check it. The worked solution appears once you have had a go. No login needed, and your progress saves in this browser.
Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
A circle has a radius of 7 cm. Work out its circumference, leaving your answer in terms of π.
A circle has a radius of 5 cm. Work out its area, leaving your answer in terms of π.
A circle has a diameter of 12 cm. Work out its area, leaving your answer in terms of π.
A circle has a circumference of 10π cm. Work out its radius.
A circle has an area of 49π cm². Work out its radius.
A circle has a radius of 4 cm. Work out its area to 1 decimal place.
Avoid these
The mistakes students make most often
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 students and parents ask
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