Circles: Area and Circumference
Section: Geometrical Reasoning, Shapes and Measurements | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
Circle Vocabulary
- Radius (r): the distance from the centre of a circle to its edge.
- Diameter (d): the distance across a circle through its centre - always twice the radius: d = 2r.
- Circumference (C): the distance around the outside of a circle - the circle's perimeter.
- π (pi): the ratio of a circle's circumference to its diameter, approximately 3.14 (or 3.14159...).
The radius, diameter, and circumference of a circle
Circumference of a Circle
The circumference of a circle can be found using either the diameter or the radius: C = πd, or equivalently C = 2πr (since d = 2r). Use π ≈ 3.14 unless told otherwise.
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Question: Find the circumference of a circle with diameter 12 cm.
- Step 1: C = πd = 3.14 × 12
- Answer: C = 37.68 cm
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Question: Find the circumference of a circle with radius 9 cm.
- Step 1: C = 2πr = 2 × 3.14 × 9
- Answer: C = 56.52 cm
Area of a Circle
The area of a circle is given by A = πr² - always use the radius, squaring it before multiplying by π.
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Question: Find the area of a circle with radius 8 cm.
- Step 1: A = πr² = 3.14 × 8² = 3.14 × 64
- Answer: A = 200.96 cm²
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Question: Find the area of a circle with diameter 20 cm.
- Step 1: Halve the diameter to find the radius: r = 10 cm
- Step 2: A = πr² = 3.14 × 10² = 3.14 × 100
- Answer: A = 314 cm²
Finding Radius from Circumference or Area
Rearranging the circle formulae - the same rearranging skills from Expressions and Formulae - lets you work backwards from a circumference or area to find the radius.
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Question: A circle has circumference 25.12 cm. Find its radius.
- Step 1: Rearrange C = 2πr to make r the subject: r = C/(2π)
- Step 2: r = 25.12/(2 × 3.14) = 25.12/6.28
- Answer: r = 4 cm
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Question: A circle has area 78.5 cm². Find its radius.
- Step 1: Rearrange A = πr² to make r the subject: r = √(A/π)
- Step 2: r = √(78.5/3.14) = √25
- Answer: r = 5 cm
Semicircles and Quarter Circles
For a semicircle, take half the full circle's area and half its circumference - but remember the perimeter also needs the straight edge (the diameter) added on.
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Question: Find the area and perimeter of a semicircle with radius 6 cm.
- Step 1 (area): Half of πr² = 0.5 × 3.14 × 36 = 56.52 cm²
- Step 2 (perimeter): Half the circumference = 0.5 × 2 × 3.14 × 6 = 18.84 cm
- Step 3: Add the diameter (straight edge): 18.84 + 12 = 30.84 cm
- Answer: Area = 56.52 cm², Perimeter = 30.84 cm
Circumference in Real-World Rotation Problems
Every full turn of a wheel or circular object covers a distance equal to its circumference - so distance travelled = circumference × number of rotations.
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Question: A bicycle wheel has a radius of 30 cm. Find the distance travelled after 200 full rotations of the wheel (use π ≈ 3.14).
- Step 1: Circumference = 2πr = 2 × 3.14 × 30 = 188.4 cm
- Step 2: Distance = circumference × number of rotations = 188.4 × 200
- Answer: 37,680 cm = 376.8 m
Real-World Applications
Circles and their measurements appear everywhere:
- Wheels and tyres: circumference determines distance travelled per rotation.
- Circular tracks and pools: perimeter for fencing or lane markings.
- Pipes and tubing: cross-sectional area for flow calculations.
- Clock faces and dials: circular design and measurement.
- Pizza sizing: comparing areas of different diameters.
Exam Tips
Common Mistakes
MistakeUsing radius in the diameter formula or vice versa, e.g. C = πr
Fixcheck which value you're given - use C = πd for diameter, or C = 2πr for radius
MistakeForgetting to square the radius in the area formula, e.g. A = πr
Fixalways square the radius first, then multiply by π: A = πr²
MistakeUsing the diameter directly in the area formula, e.g. A = πd²
Fixconvert diameter to radius (halve it) before using A = πr²
MistakeRounding too early in a multi-step calculation
Fixkeep several decimal places (or use a calculator's π button) until the final step
MistakeForgetting to add the diameter when finding the perimeter of a semicircle
Fixperimeter of a semicircle = half the circumference PLUS the diameter, not just the curved part
MistakeUsing the radius instead of the circumference when finding a distance travelled by a rotating wheel
Fixeach full rotation covers a distance equal to the CIRCUMFERENCE - find that first, then multiply by the number of rotations
For Exams
- Learn both circumference formulas: C = πd and C = 2πr.
- Learn the area formula: A = πr².
- Use π ≈ 3.14 unless told to use a more precise value or leave your answer in terms of π.
- For semicircles/quarter circles: remember to include the straight edge(s) in the perimeter.
- For rotation problems: distance = circumference × number of rotations.
- Double check whether you're given the radius or the diameter before substituting.
Interactive revision notes, videos and practice questions load below.