Circles — Area & Circumference

Area and circumference of a full circle, semicircle, and quarter circle. Perimeter and area of composite figures combining circles with squares, rectangles, and triangles.

πr² & 2πrcomposite figuressemicircle & ¼ circle P2 · calc

Area = πr² Circumference = 2πr π ≈ 3.14 (unless stated) Diameter = 2r

Part A — Basic circle

Question 1 — Short answer (2 marks) Paper 2 · calculator Area & circumference

A circular pond has a diameter of 14 m. Find (a) its circumference and (b) its area. Leave your answers in terms of π.

Radiusr = 14 ÷ 2 = 7 m

(a) C= 2πr = 2 × π × 7 = 14π m

(b) A= πr² = π × 7² = 49π m²

14π m | 49π m²

When told to leave in terms of π, do NOT substitute 3.14. Your answer should literally contain the π symbol.

Part B — Semicircle & quarter circle

Question 2 — Short answer (2 marks) Paper 1 · no calculator Semicircle perimeter

The figure below shows a semicircle with diameter 20 cm. Find its perimeter. (Take π = 3.14)

Curved= ½ × 2πr = πr = 3.14 × 10 = 31.4 cm

Straight= diameter = 20 cm

Total= 31.4 + 20 = 51.4 cm

Perimeter = 51.4 cm

Perimeter of a semicircle = curved arc + straight diameter. Students often forget to add the diameter — the flat edge IS part of the perimeter.

Question 3 — Short answer (2 marks) Paper 1 · no calculator Quarter circle area

Find the area of a quarter circle with radius 8 cm. (Take π = 3.14)

FormulaArea = ¼ × πr²

Calculate= ¼ × 3.14 × 8 × 8 = ¼ × 200.96 = 50.24 cm²

Area = 50.24 cm²

Quarter circle = ¼ of a full circle. The right-angle symbol at the corner confirms it's exactly ¼.

Part C — Composite figures (exam favourite)

Question 4 — Structured (3–4 marks) Paper 2 · calculator Composite: square + semicircles

The figure shows a square of side 10 cm with a semicircle attached to each of its four sides. Find the total area of the figure. (Take π = 3.14)

Square= 10 × 10 = 100 cm²

r for each= 10 ÷ 2 = 5 cm

4 semicircles= 2 full circles = 2 × π × 5² = 2 × 3.14 × 25 = 157 cm²

Total= 100 + 157 = 257 cm²

Total area = 257 cm²

Shortcut: 4 semicircles = 2 full circles = 2πr². Spotting this saves computation time in the exam.

Question 5 — Long answer (4–5 marks) Paper 2 · calculator Composite: rectangle − quarter circles

The figure shows a rectangle ABCD measuring 20 cm by 14 cm. Four quarter circles, each of radius 7 cm, are drawn at each corner of the rectangle. Find the shaded area. (Take π = 3.14)

Rectangle= 20 × 14 = 280 cm²

4 qtr circles= 1 full circle = π × 7² = 3.14 × 49 = 153.86 cm²

Shaded= 280 − 153.86 = 126.14 cm²

Shaded area = 126.14 cm²

Key insight: 4 quarter circles of the same radius = 1 full circle. Always look for this pattern — it dramatically simplifies the working.