Mathematics • Year 8 • Unit 3 • Lesson 8
Area of Circles in the Real World
Use A = πr² where it actually shows up: pizzas, sprinklers, circular pools, garden ponds and discus throws. Then explain your thinking in your own words.
1. Word problems
Each problem hides a circle. Identify whether you're given r or d, halve if needed, square the radius, then multiply by π.
1.1 — Pizza face-off. A circular pizza has diameter 30 cm. A square pizza measures 20 cm × 20 cm. Both cost $15.
(a) Find the area of the circular pizza to 2 d.p.
(b) Find the area of the square pizza.
(c) Which pizza gives you more food per dollar? 3 marks
1.2 — Lawn sprinkler. A garden sprinkler waters in a complete circle of radius 4 m.
(a) Find the area covered by one sprinkler (2 d.p.).
(b) A square back garden is 12 m × 12 m. How many sprinklers (rounded up) are needed to water the whole area? 3 marks
1.3 — Circular swimming pool. A round above-ground pool has diameter 4.6 m. The owner wants to buy a circular cover for the top of the pool.
(a) Find the area of the pool surface to 2 d.p.
(b) The cover costs $42 per m². Find the total cost (round up to the nearest dollar). 3 marks
1.4 — Garden pond. A semicircular garden pond has straight edge (the diameter) measuring 3 m.
(a) Find the area of the pond surface to 2 d.p.
(b) The pond is 0.4 m deep. Find its volume of water in m³ (volume = area × depth). 3 marks
1.5 — Discus circle. The throwing circle in athletics has a regulated diameter of exactly 2.5 m.
(a) Find the area of the throwing circle in m² (2 d.p.).
(b) Find the circumference of the throwing circle in m (2 d.p.). (You may need to revisit lesson 7 for circumference.) 2 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate is asked to find the area of a circle with diameter d = 10 cm. They write "A = π × 10² ≈ 314 cm²". In your own words, explain (i) what mistake they made with the diameter, (ii) what the correct area should be (show working), and (iii) by what factor their answer is wrong (e.g. 2× too big, 4× too big). Use the phrase "halve the diameter to get the radius" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Pizza face-off
(a) r = 30 ÷ 2 = 15 cm. A = π × 15² = 225π ≈ 706.86 cm².
(b) Square: 20 × 20 = 400 cm².
(c) Circular pizza gives much more area for the same $15 (706.86 vs 400 cm² — about 77% more), so the circular pizza is the better deal per dollar.
1.2 — Lawn sprinkler
(a) A = π × 4² = 16π ≈ 50.27 m².
(b) Garden = 12 × 12 = 144 m². Sprinklers = 144 ÷ 50.27 ≈ 2.86 → round up to 3 sprinklers (assuming non-overlapping ideal placement).
1.3 — Pool cover
(a) r = 4.6 ÷ 2 = 2.3 m. A = π × 2.3² = π × 5.29 ≈ 16.62 m².
(b) Cost = 16.62 × $42 = $697.83 → round up to $698.
1.4 — Semicircular pond
(a) r = 3 ÷ 2 = 1.5 m. A = ½ × π × 1.5² = ½ × 2.25π = 1.125π ≈ 3.53 m².
(b) V = A × depth = 3.53 × 0.4 ≈ 1.41 m³.
1.5 — Discus circle
(a) r = 1.25 m. A = π × 1.25² = 1.5625π ≈ 4.91 m².
(b) C = π × 2.5 = 2.5π ≈ 7.85 m.
2.1 — Explain your thinking (sample response)
The classmate has plugged the diameter (10 cm) directly into A = πr², when they should first halve the diameter to get the radius: r = 10 ÷ 2 = 5 cm. The correct working is A = π × 5² = 25π ≈ 78.54 cm². Their answer of 314 cm² is exactly 4 times too big, because using d instead of r squared the diameter, which is the same as using (2r)² = 4r² — so the answer balloons by a factor of 4 every time you make this mistake.
Marking: 1 mark for spotting the "used diameter, didn't halve" mistake; 1 mark for correct answer 78.54 cm² with working; 1 mark for "4× too big" (or equivalent quantification); 1 mark for clear full-sentence explanation using "halve the diameter to get the radius".