What fraction of the rectangle is covered by the circles?
- (a) What fraction of the rectangle is covered by the circles?
- (b) What fraction of the rectangle is covered by the circles?
Answer: (a) π4 ≈ 0.785 (b) π4 ≈ 0.785: the same fraction, 1114 with π = 227.
Step-by-step solution
Idea: Each circle sits in its own 2r × 2r square, so the fraction is the same as one circle in its square: πr2 ÷ 4r2.
(a) What fraction of the rectangle is covered by the circles?
- Radius r: the rectangle is 3 × 2r = 6r long and 2r wide, area 12r2. The circles cover 3πr2.1 mark
- Fraction = 3πr212r2 = π4 ≈ 0.785 (with π = 227: 1114).½ mark
π4 ≈ 0.785
(b) What fraction of the rectangle is covered by the circles?
- Rectangle 8r × 2r = 16r2; circles 4πr2.1 mark
- Fraction = 4πr216r2 = π4 ≈ 0.785, the same as before.½ mark
π4 ≈ 0.785
In both rectangles the circles cover π/4 ≈ 0.785 (about 78.5%) of the area.
Answer to write in the exam
(a)
Rectangle = 6r × 2r = 12r2; circles = 3πr2
∴ Fraction = 3πr212r2 = π4 ≈ 0.785
(b)
Rectangle = 8r × 2r = 16r2; circles = 4πr2
∴ Fraction = π4 ≈ 0.785
Common mistakes that cost marks
- Taking the rectangle’s width as r instead of 2r.
- Thinking more circles cover a bigger fraction; the fraction stays π4.
How this can come in the exam
MCQ (1 mark)
A circle fits exactly inside a square of side 14 cm. The area of the square not covered by the circle is (use π = 227)
- 42 cm2
- 154 cm2
- 196 cm2
- 38.5 cm2
Show answer
(A) 42 cm2
196 − 154 = 42 cm2.
Try one yourself
Two circles of radius 7 cm fit side by side in a rectangle. Find the area of the rectangle not covered. (Use π = 227.)
Show answer
Rectangle 28 × 14 = 392; circles 2 × 154 = 308; uncovered 84 cm2.
More questions like this
- Use the above to make a conjecture about the area occupied by circles fitted into a rectangle in the manner shown. Test your conjecture for particular cases: 10 circles; 20 circles; 50 circles. Then prove your conjecture!
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