Problems — 8.13 Circles and Angles | Mathematics Department

1

**Circumference and area together.** A circular garden has radius 4 m. (a) Find the circumference and area (use π = 3.14; 1 d.p.). (b) A path of width 1 m surrounds the garden. Find the area of the path. (c) The path is paved at 20 chf/m². Find the total cost.

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2

**Pizza pricing.** A pizza shop sells round pizzas: - 8" (20 cm diameter) for 10 chf - 12" (30 cm diameter) for 18 chf - 16" (40 cm diameter) for 28 chf (a) Find the area of each pizza (π = 3.14; nearest cm²). (b) Find price per cm² for each. (c) Which is the best value per cm²?

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3

**Circle in a square.** A circle is inscribed in a square of side 10 cm. (a) Find the diameter of the circle. (b) Find the area of the circle. (c) Find the area of the square not covered by the circle (use π = 3.14). (d) What fraction of the square is the circle's area?

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4

**Compound shape.** A 2D shape is made by joining a square of side 8 cm to a semicircle whose diameter equals one side of the square. (a) Find the perimeter to 1 d.p. (π = 3.14). (b) Find the area in terms of π.

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5

**Hexagonal patio.** A patio is a regular hexagon with side 4 m. (a) Find the size of each interior angle. (b) A regular hexagon = 6 equilateral triangles. Use this to find the area. (c) Paving at 35 chf/m². Find total cost.

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6

**Quarter-circle problem.** A quarter-circle has radius 6 cm. (a) Find the area. (b) Find the perimeter (arc + two radii). (c) If the quarter-circle is part of a square 6 cm × 6 cm, find the area outside the quarter-circle but inside the square.

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7

**π estimation.** Archimedes estimated π by inscribing and circumscribing regular polygons in a circle. (a) A regular hexagon inscribed in a circle of radius 1 has perimeter 6. What does this say about π? (b) Using a regular 12-gon inscribed (perimeter $\approx 6.21$), what is the improved bound for π? (c) Compare with the actual value of π.

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8

**Bicycle wheel.** A wheel has radius 35 cm. (a) Find the circumference. (b) How far does the bike travel in 100 revolutions? (c) The bike rides 220 m. How many revolutions does the wheel make?

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9

**Two concentric circles.** Two circles share the same centre. The inner has radius 5 cm, the outer 8 cm. (a) Find the area of each. (b) Find the area of the annulus (ring). (c) Find the ratio of the inner to outer area.

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10

**Sector problems.** A sector of a circle has radius 8 cm and angle 45°. (a) Find the area (use π = 3.14). (b) Find the arc length. (c) Find the total perimeter (arc + 2 radii).

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11

**Race track design.** Design a 400 m running track that fits inside a 100 m × 60 m rectangle, with two straight sections and two semicircular ends. (a) Find the straight-section length. (b) Find the radius of each end. (c) Verify the total perimeter equals 400 m.

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12

**Investigating π experimentally.** A student measures the circumference (C) and diameter (d) of 5 cylindrical objects: | d (cm) | 5 | 8 | 10 | 12 | 15 | |--------|---|---|----|----|-----| | C (cm) | 16 | 25 | 31 | 38 | 47 | (a) Find C/d for each. (b) Compute the mean of C/d. (c) What does this experiment estimate? Comment on accuracy.

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