Corrigé
8.13 Circles and Angles
Pack A — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Find the circumference of a circle with radius 5 cm. Use $\pi = 3.14$; 1 d.p. | 31.4 cm |
| 2 | Find the area of a circle with radius 4 cm. Use $\pi = 3.14$; 1 d.p. | 50.2 cm² |
| 3 | Identify each as acute, right, obtuse, straight, or reflex: 35°, 90°, 120°, 180°, 250°. | Acute, right, obtuse, straight, reflex |
| 4 | Find the diameter of a circle with circumference 31.4 cm. Use $\pi = 3.14$; 1 d.p. | 10 cm |
| 5 | A pizza has diameter 30 cm. Find its area. Use $\pi = 3.14$; 1 d.p. | 706.5 cm² |
| 6 | Is the shape concave or convex? Octagonal shape with all interior angles < 180°. | Convex |
| 7 | A circle has radius 7 cm. Find the (a) diameter, (b) circumference, (c) area. Use $\pi = 22/7$. | (a) 14 cm (b) 44 cm (c) 154 cm² |
| 8 | What is the value of $\pi$ to 3 decimal places? | 3.142 |
| 9 | A circle has radius 10 cm. Find the (a) circumference and (b) area. Use $\pi$ in your answer. | (a) $20\pi$ cm (b) $100\pi$ cm² |
| 10 | Classify each angle: (a) 35°, (b) 95°, (c) 175°, (d) 200°. | (a) acute (b) obtuse (c) obtuse (d) reflex |
| 11 | A circle has diameter 10 cm. Find (a) radius, (b) circumference, (c) area. Use $\pi = 3.14$; 1 d.p. | (a) 5 cm (b) 31.4 cm (c) 78.5 cm² |
| 12 | A circular pizza has area 314 cm² (use $\pi = 3.14$). Find its radius. | 10 cm |
| 13 | Find the perimeter of a semicircle of radius 5 cm. Use $\pi = 3.14$; 1 d.p. | 25.7 cm |
| 14 | A circle is inscribed in a square of side 8 cm. Find the area of the circle (use $\pi = 3.14$). | 50.24 cm² |
| 15 | Find the area of the sector with angle 90° and radius 6 cm. | $9\pi$ cm² ≈ 28.3 cm² |
| 16 | Find the arc length of a sector with angle 90° and radius 6 cm. | $3\pi$ cm ≈ 9.4 cm |
| 17 | A bicycle wheel has diameter 60 cm. Find the distance travelled in 1 revolution (in m). | ≈ 1.88 m |
| 18 | A clock's minute hand is 12 cm long. Find the distance the tip travels in 15 minutes. | ≈ 18.85 cm |
| 19 | A circular pond has diameter 10 m. Find (a) the perimeter, (b) the area covered by the pond. (Use $\pi = 3.14$.) | (a) 31.4 m (b) 78.5 m² |
| 20 | Classify the shape: a polygon has 6 angles, all measuring 120°. Is it convex or concave? Identify the shape. | Convex regular hexagon |
| 21 | A circle has radius 2 m. Find its area in (a) m² and (b) cm². Use $\pi = 3.14$; 1 d.p. | (a) 12.6 m² (b) 125 600 cm² |
| 22 | A circle has radius $(x + 2)$ cm and area $9\pi$ cm². Find $x$. | $x = 1$ |
| 23 | A circular flower bed has area $25\pi$ m². (a) Find the radius. (b) Find the circumference (use $\pi = 3.14$; 1 d.p.). | (a) 5 m (b) 31.4 m |
| 24 | A composite shape: a rectangle 10 cm by 6 cm with a semicircle of diameter 6 cm attached to one short side. Find the perimeter (use $\pi = 3.14$; 1 d.p.). | ≈ 35.4 cm |
| 25 | A square is inscribed inside a circle. The circle has area $50\pi$ cm². Find (a) the radius, (b) the side of the square. | (a) $\sqrt{50} = 5\sqrt{2}$ cm (b) 10 cm |
| 26 | A circle is inscribed in a square of side 10 cm. Find the area between the square and the circle. | ≈ 21.5 cm² |
| 27 | A semicircle has diameter 10 cm. Find its (a) area, (b) perimeter (use $\pi = 3.14$; 1 d.p.). | (a) 39.3 cm² (b) 25.7 cm |
| 28 | Two concentric circles have radii 8 cm and 5 cm. Find the area of the annulus (the ring between them). | $39\pi \approx 122.5$ cm² |
| 29 | Three congruent circles, each of radius 5 cm, are arranged in a row touching each other. Find the total perimeter (of the visible outline if joined as one shape). | ≈ 62.8 cm (each circle's full perimeter = $10\pi$; touching only at points, so total is $30\pi$) |
| 30 | A pizza is cut into 8 equal slices. Find the angle of each slice. If the pizza has radius 15 cm, find the area of one slice. | 45°; area $\approx 88.4$ cm² |
| 31 | A flower bed is a square with a quarter-circle attached to one corner. The square is 6 m by 6 m, and the quarter-circle has radius 6 m (occupying the area outside the square). Find the total area and total perimeter. | Area $\approx 64.3$ m²; perimeter $\approx 24 + 9.4 = 33.4$ m |
| 32 | A running track consists of a rectangle 80 m by 40 m, with semicircles of radius 20 m at each short end. Find the total perimeter and area. | Perimeter ≈ 285.6 m; Area ≈ 4456 m² |
| 33 | A square has area equal to that of a circle of radius 6 cm. Find the side length of the square. | $6\sqrt{\pi} \approx 10.6$ cm |
| 34 | A circle has circumference $C$. By how much does its circumference increase if its radius increases by 1 cm? | $2\pi$ cm ≈ 6.28 cm — independent of original radius |
| 35 | Three identical circles of radius 5 cm are arranged so each touches the other two. Find the area of the curvilinear triangle in the middle (between the three circles). | ≈ 4.0 cm² |
| 36 | A circle's area increases by 21%. By what percentage does its radius increase? | ≈ 10% (since $1.21 = 1.1^2$) |
| 37 | A circular pond of radius 10 m is to be surrounded by a path of width 1 m. Find the area of the path. | ≈ 66.0 m² ($21\pi$) |
| 38 | A circle is inscribed in a regular hexagon of side 6 cm. (a) Find the radius of the circle. (b) Find the area of the hexagon and the circle, and the area between them. | (a) $3\sqrt{3}$ cm (b) Hexagon ≈ 93.5; circle ≈ 84.8; difference ≈ 8.7 |
| 39 | A bicycle wheel has diameter 70 cm. The bike travels 1 km. How many full revolutions does the wheel make? | ≈ 455 revolutions |
| 40 | A clock's minute hand is 8 cm long. (a) Find the distance the tip travels in 1 hour. (b) Find the area swept by the hand in 15 minutes. | (a) $16\pi \approx 50.27$ cm (b) $16\pi \approx 50.27$ cm² |
Pack B — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Find the circumference of a circle with radius 8 cm. Use $\pi = 3.14$; 1 d.p. | 50.2 cm |
| 2 | Find the area of a circle with radius 6 cm. Use $\pi = 3.14$; 1 d.p. | 113.0 cm² |
| 3 | Identify each as acute, right, obtuse, straight, or reflex: 35°, 90°, 120°, 180°, 250°. | Same. |
| 4 | Find the diameter of a circle with circumference 62.8 cm. Use $\pi = 3.14$; 1 d.p. | 20 cm |
| 5 | A pizza has diameter 30 cm. Find its area. Use $\pi = 3.14$; 1 d.p. | 1256.0 cm² |
| 6 | Is the shape concave or convex? Octagonal shape with all interior angles < 180°. | Concave |
| 7 | A circle has radius 7 cm. Find the (a) diameter, (b) circumference, (c) area. Use $\pi = 22/7$. | (a) 28 cm (b) 88 cm (c) 616 cm² |
| 8 | What is the value of $\pi$ to 3 decimal places? | Same. |
| 9 | A circle has radius 10 cm. Find the (a) circumference and (b) area. Use $\pi$ in your answer. | (a) $12\pi$ cm (b) $36\pi$ cm² |
| 10 | Classify each angle: (a) 35°, (b) 95°, (c) 175°, (d) 200°. | (a) acute (b) acute (c) obtuse (d) reflex |
| 11 | A circle has diameter 14 cm. Find (a) radius, (b) circumference, (c) area. Use $\pi = 3.14$; 1 d.p. | (a) 7 cm (b) 44.0 cm (c) 153.9 cm² |
| 12 | A circular pizza has area 314 cm² (use $\pi = 3.14$). Find its radius. | ≈ 14.14 cm |
| 13 | Find the perimeter of a semicircle of radius 7 cm. Use $\pi = 3.14$; 1 d.p. | 36.0 cm |
| 14 | A circle is inscribed in a square of side 8 cm. Find the area of the circle (use $\pi = 3.14$). | 78.54 cm² |
| 15 | Find the area of the sector with angle 60° and radius 9 cm. | $13.5\pi$ cm² ≈ 42.4 cm² |
| 16 | Find the arc length of a sector with angle 120° and radius 9 cm. | $6\pi$ cm ≈ 18.8 cm |
| 17 | A bicycle wheel has diameter 60 cm. Find the distance travelled in 1 revolution (in m). | ≈ 2.2 m |
| 18 | A clock's minute hand is 12 cm long. Find the distance the tip travels in 15 minutes. | ≈ 47.12 cm |
| 19 | A circular pond has diameter 10 m. Find (a) the perimeter, (b) the area covered by the pond. (Use $\pi = 3.14$.) | (a) 25.12 m (b) 50.24 m² |
| 20 | Classify the shape: a polygon has 6 angles, all measuring 120°. Is it convex or concave? Identify the shape. | Convex regular pentagon |
| 21 | A circle has radius 3 m. Find its area in (a) m² and (b) cm². Use $\pi = 3.14$; 1 d.p. | (a) 28.3 m² (b) 282 600 cm² |
| 22 | A circle has radius $(x + 2)$ cm and area $9\pi$ cm². Find $x$. | $x = 2$ |
| 23 | A circular flower bed has area $25\pi$ m². (a) Find the radius. (b) Find the circumference (use $\pi = 3.14$; 1 d.p.). | (a) 7 m (b) 44.0 m |
| 24 | A composite shape: a rectangle 10 cm by 6 cm with a semicircle of diameter 6 cm attached to one short side. Find the perimeter (use $\pi = 3.14$; 1 d.p.). | ≈ 44.6 cm |
| 25 | A square is inscribed inside a circle. The circle has area $50\pi$ cm². Find (a) the radius, (b) the side of the square. | (a) $\sqrt{32} = 4\sqrt{2}$ cm (b) 8 cm |
| 26 | A circle is inscribed in a square of side 10 cm. Find the area between the square and the circle. | ≈ 30.9 cm² |
| 27 | A semicircle has diameter 10 cm. Find its (a) area, (b) perimeter (use $\pi = 3.14$; 1 d.p.). | (a) 77.0 cm² (b) 36.0 cm |
| 28 | Two concentric circles have radii 8 cm and 5 cm. Find the area of the annulus (the ring between them). | $64\pi \approx 201.1$ cm² |
| 29 | Three congruent circles, each of radius 5 cm, are arranged in a row touching each other. Find the total perimeter (of the visible outline if joined as one shape). | ≈ 75.4 cm |
| 30 | A pizza is cut into 8 equal slices. Find the angle of each slice. If the pizza has radius 15 cm, find the area of one slice. | 60°; area $\approx 209.4$ cm² |
| 31 | A flower bed is a square with a quarter-circle attached to one corner. The square is 6 m by 6 m, and the quarter-circle has radius 6 m (occupying the area outside the square). Find the total area and total perimeter. | Area $\approx 114.3$ m²; perimeter $\approx 44.6$ m |
| 32 | A running track consists of a rectangle 80 m by 40 m, with semicircles of radius 20 m at each short end. Find the total perimeter and area. | Perimeter ≈ 357.1 m; Area ≈ 6963.5 m² |
| 33 | A square has area equal to that of a circle of radius 6 cm. Find the side length of the square. | $10\sqrt{\pi} \approx 17.7$ cm |
| 34 | A circle has circumference $C$. By how much does its circumference increase if its radius increases by 1 cm? | Same. |
| 35 | Three identical circles of radius 5 cm are arranged so each touches the other two. Find the area of the curvilinear triangle in the middle (between the three circles). | ≈ 5.8 cm² |
| 36 | A circle's area increases by 21%. By what percentage does its radius increase? | ≈ 20% (since $1.44 = 1.2^2$) |
| 37 | A circular pond of radius 10 m is to be surrounded by a path of width 1 m. Find the area of the path. | ≈ 113.1 m² ($36\pi$) |
| 38 | A circle is inscribed in a regular hexagon of side 6 cm. (a) Find the radius of the circle. (b) Find the area of the hexagon and the circle, and the area between them. | Same. |
| 39 | A bicycle wheel has diameter 70 cm. The bike travels 1 km. How many full revolutions does the wheel make? | ≈ 398 revolutions |
| 40 | A clock's minute hand is 8 cm long. (a) Find the distance the tip travels in 1 hour. (b) Find the area swept by the hand in 15 minutes. | Same. |
Problèmes — Solutions détaillées
**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.
(a) C ≈ 25.1 m, A ≈ 50.2 m² (b) Path area ≈ 28.3 m² (c) ≈ 565 chf
**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?
(a) ≈ 314, 707, 1256 cm² (b) 3.18, 2.55, 2.23 c/cm² (c) 16" pizza
**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?
(a) 10 cm (b) 78.5 cm² (c) 21.5 cm² (d) ≈ 78.5%
**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.
(a) ≈ 36.6 cm (b) $(64 + 8\pi)$ cm²
**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.
(a) 120° (b) ≈ 41.6 m² (c) ≈ 1456 chf
**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.
(a) $9\pi$ ≈ 28.3 cm² (b) $3\pi + 12$ ≈ 21.4 cm (c) $36 - 9\pi$ ≈ 7.7 cm²
**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.
(a) $\pi > 3$ (since perim < circumference $2\pi$, but here perim 6 < $2\pi$, so $\pi > 3$) (b) $\pi > 3.105$ (c) $\pi \approx 3.14159$
**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?
(a) ≈ 219.9 cm (b) ≈ 219.9 m (c) ≈ 100
**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.
(a) Inner $25\pi$ ≈ 78.5; outer $64\pi$ ≈ 201.1 (b) $39\pi$ ≈ 122.5 (c) 25 : 64
**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).
(a) ≈ 25.1 cm² (b) ≈ 6.3 cm (c) ≈ 22.3 cm
**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.
(a) Straight: ≈ 86 m (b) Radius 30 m (c) $2 \times 86 + 2 \times \pi \times 30 \approx 172 + 188 = 360$ — doesn't hit 400. Adjust radius.
**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.
(a) 3.2, 3.125, 3.1, 3.17, 3.13 (b) ≈ 3.145 (c) An estimate of π ≈ 3.142