Find the total perimeter of all the petals in each of the given flowers.
- (i) Flower in a square of side 14 cm (the centres of the arcs are the midpoints of the sides of the square)
- (ii) Flower in a regular hexagon of side 42 cm (the centres of the arcs are the vertices of the hexagon)
Step-by-step solution
Idea: Each petal is bounded by two equal arcs. Find the radius and the angle each arc subtends at its centre, then count the arcs.
(i) Flower in a square of side 14 cm (the centres of the arcs are the midpoints of the sides of the square)
- Each arc has its centre at the midpoint of a side and passes through a corner and the centre of the square, so its radius is 7 cm. The radius to the corner lies along the side and the radius to the centre is perpendicular to it, so each arc is a quarter circle.1 mark
- Each petal has 2 such arcs, so 4 petals have 8 quarter circles = 2 full circles of radius 7 cm. Total = 2 × 2 × 227 × 7 = 88 cm.1 mark
(ii) Flower in a regular hexagon of side 42 cm (the centres of the arcs are the vertices of the hexagon)
- Join the centre O of the hexagon to the vertices: this makes 6 equilateral triangles of side 42 cm. Each petal joins O to a vertex, and each of its two arcs is centred at a neighbouring vertex with radius 42 cm. The angle at that centre is an angle of an equilateral triangle, 60°.1 mark
- Each arc = 2 × 227 × 42 × 60360 = 264 × 16 = 44 cm. 6 petals × 2 arcs = 12 arcs. Total = 12 × 44 = 528 cm.1 mark
Check: (i) The 8 quarter arcs are exactly the 4 semicircles drawn on the sides, 4 × 22 = 88 ✓. (ii) 12 arcs of 60° make 720° = 2 full circles of radius 42: 2 × 264 = 528 ✓.
Answer to write in the exam
(i)
Radius = 142 = 7 cm; each arc is a quarter circle
8 quarter arcs = 2 full circles
Total = 2 × 2π × 7 = 2 × 44
∴ Total perimeter = 88 cm
(ii)
Radius = 42 cm; each arc subtends 60° (equilateral triangles)
One arc = 2 × 227 × 42 × 60°360° = 44 cm
Total = 12 × 44
∴ Total perimeter = 528 cm
Common mistakes that cost marks
- In (i), taking the radius as 14 cm (the side) instead of 7 cm (half the side).
- In (ii), taking each arc as 120° instead of 60°.
- Counting only one arc per petal.
How this can come in the exam
A petal is made of two quarter circles of radius 14 cm. Its perimeter is (use π = 227)
- 22 cm
- 88 cm
- 154 cm
- 44 cm
Show answer
(D) 44 cm
Each quarter circle = 14 × 2 × 227 × 14 = 22 cm; two of them = 44 cm.
Try one yourself
Find the total perimeter of the petals in the square flower of part (i) if the square has side 28 cm. (Use π = 227.)
Show answer
Radius 14 cm; 8 quarter arcs = 2 circles = 2 × 88 = 176 cm.
More questions like this
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