Find the perimeters of the following shapes (taking the arcs to be quarter or half or three-quarters of a circle, as appropriate) (Fig. A to I):                                                                                                                                                            [Page No. 129]

(A) Diameter of each semicircle = 60 m

Radius = 30 m

Straight part = 80 m (top and bottom)

Perimeter = $2 \times 80 + 2\pi r$

$$= 160 + 2 \times \frac{22}{7} \times 30$$

$$= 160 + \frac{1320}{7}$$

$$= \frac{2440}{7} = 348.57 \text{ m}$$

(B) Outer semicircle diameter = 12 cm

Outer radius, $R = 6 \text{ cm}$

Inner semicircle diameter = 8 cm

Inner radius, $r = 4 \text{ cm}$

Perimeter = $\pi R + \pi r + 2(R - r)$

$$= \frac{22}{7}(6) + \frac{22}{7}(4) + 2(2)$$

$$= \frac{220}{7} + 4 = \frac{248}{7} = 35.43 \text{ cm}$$

(C) Side of square = 10 cm

Four semicircles are drawn externally.

Radius of each semicircle = 5 cm

Total curved length = $4 \times \pi r$

$$= 4 \times 227 \times 5$$

$$= \frac{440}{7}$$

$$= 62.86 \text{ cm}$$

(D) Triangle side = 12 cm

Three semicircles are drawn on each side.

Radius of each semicircle = 6 cm

Perimeter = $3 \times \pi r$

$$= 3 \times \frac{22}{7} \times 6$$

$$= 3967 = 56.57 \text{ cm}$$

(E) The figure has 4 semicircles, each with diameter 14 cm and 4 quarter circle, each with radius 14 cm.

Radius of each semicircle = 7 cm

Perimeter = 4 × semicircle length

+ 4 × quarter circle length

$$= 4\pi r + 4 \left( \frac{1}{2} \pi R \right)$$

$$= 4\pi \times 7 + 2\pi \times 14$$

$$= 28\pi + 28\pi = 56\pi$$

$$= 56 \times \frac{22}{7}$$

$$= 176 \text{ cm}$$

(F) Diameter of big semicircle = 28 cm

Radius = 14 cm

Bottom contains 4 equal semicircles.

$$\text{Each diameter} = \frac{28}{4} = 7 \text{ cm}$$

Radius of each small semicircle = 3.5 cm

$$Perimeter = \pi(14) + 4\pi(3.5)$$

$$= \frac{22}{7}(14) + 4 \times \frac{22}{7}(3.5)$$

$$= 44 + 44 = 88 \text{ cm}$$

(G) Right triangle with sides 8 cm and 6 cm.

Hypotenuse = √8² + 6² = 10 cm

Three semicircles have diameters 8 cm, 6 cm and 10 cm.

$$\text{Perimeter} = \frac{\pi}{2}(8 + 6 + 10)$$

$$= \frac{22}{7} \times 12 = \frac{264}{7} = 37.71 \text{ cm}$$

(H) Large semicircle diameter

$$= 4 + 4 + 4 = 12 \text{ cm}$$

Large radius = 6 cm

Three small semicircles each have diameter 4 cm.

Radius = 2 cm

$P = \pi(6) + 3\pi(2)$

$$= \frac{22}{7}(6) + 3 \times \frac{22}{7}(2)$$

$$= \frac{264}{7} = 37.71 \text{ cm}$$

(I) Large semicircle diameter = 10 + 10 = 20 cm

Large radius = 10 cm

Two small semicircles each diameter = 10 cm

Radius = 5 cm

Perimeter$=\pi(10)+2\pi(5)$

$$= \frac{22}{7}(10) + 2 \times \frac{22}{7}(5)$$

$$= \frac{440}{7} = 62.86 \text{ cm}$$