Question 5

We will find the perimeter of each shape by adding the lengths of its curved and straight parts.

Step 1 — Perimeter of shape (i)

This shape has two straight sides. Each straight side is 80 m long. It has two semicircular ends. The diameter of each semicircle is 60 m. The two semicircles form one full circle.

$\text{Perimeter} = 2 \times \text{straight length} + \pi \times \text{diameter}$

$= 2 \times 80 + \frac{22}{7} \times 60$

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

$= \frac{1120 + 1320}{7}$

$= \frac{2440}{7}$

$\boxed{348.57 \text{ m}}$

Step 2 — Perimeter of shape (ii)

This shape has an outer semicircle and an inner semicircle. The outer diameter is 12 cm, so its radius is 6 cm. The inner diameter is 8 cm, so its radius is 4 cm. There are two straight connecting parts. Each part is 6 cm - 4 cm = 2 cm long.

$\text{Perimeter} = \pi \times \text{outer radius} + \pi \times \text{inner radius} + 2 \times \text{straight part}$

$= \pi \times 6 + \pi \times 4 + 2 \times 2$

$= 10\pi + 4$

$= 10 \times \frac{22}{7} + 4$

$= \frac{220}{7} + \frac{28}{7}$

$= \frac{248}{7}$

$\boxed{35.43 \text{ cm}}$

Step 3 — Perimeter of shape (iii)

This figure has four semicircles. Each semicircle has a diameter of 10 cm. So, the radius of each semicircle is 5 cm.

$\text{Perimeter} = 4 \times (\pi \times \text{radius})$

$= 4 \times \pi \times 5$

$= 20\pi$

$= 20 \times \frac{22}{7}$

$= \frac{440}{7}$

$\boxed{62.86 \text{ cm}}$

Step 4 — Perimeter of shape (iv)

This figure has three semicircles. Each semicircle has a diameter of 12 cm. So, the radius of each semicircle is 6 cm.

$\text{Perimeter} = 3 \times (\pi \times \text{radius})$

$= 3 \times \pi \times 6$

$= 18\pi$

$= 18 \times \frac{22}{7}$

$= \frac{396}{7}$

$\boxed{56.57 \text{ cm}}$

Step 5 — Perimeter of shape (v)

This figure has four small semicircles. Each small semicircle has a diameter of 14 cm. So its radius is 7 cm. It also has four large quarter circles. The radius of each large quarter circle is 14 cm.

$\text{Perimeter} = 4 \times (\pi \times \text{small radius}) + 4 \times (\frac{1}{2} \pi \times \text{large radius})$

$= 4 \times \pi \times 7 + 4 \times \frac{1}{2} \times \pi \times 14$

$= 28\pi + 28\pi$

$= 56\pi$

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

$= 8 \times 22$

$\boxed{176 \text{ cm}}$

Step 6 — Perimeter of shape (vi)

This figure has one large upper semicircle. Its diameter is 28 cm, so its radius is 14 cm. It has four small lower semicircles. The total base length is 28 cm. Each small semicircle's diameter is 28 cm / 4 = 7 cm. So, the radius of each small semicircle is 3.5 cm.

$\text{Perimeter} = \pi \times \text{large radius} + 4 \times (\pi \times \text{small radius})$

$= \pi \times 14 + 4 \times \pi \times 3.5$

$= 14\pi + 14\pi$

$= 28\pi$

$= 28 \times \frac{22}{7}$

$= 4 \times 22$

$\boxed{88 \text{ cm}}$

Step 7 — Perimeter of shape (vii)

This figure has three semicircles. They are on the sides of a right-angled triangle. The two perpendicular sides are 6 cm and 8 cm. Let's find the hypotenuse using the Pythagorean theorem.

$\text{Hypotenuse}^2 = 6^2 + 8^2$

$= 36 + 64$

$= 100$

$\text{Hypotenuse} = \sqrt{100} = 10 \text{ cm}$

The diameters of the semicircles are 6 cm, 8 cm, and 10 cm. Their radii are 3 cm, 4 cm, and 5 cm.

$\text{Perimeter} = \pi \times 3 + \pi \times 4 + \pi \times 5$

$= \pi \times (3 + 4 + 5)$

$= 12\pi$

$= 12 \times \frac{22}{7}$

$= \frac{264}{7}$

$\boxed{37.71 \text{ cm}}$

Step 8 — Perimeter of shape (viii)

This figure has one large upper semicircle. Its diameter is 4 cm + 4 cm + 4 cm = 12 cm. So its radius is 6 cm. It has three small lower semicircles. Each small semicircle has a diameter of 4 cm. So its radius is 2 cm.

$\text{Perimeter} = \pi \times \text{large radius} + 3 \times (\pi \times \text{small radius})$

$= \pi \times 6 + 3 \times \pi \times 2$

$= 6\pi + 6\pi$

$= 12\pi$

$= 12 \times \frac{22}{7}$

$= \frac{264}{7}$

$\boxed{37.71 \text{ cm}}$

Step 9 — Perimeter of shape (ix)

This figure has one large outer semicircle. Its diameter is 10 cm + 10 cm = 20 cm. So its radius is 10 cm. It has two inner semicircles. Each inner semicircle has a diameter of 10 cm. So its radius is 5 cm.

$\text{Perimeter} = \pi \times \text{large radius} + 2 \times (\pi \times \text{small radius})$

$= \pi \times 10 + 2 \times \pi \times 5$

$= 10\pi + 10\pi$

$= 20\pi$

$= 20 \times \frac{22}{7}$

$= \frac{440}{7}$

$\boxed{62.86 \text{ cm}}$

Answer

(i) The perimeter is 348.57 m. (ii) The perimeter is 35.43 cm. (iii) The perimeter is 62.86 cm. (iv) The perimeter is 56.57 cm. (v) The perimeter is 176 cm. (vi) The perimeter is 88 cm. (vii) The perimeter is 37.71 cm. (viii) The perimeter is 37.71 cm. (ix) The perimeter is 62.86 cm.