Question 7

We need to find the total perimeter of all petals in each figure.

Step 1 — Calculate perimeter for Figure (i)

Let's analyze the first figure. The figure is a square. Its side length is 14 cm. The centers of the arcs are the midpoints of the sides. The radius of each arc is half the side length. So, the radius $r = \mathbf{14 \text{ cm}} / 2 = \mathbf{7 \text{ cm}}$. Each petal consists of two quarter-circle arcs. The perimeter of one petal is the length of two quarter-circles. This is equal to the length of one semicircle. The length of a semicircle is $\pi r$. There are 4 petals in total. The total perimeter is 4 times the perimeter of one petal.

$\text{Total perimeter} = 4 \times (\pi r)$

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

$= 4 \times 22$

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

Step 2 — Calculate perimeter for Figure (ii)

Let's analyze the second figure. The figure is a regular hexagon. Its side length is 42 cm. The centers of the arcs are the vertices of the hexagon. The radius of each arc is the side length of the hexagon. So, the radius $r = \mathbf{42 \text{ cm}}$. Each petal consists of two arcs. Each arc subtends an angle of 60° at its center. This is because a regular hexagon can be divided into six equilateral triangles. The length of an arc is given by $(\theta/360^\circ) \times 2\pi r$. For one arc, the length is $(60/360) \times 2\pi r = \pi r / 3$. The perimeter of one petal is the length of two such arcs. So, the perimeter of one petal is $2 \times (\pi r / 3) = 2\pi r / 3$. There are 6 petals in total. The total perimeter is 6 times the perimeter of one petal.

$\text{Total perimeter} = 6 \times \left(\frac{2\pi r}{3}\right)$

$= 4\pi r$

$= 4 \times \frac{22}{7} \times 42$

$= 4 \times 22 \times 6$

$= 88 \times 6$

$\boxed{528 \text{ cm}}$

Answer

(i) The total perimeter of all the petals is 88 cm. (ii) The total perimeter of all the petals is 528 cm.