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In the figure below, the boundary of the shaded region consists of four semi-circular arcs, the smallest two being equal. If the diameter of the largest is $ 14 \mathrm{~cm} $ and of the smallest is $ 3.5 $ $ \mathrm{cm} $, find the length of the boundary."

Given:

The boundary of the shaded region consists of four semi-circular arcs, the smallest two being equal.

The diameter of the largest is \( 14 \mathrm{~cm} \) and of the smallest is \( 3.5 \) \( \mathrm{cm} \),

To do: 

We have to find the length of the boundary.

Solution:

Diameter of the largest semicircle $= 14\ cm$

This implies,

Radius $R =\frac{14}{2}$

$ = 7\ cm$
Diameter of the semicircle with point $D= 7\ cm$

This implies,

Radius $r_1 =\frac{7}{2}\ cm$

Diameter of each smaller circles $= 3.5\ cm$

This implies,

Radius $r_{2}=\frac{3.5}{2}=1.75 \mathrm{~cm}$

Therefore,

Length of the boundary $=$ Circumference of the largest semicircle $+$ Circumference of middle semicircle $+$ Circumference of two smaller semicircles

$=\pi \mathrm{R}+\pi r_{1}+2 \pi r_{2}$

$=\pi(\mathrm{R}+r_{1}+2 r_{2})$

$=\frac{22}{7}(7+3.5+2 \times \frac{3.5}{2})$

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

$=\frac{22}{7} \times 14$

$=44 \mathrm{~cm}$

The length of the boundary is $44\ cm$.

Updated on: 10-Oct-2022

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