Question

The given figure consists of a square, a quarter circle and a semicircle. Area of sector is $${\displaystyle \frac{\pi r^2\theta }{{360}^{\circ }}}$$ . All the sides of a square are equal. Find the perimeter of the shaded portion.

$\begin{array}{rl}& a)5\pi \\ & b)10\pi \\ & c)10(1+\pi )\\ & d)10(1-\pi )\end{array}$

Answer 1

Hint:In the figure we can see that the perimeter of the shaded portion is the addition of the circumference of quarter circle and semicircle. We know that circumference of a semicircle is ${\displaystyle \frac{2\pi r}{2}}$ and circumference of the quarter circle is ${\displaystyle \frac{2\pi r}{4}}$ . Now we can easily find the values of the two circumference and hence the required perimeter

Complete step by step answer:
Now consider the figure given

We have two parts of circles here one with radius AB and another with diameter AB. We can see that the Perimeter of the shaded portion will be the addition of circumference of both radius.
Now consider the Quarter circle with a radius 10cm. Now we know that circumference of the quarter circle is given by ${\displaystyle \frac{2\pi r}{4}}$.
Now here the radius is AB, BC and is equal to 10cm hence we get
${\displaystyle \frac{2\pi \left(10\right)}{4}}={\displaystyle \frac{20\pi }{4}}=5\pi$
Hence we have the circumference of the quarter-circle that is arc AC is $5\pi ........................\left(1\right)$
Now consider the semicircle with diameter AB.
Now we know that circumference of the semicircle is given by ${\displaystyle \frac{2\pi r}{2}}$
Now we know that diameter = 2 × radius. Hence we have 2r = d
Substituting this we get
${\displaystyle \frac{2\pi r}{2}}={\displaystyle \frac{\pi d}{2}}$
Now we know that the diameter of the semicircle is AB and is equal to 10cm
Hence we get the circumference of the semicircle is
${\displaystyle \frac{\pi \left(10\right)}{2}}=5\pi$
Hence we have a circumference of a semicircle that is arc AB = $5\pi ............................\left(2\right)$
Now we know the perimeter of the shaded portion will be the addition to the circumference of both radii.
Hence from equation (1) and (2) we have
Perimeter of shaded portion is $5\pi +5\pi =10\pi$
Hence we get the perimeter of the shaded portion is $10\pi$.
Option b is the correct option.

Note:
Note that the area of circle is given by $\pi r^2$ and circumference is given by $2\pi r$ . Also we have in general if the angle subtended by arc is $\theta$ then circumference of the arc is ${\displaystyle \frac{2\pi r\left(\theta \right)}{{360}^{\circ }}}$ and similarly area of the part will be equal to ${\displaystyle \frac{\pi r^2\left(\theta \right)}{{360}^{\circ }}}$