Question

Calculate the area and perimeter of a quadrant of a circle of radius 21 cm.

Answer 1

Hint: A quadrant is a quarter of a circle. First work out the area of the whole circle and then divide the answer by 4. Second work out the perimeter of the whole circle, then divide by 4 and add length of the two sides.

Complete step-by-step solution -

First work out the area of the whole circle by substituting the radius of r = 21 cm into the formula for the area of the circle.
The area of the circle $=\pi \times r^2.............(1)$
The area of the circle $=\pi \times {(21)}^2$
Put the value of the $\pi ={\displaystyle \frac{22}{7}}$ in the equation (1), we get
The area of the circle $={\displaystyle \frac{22}{7}}\times 441=3.142\times 441=1385.62$
A quadrant is one-fourth of a circle. If a circle is evenly divided into four sections by two perpendicular lines, then each of the four areas is a quadrant.

So the area of the circle is divided by 4.
The area of a quadrant of the circle $={\displaystyle \frac{1385.62}{4}}=346.40$
Hence the area of a quadrant of the circle is 346. 40 $c{m}^2$.
Second work out the perimeter of the whole circle by substituting the radius of r = 21 cm into the formula for the circumference of the circle.
The circumference of the circle $=2\pi r.................(2)$
The circumference of the circle $=2\pi (21)$
Put the value of the $\pi ={\displaystyle \frac{22}{7}}$ in the equation (2), we get
The circumference of the circle $=2\times {\displaystyle \frac{22}{7}}\times (21)=2\times 3.142\times 21=131.96$
A quadrant of a circle is a sector of the circle whose sectorial angle is 90 degree.
The perimeter of a quadrant of the circle is one fourth of the circumference and 2 times of the radius of the circle.
The perimeter of a quadrant of the circle $={\displaystyle \frac{2\pi r}{4}}+2(r)$
The perimeter of a quadrant of the circle $={\displaystyle \frac{131.96}{4}}+2(21)$
The perimeter of a quadrant of the circle $={\displaystyle \frac{131.96}{4}}+42=74.99=75$
Hence the perimeter of a quadrant of the circle is 75 cm.

Note: Alternatively, you could substitute the radius of the quadrant directly into the formula of area $(A={\displaystyle \frac{1}{4}}\pi r^2)$ and perimeter $[P=({\displaystyle \frac{\pi }{2}}+2)r]$ of a quadrant of the circle.