Question

i) Find the area of sector of a circle with the radius $6cm$, if the angle of the sector is ${60^o}$
ii) Find the area of a quadrant of the circle whose circumference is $22cm$.
iii) The length of the minute hand of a clock is \[14cm\]. Find the area swept by the minute in 5 minutes.

Answer 1

Hint: Use formula for area of the circles and per minute angles changes by ${6^o}$
Since, we are given the radius and also the angle of sector, we can directly substitute them into the area of sector formula and get the answer. To find the area of the quadrant we need to find the radius first from the circumference given in the question. Then we find the area of the quadrant. The length of a minute hand represents the radius and we can calculate the angle by multiplying 5 minutes to the change in angle every minute and apply the formula of area of sector to get the answer.

Complete step by step solution:
i)In the question, we are given the radius of the circle and also the angle of the sector.
We have a direct formula, for finding the area of sector

$Area = \dfrac{\theta }{{360}} \times \pi {r^2}$
We substitute the value of r and $\theta $ in the formula
$
   = \dfrac{{60}}{{360}} \times \dfrac{{22}}{7} \times {6^2} \\
   = \dfrac{1}{6} \times \dfrac{{22}}{7} \times 36 \\
   = \dfrac{{6 \times 22}}{7} \\
   = \dfrac{{132}}{7}c{m^2} \\
$
We have the area of the sector.
ii) We are given a circle whose circumference is $22cm$
With circumference, we can find a radius which can be used to calculate the area of the quadrant.
$circumference = 2\pi r$
We substitute here and get the radius
$
  2\pi r = 22 \\
  r = 22 \times \dfrac{7}{{22}} \times \dfrac{1}{2} \\
  r = \dfrac{7}{2} \\
$
Area of quadrant is
$Area = \dfrac{1}{4} \times Area\,of\,circle$
\[area = \dfrac{{154}}{3}c{m^2}\]
We have found the area of the quadrant.
iii) Since, we know that per minute angles changes by ${6^o}$
So, for 5 minutes, the minute hands cover an angle of $5 \times {6^o} = {30^o}$ and also the minute hand denotes the radius of the circle.
We have a direct formula, for finding the area of sector

$Area = \dfrac{\theta }{{360}} \times \pi {r^2}$
On substituting
\[
   = \dfrac{{30}}{{360}} \times \pi \times 14 \times 14 \\
    \\
\]
If we simplify this, we get
\[area = \dfrac{{154}}{3}c{m^2}\]

Note: We have to consider only the length of the minute hand for the purpose of finding radius not the hours hand. While calculating the area of the quadrant , we have to divide the area of the circle by 4 and not by 2 , as four quadrants make a circle.