Question

The perimeter of sector of circle of area 25sq.cm is 36cm.Then find the area of sector.

Answer 1

Hint: Here we will use the formula for perimeter of sector to find angle made by sector and then we will find area.

Complete step by step solution:
Given: Area of circle is given as 25 cm2 and perimeter of sector is given as 36cm

Area of circle is, \[\pi {r^2} = 25\] cm²
We can find radius ‘r’ of the circle from above formula
 \[{r^2} = 25/\pi \]
r =25
r = 2.82cm ………. (i)
Formula for the perimeter of the sector = \[\alpha r{\rm{ }} + {\rm{ }}2r\]
Where, α = angle subtended by sector at centre. Perimeter of sector is given as 36 cm
\[\alpha r{\rm{ }} + {\rm{ }}2r\] = 36
Putting value of r from equation (i), we can find angle subtended by the sector at the centre of the circle as:
\[\alpha {\rm{ }} = {\rm{ }}\left( {36 - {\rm{ }}2r} \right)/r\]
\[\alpha {\rm{ }} = {\rm{ }}\left( {36{\rm{ }} - {\rm{ }}2 \times 2.82} \right)/2.82\] …….. (ii)
Area of the sector is given by = 12 \[\alpha {r^2}\]
Using the value of α and r from equation (ii) and equation (i) respectively, we will get area of the sector as:
Area of the sector = 12 × (36 - 2×2.82)2.82 × (2.82)2
After solving this we will get:

Area of the sector = 42.82 cm²

Note: In this kind of problem where a sector of a circle is involved, firstly start with finding the angle subtended by the sector at the center of the circle and then find the required in the problem.