Question

The Perimeter of a sector of a circle is 56 cm and the area of the circle is 64 $\pi $ Sq cms. Find the area of the sector.
A.360 cm$^2$ B. 260 cm$^2$ C. 160 cm$^2$ D. None of these

Answer 1

Hint: We know the area of the circle is 64$\pi $ sq. cms. From here, we can get the radius of the circle
(Since, Area of circle =$\pi \,{{\text{r}}^2}$ )
When we know the radius of circle, we can get the arc length i.e. l (in the fig)
Once we get the arc length, we can get the area of the sector.

Complete step-by-step answer:

It is given that the area of the circle is 64 $\pi $ sq. cm.
But Area of circle = $\pi $r$^2$(Standard formula)
$ \Rightarrow $ $\pi $r$^2$ = 64$\pi $ sq. cms
  r$^2$= 64 sq. cm
 $\therefore $ r = 8 cm
We get the radius of circle as 8cm
It is also given that perimeter of a sector of a circle is 56 cm
Let C be centre of circle
Let AB be length of part of circle
AB=l
Perimeter of sector of circle = AC+ BC+AB
= r+r+l
= 2r+l
But 2r+l = 56 cm
And also r = 8cm
$\therefore $ 2 $ \times $ 8cm + l = 56 cm
$\therefore $ l = 56 cm - 16cm
 = 40 cm
AB = l = 40 cm
We know that area of sector of circle
= $\dfrac{1}{2}$$ \times $ l $ \times $ r
= $\dfrac{1}{2}$$ \times $AB $ \times $r
= $\dfrac{1}{2}$$ \times $40cm $ \times8 $
= 160 sq.cm
Where l = length of arc
 R = radius of circle
$\therefore $ Area of sector of circle = 160 sq. cm
Option (C)

Note: We can also find the area of sector of circle by alternate formula:
If $\theta $ is the angle of the sector
When $\theta $ is in radians:
Area of sector = \[\dfrac{\theta }{2} \times \,{{\text{r}}^2}\]
Area of sector = \[\dfrac{{\theta \times \pi }}{{360}} \times \,{{\text{r}}^2}\]
Where r = radius of circle