The perimeter of a certain sector of a circle of radius 6.5 cm is 31 cm. Find the area of sector.

Given:

Radius of circle = 6.5 cm

Perimeter of sector = 31 cm

Now, Perimeter of sector = 2 × radius + Length of arc

∵ Length of arc = × 2r × 2πr

∴ Perimeter of sector = 2 × r + × 2r × π

= 2r × [1 + × π]

31 = 2 × 6.5 × [1 + × ]

31 = 13 × [1 + × ]

= 1 + ×

- 1 = ×

= ×

θ = × 360 × …………………. (i)

∵ Area of sector = × πr²

∴ using (i), we have

Area = × 360 × × × × 6.5 × 6.5

= 18 × 3.25 = 58.5 cm²

Hence, the area of the sector is 58.5 cm².