The perimeter of a certain sector of a circle of radius 5.6 m is 20 m

Q) The perimeter of a certain sector of a circle of radius 5.6 m is 20.0 m. Find the area of the sector.

Ans:

Given that the radius = 5.6 m

In a circle, perimeter of the sector is given by, P = 2 r + L

Here, L is the arch length of a sector

We are given the values: P = 20 m, r = 5.6 m,

∴ 20 = 2 x 5.6 + L

∴ L = 20 – 11.2 = 8.8 m

Next, we need to find area of the sector

and area of circle’s sector is given by, A = $\frac{1}{2}$ r L (check derivation below)

By substituting values of r and L, we get:

A = $\frac{1}{2}$ x (5.6) (8.8)

= 5.6 x 4.4 = 24.64 m²

Therefore, The area of the sector is 14 m²

[ Derivation: Length of the arch is given by, L = 2 π r $(\frac{\theta}{360})$

hence $\frac{1}{2} L = \pi r (\frac{\theta}{360})$ ……………. equation (i)

Next, Area of the circle’s sector is given by, A = π r ² $\frac{\theta}{360}$

∴ A = r (π r $\frac{\theta}{360}$ )

By substituting values of L from equation (i), we get:

A = r ( $\frac{1}{2}$ L)

∴ A = $\frac{1}{2}$ r L ]

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