Question

the perimeter of a sector of a circle is equal to the length of the arc of a semicircle having the same radius then the angle of sector is

Answer 1

AnswEr:

Let r be the radius of the circle and ∅ be the angle of the sector.

Then,

$\qquad \sf \: Perimeter \: of \: the \: sector = 2r + r \: \theta$

Length of the arc of a semi-circle of radius r = πr

It is given that -

$\qquad \sf \: 2r + r \: \theta = \pi \: r \\ \\ \implies \sf \: 2 + \theta = r \\ \\ \\ \implies \tt \: \theta = (\pi - 2) \: \sf radians \\ \\ \\ \longrightarrow \tt \: (\pi - 2) \times \frac{180}{\pi} \degree \\ \\ \\ \longrightarrow \tt \: 180 \degree - ( \frac{360}{\pi} ) \degree \\ \\ \\ \longrightarrow \tt \: 180 - 114 \degree \: 32'44' \\ \\ \\ \longrightarrow \tt \blue {65 \degree \: 27'16'}$