Question

A regular pentagon is inscribed in a circle. find the area of sector which each side of the pentagon subtends at the centre​

Answer 1

Answer:

A regular pentagon is inscribed in a circle,

Total angle subtended by five sides at the centre is 360∘

So angle subtended by one side at the centre is =1/5x360=72∘

The angle subtended by each side at the centre of the circle is 72∘

So as per the property of the circle, the sum of the entire central angle will be 360.

Its now clear that the value of central angle of each triangle will be 72. Therefore the sum of the rest of the angles will be is the sum of two equal angles, hence each angle will be equal to .

The value of the angle OAB is 54.

Side of Pentagon

$a = r \times sin36$

Area of a Regular Pentagon

$A = \frac{ {a}^{2} }{4}. \sqrt{5(5 + 2 \sqrt{5}) }$

$A = \frac{ {(r \times sin36)}^{2} }{4}. \sqrt{5(5 + 2 \sqrt{5}) }$

$A = 0.6 \times {r}^{2}$