Question

The perimeter of a certain sector is equal to the length of Arc of a semicircle having the same radius. Express the angle of the sector in degrees, minutes and seconds.

Answer 1

To solve this problem let us draw a circle of radius 'r'. now mark a sector as shown in fig. above. Let us assume that the arc lenght of sector is ''.

and the angle between them is '' as also shown in fig.
NOTE- '' is in radians

and we know that the arc lenght is equal to

Now let us assume that perimeter of sector is
so, =
=
=
Now as given in the question there is a another semi-circle with same radius, so lets assume that perimeter of semi-circle is .
=
And it is also given in question that =
therefore put values of and in above equation
=
=
= radians
but we need the solution in degree, minute and second
we know