# given: segment area = triangle area for a sector with radius r Find theta (angle for the sector)?The answer is 109 dgree. I managed to solve by write equation for area triangle = area segment I...

given: segment area = triangle area for a sector with radius r

Find theta (angle for the sector)?

The answer is 109 dgree. I managed to solve

by write equation for area triangle = area segment

I have: theta = 2 sin theta. How do i solve the rest?

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You should remember the formula that gives the area of a segment of a circle such that:

`A = r^2/2*(pi/180^o*alpha - sin alpha)`

`A = (r^2/2)(alpha - sin alpha)`

r represents the radius of circle

`alpha`  is central angle in degrees

The problem provides the information that segment area is equal to triangle area for the sector having the radius r, hence, you should evaluate the area of triangle using the following formula such that:

`A_Delta = (r*r*sin alpha)/2`

Setting the equation of segment area equal to triangle area yields:

`(r^2/2)sin alpha = (r^2/2)(alpha - sin alpha)`

Reducing by `(r^2/2)`  yields:

`sin alpha = (alpha - sin alpha) => 2sin alpha = alpha`

Considering the functions `y = 2 sin alpha`  and `y = alpha` , you should draw the graphs of the functions such that:

You should know that the sine function  reaches its maximum at alpha = pi/2.

Notice that the graph of function `y = 2 sin alpha`  intersects the red line that represents the graph of the function `y=alpha `, somewhere between `pi/2`  and `pi` , nearly `pi/2` .

Hence, using the graphical method yields that `theta in (pi/2, pi). `