Sample Problem

If the area of the shaded region is, then what is the radius of the whole sector?



The shaded region is the difference of the sector and the right isosceles triangle. Let’s assume the radius of the sector is r, so its area is


If we draw the height of the triangle from the hypotenuse, the height is r/2. Thus, the area of the right triangle is


Thus, the area is represented as


Eliminating  from both sides of the equation, we have , so  = 4 cm. We see that 42/4 also equals 4 so it works.