Question

Express answers in simplest exact form.

A square with sides of 3sqrt2 is inscribed in a circle. What is the area of one of the sectors formed by the radii to the vertices of the square

Answer 1

For such an inscribed square, the radius "R" of the circle is given by the formula

a²
R² = ----- where a is the length of the side of the rectangle
2

In this case

(3√2)² 9*2
R² = ------ = ----- = 9
2 2

so R = √9 => R = 3

Answer 2

Now radii to the vertices form an angle of 90°

so we have to find the area of a sector of angle 90°

angle 90° 22
area of sector = -------- * πR² = ------ * ----- * 3²
360° 360° 7
1 22 22 * 9 198
= ----- * ----- * 9 = -------- = ------- = 7.07 sq units
4 7 4 * 7 28