Chapter 1.5, Problem 50AYU

50. Find the area of the blue shaded region in the figure, assuming the quadrilateral inside the circle is a square.

To determine

To find: The area of the shaded region given in the figure

Answer to Problem 50AYU

Solution:

Area of the shaded region $\text{= }\text{(36□ - 72) square units}$

Explanation of Solution

Given:

The following figure:

Radius of the circle $\text{= 6 units}$

Diameter of the circle $\text{= 12 units}$

Formula used:

Pythagoras Theorem:

$\left(\text{base}\right)\text{² + }\left(\text{height}\right)\text{² = }\left(\text{hypotenuse}\right)\text{²}$

Let the side of the square is $a$. The diagonal divides the square into two right triangles, with the base and height equal to $a$. The diagonal of the square is the diameter of the circle and hence the hypotenuse of the right triangle is 12 units.

Using Pythagoras theorem,

$\left(\text{base}\right)\text{² + }\left(\text{height}\right)\text{² = }\left(\text{hypotenuse}\right)\text{²}$

$\left(a\right)²+\left(a\right)²=\left(12\right)²$

$2a²=144$

$a²=72$

Area of the square $=a²$

$\text{= 72 square units}$

$\text{Area of the shaded region = area of the circle - area of the square}$

$=\square r²-a²$

$=(36\square -72)\text{ square units}$