Question

Find the area of each figure:

A square inscribed in a circle with radiusr.

Short answer

The area of given figure is $2r^2$square units.

Step-by-step solution

Step 1. Analyze the given information.

Let the side of the square is x and the radius of circle r ,

Then, $DB=r+r=2r$

Step 2. Concept Used.

We can use Pythagoras theorem.

Pythagorean Theorem define that in a right angle triangle, the square of the hypotenuse is equal to the sum of squares of other sides.

In$△BCD$

$D{B}^2=B{C}^2+C{D}^2$

${\left(2r\right)}^2=x^2+x^2$

$2x^2=4r^2$

$x^2=2r^2$

$x=\sqrt{2}r$

Thus, the side of square is$\sqrt{2}r$

Step 3. Find the area of given figure.

Area of square can be found using the formula

$A=s^2$

Where,

s= Side of square

$A={\left(\sqrt{2}r\right)}^2$

$=2r^2$Square units.

Therefore, the area of given figure is $2r^2$square units.