Q19. - Geometry

Question

Find the area of each figure:

A square inscribed in a circle with radius r.

Step-by-Step Solution

Answer

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

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.