How to find the area of a circle with a square drawn inside it


In the previous post we explored finding the area of a circle drawn inside a square. In this post we will find the area of a circle having a square drawn inside it. Consider there is a circle having a square drawn inside it and the side length of the square is given.


Look at the diagram given below to understand the problem better. The area of the circle is unknown and we even don't know its radius. What we are given with is only the side length of the square inside the given circle.


The side length "s" of the square = 13 cm


A circle with a square inside it. Tips to find the area of this circle.




We are given with the side length of the circle, the only piece of information to solve for area of the circle. We will go through the following steps to find the circle area:


  • Find the diagonal of the square from its sides by using the Pythagoras Theorem.
  • The diagonal of the square drawn inside a circle is always equal to the diameter of the circle, hence equate the diagonal of the square to the diameter of the circle.
  • Once we know the diameter of the circle, divide it by 2 to get the radius of the circle.
  • Use the formula pi x radius x radius  to find the area of the circle.


So, let's follow all the above steps to to find the area of the given circle:


Find the diagonal "d" of the square from its given side length "s" = 13 cm, using Pythagorean Theorem


Finding diagonal of a square using Pythagorean Theorem



Hence the diagonal of the square is found to be equal to 18.38 cm.


Now as said earlier, the diameter of a circle is always equal to the diagonal of the square drawn inside it.


So the diameter of the circle is equal to 18.38 and now we can find the radius "r" of the circle by dividing its diameter by 2 as shown below:


Radius of the circle r = 18.38 2 = 9.19 cm


Once you know the radius "r = 9.19", use it in the formula to find the area of a circle to solve the problem.


Area of the circle A = pi x rad. x rad.  

A = 3.14 x 9.19 x 9.19

A = 3.14 x 84.46

A = 265.20


Hence the area of the circle, with a square of side length equal to 13cm, is found to be 265.20 sq.cm. You can try the same kind of problems with the different side lengths of square drawn inside the circle.



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