Question

If the circumference of a circle is equal to the perimeter of a square what is the ratio of the area of the circle to the area of the square?

Answer 1

Hint:First assume the perimeter of the square and circle and then equate the perimeters and then find the value of the sides \[s\] of the square and then equate the area of the square and circle. The formula for the square and circle as:
Area of square: \[{{s}^{2}}\]
Area of circle: \[\pi {{r}^{2}}\] (\[r\]: radius)

Complete step by step solution:
As given in the question, the circumference of the circle is equal to the value of the perimeter of the square, both square and circumference are the measurement of the outer boundary of the figure. Let us see the perimeter of the square which can be defined by the formula as:
\[\Rightarrow 4s\]
Whereas the perimeter of the circle is given as:
\[\Rightarrow 2\pi r\]
Now equating the perimeter of the square and circle together, we get the value of the sides of the square as:
\[\Rightarrow 4s=2\pi r\]
\[\Rightarrow s=\dfrac{2\pi r}{4}\]
\[\Rightarrow s=\dfrac{\pi r}{2}\]
Now as we have gotten the value of the sides of the square in terms of radius, we now equate the area of the square with the area of the circle and write the equation as:
Area of square \[=\] Area of circle
Placing the values in the above equation, we get the area of the circle to the area of the square as:
\[\Rightarrow {{\left( \dfrac{\pi r}{2} \right)}^{2}}=\pi {{r}^{2}}\]
\[\Rightarrow \dfrac{\pi {{r}^{2}}}{4}=\pi {{r}^{2}}\]
\[\Rightarrow \pi {{r}^{2}}\times \dfrac{4}{\pi {{r}^{2}}}\]
\[\Rightarrow \dfrac{4}{1}\]
Therefore, the ratio of the area of the circle and square as \[4:1\].

Note:
The circumference is same as the perimeter but for circular objects whereas both the perimeter and circumference is same as both of them are the measurement of the outer length of the object.