If the perimeter 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?

medium 2 marks Class 5 · Mathematics · Does it Look the Same

Answer

4/π

The correct answer is 4/π — Hence the required ratio is $\frac{4}{π}$ .

Step-by-step solution

Let the radius of the circle be $r$ and the side of the square be $s$ .
Equate the perimeters: $2 p i r = 4 s$ .
Solve for $s$ : $s = \frac{π}{2} r$ .
Area of the circle: $A_{c} = π r^{2}$ .
Area of the square: $A_{s} = s^{2} = (\frac{π}{2} r)^{2} = \frac{π ^{2} r ^{2}}{4}$ .
Take the ratio $A_{c} / A_{s} = \frac{π r ^{2}}{π ^{2} r ^{2} /4}$ .
Cancel $r^{2}$ and simplify: $= \frac{4}{p i}$ .
Hence the required ratio is $\frac{4}{π}$ .

Common mistake: Not canceling

r^{2}

or forgetting to square the side

Concept tested: Perimeter equality leads to area ratio