If the perimeter of a circle is equal to that of a square, find the ratio of their areas.

medium 2 marks Class 6 · Mathematics · Playing with Numbers

Answer

4:π

The correct answer is 4:π — Therefore, the ratio of the areas is $4 : π$ .

Step-by-step solution

Let the side of the square be $a$ . The perimeter of the square is $4 a$ .
Let the radius of the circle be $r$ . The perimeter (circumference) of the circle is $2 π r$ .
Set them equal: $4 a = 2 π r$ so $r = \frac{2 a}{π}$ .
Area of the circle: $π r^{2} = π (\frac{2 a}{π})^{2} = \frac{4 a ^{2}}{π}$ .
Area of the square: $a^{2}$ .
Take the ratio: $(4 a^{2} / π) : a^{2} = 4 : π$ .
Therefore, the ratio of the areas is $4 : π$ .

Common mistake: Assuming equal perimeters imply equal radii or equal side lengths, leading to an incorrect ratio.

Concept tested: Relationship between perimeter and area of circle and square