The ratio of the perimeters of two circles is . What is the ratio of their radii?

Question

Homi · Accepted Answer

Understand the Question

The perimeter (circumference) of a circle with radius $r$ is given by $P = 2\pi r$ .
Since $2\pi$ is a constant, the perimeter of a circle is directly proportional to its radius.
Therefore, the ratio of the perimeters of two circles is equal to the ratio of their radii.

Step 1 · Set Up Perimeter Ratio

Let the radius of the first circle be $R$ and the radius of the second circle be $r$ .Given that the ratio of their perimeters is $5 : 4$ :

\begin{aligned} \dfrac{\text{Perimeter of first circle}}{\text{Perimeter of second circle}} &= \dfrac{5}{4} \\[0.6em] \dfrac{2\pi R}{2\pi r} &= \dfrac{5}{4} \end{aligned}

Cancelling the common factor $2\pi$ from the numerator and denominator: $\dfrac{R}{r} = \dfrac{5}{4}$

$\implies R : r = 5 : 4$

Answer

$5 : 4$

Common Mistakes

Confusing Perimeter with Area: Do not confuse perimeter ratio with area ratio. The ratio of perimeters is linear ( $R : r$ ), whereas the ratio of areas involves squaring the radii ( $R^2 : r^2 = 25 : 16$ ).

Question 8