Ellipse and equivalent circles

Calculus Level pending

Given an ellipse

$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$

where $a \gt b \gt 0$ , you want to find the radius $r_1$ of the circle which has the same area as the ellipse, and also the radius $r_2$ of the circle that has the same perimeter as the ellipse. Which radius is larger, $r_1$ or $r_2$ ?

r_1 = r_2

None of the other choices

r_1

r_2

1 solution

Chris Lewis
May 24, 2021

Call the ellipse $E$ and say circle $C_1$ has radius $r_1$ and circle $C_2$ has radius $r_2$ .

Among all shapes with the same perimeter, the circle has the largest area. So $\text{Area}\left[C_2 \right] > \text{Area}\left[E \right] = \text{Area}\left[C_1 \right]$

from which we immediately have $r_2>r_1$ .

A very convincing argument. Thanks for sharing this solution.

Hosam Hajjir - 2 weeks, 5 days ago

Ellipse and equivalent circles

1 solution

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