Question

What is the area of the largest ellipse you can inscribe into a triangle with side lengths 3,4 , and 5 ? Hint: The largest ellipse you can inscribe into an equilateral triangle is a circle.

Short answer

Therefore, the area of the circle inscribed into this triangle is

$1^2.\pi =\pi$.

Step-by-step solution

Given

The length of the sides of triangle are 3, 4, and 5.

What is the area of the largest ellipse you can inscribe into a triangle.

We know that the largest ellipse that can be inscribed into any kind of triangle is a circle.

Same applies particularly for right triangles, as well.

We also know that,

For any kind of triangle with side lengths a,b, and c, if r is the radius of the circle inscribed into the triangle, and $s=\frac{a+b+c}{2}$, then for the triangle's area applies $p=rs$, which leads to $r=\frac{s}{p}$.

A triangle with side lengths 3, 4, and 5 is a right triangle, so we have

$$\begin{gathered} r=\frac{{\displaystyle \frac{3+4+5}{2}}}{{\displaystyle \frac{3.4}{2}}} \\ r=\frac{6}{6} \\ r=1, \end{gathered}$$

So, the area of the circle inscribed into this triangle is

${\mathrm{I}}^2.\mathrm{\pi }=\mathrm{\pi }$.