The largest triangle inside a circle

Question

A triangle is drawn inside a circle of radius 1.

Which triangle has the largest possible area, and what is that area? Give an argument that does not rely on calculus, and explain why the answer is not the right-angled triangle on a diameter.

Accepted Answer

The equilateral triangle inscribed in the circle, with area 3 root 3 / 4, about 1.30. Corners on the circle. If a corner were strictly inside the circle, you could push it outward (away from the opposite side) until it reached the circle, increasing the height and hence the area. So the largest triangle has all corners on the circle. Each corner opposite the middle of its arc. Fix two corners A and B. The third corner C should be as far as possible from the line AB, which puts it at the midpoint of the larger arc from A to B, so that CA = CB. Apply this to every corner: in the largest triangl…

The largest triangle inside a circle

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