# Least possible area of a triangle with vertices on…

Assume you have a regular polygon of $$n$$ sides and its circumcircle $$(n>3)$$. Assume that $$A,B$$ and $$C$$ are $$3$$ different vertices of the polygon. Is the triangle $$ABC$$ with least possible area the one that is formed by $$3$$ consecutive vertices? Can this be applied to a polygon of any number of sides? What would be the traingle with most area?

A regular convex polygon $$P$$ has a circumscribed circle. A triangle whose vertices are a subset of the vertices of $$P$$ has the same circumscribed circle. The area $$a$$ of a triangle with angles $$\alpha, \beta, \gamma$$ and circumscribed circle with radius $$r$$ is [ref] $$$$a=2r^2 sin(\alpha) \sin(\beta) \sin(\gamma)\,.$$$$
The radius $$r$$ is the same for all triangles whose vertices are a subset of the vertices of $$P$$. Hence, the triangle with the smallest area is the triangle that minimizes $$sin(\alpha) \sin(\beta) \sin(\gamma)$$ and the triangle with the largest area is the triangle that maximizes $$sin(\alpha) \sin(\beta) \sin(\gamma)$$.
Every angle in a triangle is smaller than 180° and the sum of the three angles is 180°. Looking at $$\sin(x)$$ on the interval [0°,180°], one can see that one minimizes $$sin(\alpha) \sin(\beta) \sin(\gamma)$$ when choosing one angle to be close to 180° as possible and the other two angles as small as possible. The largest angle formed by a triplet of vertices of a regular polygon is $$\angle ABC$$, where $$A, B, C$$ are consecutive vertices. The smallest angle formed by a triplet of vertices of a regular polygon is $$\angle BAC$$. So a triangle whose vertices are a subset of the vertices of a regular convex polygon has a minimal area if its vertices $$A,B,C$$ are consecutive vertices of that polygon.
• Unfortunately, the proof in my answer works only for triangles, because every triangle has a circumscribed circle that touches all of its vertices. If you take 5 consecutive vertices of a regular convex $n$-polygon ($n>5$) to construct a pentagon that pentagon will not be regular and may be that some of the pentagon's vertices do not lie on its circumscribed circle. – Alice Schwarze Oct 14 '18 at 3:35