What Are Neusis Constructions?

Geometry Level 1

I want to draw a regular pentagon and a regular hexagon such that their perimeter is equal and they all have integer side lengths.

What is the minimum perimeter (each) of these two polygons?

25 30 24 35

1 solution

Sam Bealing
May 8, 2016

Let the side length of the hexagon be $x$ and the pentagon be $y$ :

$5x=6y$

$5 \vert 6y \Rightarrow y \geq 5,6 \vert 5x \Rightarrow x \geq 6$

This is because $\gcd(5,6)=1$ so our minimum perimeter is:

$5 \times 6=\boxed{30}$

Simple standard approach.

@Chung Kevin What are neusis constructions?

Calvin Lin Staff - 5 years ago