Construct a regular hexagon of specific height?

Question

Is it possible to construct a hexagon of particular height, meaning distance between the faces (not vertices)? I have seen various methods of constructing a hexagon (ruler and compass only) which are based on the length of a side, but have not seen any methods based on the distance between faces.

Just to make the question perfectly clear: Given a line segment of fixed length, show step-by-step how to construct a hexagon using only a straightedge and compass for which that line segment is the distance between any two faces of the hexagon.

Answer 1

1. Start with drawing the two parallel (infinite) lines you want to be opposite sides of your regular hexagon.

2. Draw a line intersecting them at a 60° angle, by constructing an equilateral triangle with one of two parallel lines coincident with a side of the triangle. This will become a diameter in the hexagon. This can be done by setting the compass at width greater than that between the two lines, then drawing a circle with its diameter along one of the two parallel lines. Then set the compass where the circle intersects the parallel line which lies along its diameter, and draw another circle. Where the two circles intersect beyond the other parallel line is the apex of the equilateral triangle. Draw one side of this triangle between the two parallel lines, thereby forming a 60-degree angle.

3. Bisect the 60-degree line segment to find the center of both the line segment and the hexagon. Note that where the line segment intersects with the two parallel lines are two of the vertices of the hexagon.

4. Draw the circle centered at the midpoint you just found, which passes through the intersections between the sides and the diameter. This becomes the circumcircle of your hexagon.

5. Find the two last corners by marking equal distances from the existing vertices using the compass. Note that because a hexagon contains 6 equilateral triangles, the distance from the center to each vertex will be the same as the length of each side.

Answer 2

I assume you wish to construct a regular hexagon, i.e. a hexagon where all the sides have the same length. Also, from your question I assume that you know how to construct a hexagon with a given side length. Thus I will show you how to compute the side length of a regular hexagon of given height.

Observe that the diagonals of a hexagon cut it into $6$ regular triangles, all with the same side length.

In particular, since the height of an equilateral triangle of side $s$ is $\frac{\sqrt{3}}{2} s$, it follows that the height of a regular hexagon of side $s$ is $\sqrt{3} s$. Thus a regular hexagon of prescribed height $h$ will have side length $$ \frac{h}{\sqrt{3}} $$

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To construct a segment of the required length with compass and straight-edge you can for example construct an equilateral triangle with side $$ \frac{2}{3} h $$ because its height will have the required length.

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