A Survey of Right Angles in Convex Heptagons

Posted on 26 July 2014 by RobertLovesPi

In how many ways can different numbers of right angles appear in convex heptagons?

Heptagons do not have to have right angles at all, of course. If a heptagon has exactly one right angle, only one arrangement is possible: a right angle, and six oblique angles (“oblique” means non-right, so it includes both acute and obtuse angles).

With two right angles, there are three possibilities. In the first one shown, the right angles are consecutive. In the second, one oblique angle appears between the right angles. In the third, two oblique angles appear between the right angles. Increasing two oblique angles to three is simply a repeat of the third three-right angle heptagon, so this set stops with exactly three members.

With three right angles, I have found four possibilities: (1) all three right angles appear consecutively; (2) two right angles are consecutive, and one of them has one oblique angle between it and the third right angle; (3) two right angles are consecutive, and each of them has two oblique angles between it and the third right angle; and (4) None of the three right angles is consecutive. In this heptagon, the number of oblique angles which appear between the three different right-angle pairs are one, one, and two.

I have found no others, and, after searching to exhaustion, I do not think any other arrangement of right and oblique angles in a convex heptagon is possible. However, this is a conjecture, not a proof, and may, in fact, be incorrect. If you can provide proof that this listing of possibilities is complete, or a counterexample to show that it is not, please leave a comment with details.

A Survey of Right Interior Angles in Hexagons

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A Survey of Right Interior Angles in Hexagons

A regular hexagon, of course, has no right angles, but irregular, convex hexagons can have one, two, or three right angles.

With one right angle, there is only one basic configuration, but, with two right angles, there are three: the right angles may be consecutive, have one non-right angle between them, or be opposite angles.

There are also three possible configurations with three right angles: the three angles can be consecutive, or two can be consecutive with one non-right angle separating the other right angle from the consecutive pair, or every other angle can be a right angle.

Four right angles cannot exist in a convex hexagon, nor can five, nor, of course, six. Four right interior angles are possible, however, for non-convex hexagons, and, again, there are three possible configurations. In the first, the four right angles are consecutive. In the second, three are consecutive, then a non-right angle separates the fourth right angle from the other three. In the third, there are two pairs of consecutive right angles, with single non-right angles separating the pairs on opposite sides of the hexagon.

A Survey of Right Angles in Convex Pentagons

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A Survey of Right Angles in Pentagons

A regular pentagon, of course, has no right angles, but irregular pentagons can have one, two, or three (but not four, nor five). There are two varieties for both two and three right angles in pentagons — the right angles can be consecutive, or non-consecutive.