Regular polygons: angles and two different radii

If every corner of a planned octagonal shape must lie 2 m from its centre, the midpoint of each side will be closer. Confusing the two radii gives the wrong distance between opposite sides.

A regular polygon has all sides equal and all interior angles equal. Its circumcircle passes through the vertices; its incircle touches the sides. The circles share a centre but have different radii. You will use circle angles, similarity and sine and cosine.

Not every closed chain is our figure

Here a polygon is simple: its sides form a closed boundary without self-intersections, and adjacent sides are not collinear. An n-gon has n vertices and n sides; its perimeter is the sum of their lengths.

A convex polygon lies on one side of every line containing one of its sides. Every interior angle is then below 180°. A concave simple figure has an interior angle above 180°; it is another simple shape, rather than a self-crossing star. OpenStax on polygon types.

Equal sides alone do not guarantee regularity: a rhombus with 60° and 120° angles is not a square. Equal angles alone are also insufficient: a rectangle with unequal adjacent sides is not regular. Both conditions are needed.

Where the angle sum comes from

Draw every diagonal from one vertex of a convex n-gon. They divide the interior into n−2 triangles without overlap. Their angles together make the polygon’s angles, so:

$S_{\text{angles}}=(n-2)\cdot180\text{°}.$

This gives 720° for a hexagon and 1080° for an octagon. Dividing by n to obtain one angle is valid only when the angles are equal. Each interior angle of a regular n-gon is 180°−360°/n. The exterior turn when walking around it is 360°/n, and the turns total 360°.

For an octagon the interior angle is 135° and the exterior turn 45°. The angle between radii to adjacent vertices is also 45°, but at the centre rather than a vertex: this is the central angle.

Convex and concave pentagons

n = 6, R = 2, r ≈ 1.732

n = 8, R = 2, r ≈ 1.848

Half of the central triangle

First: the convex pentagon at left is split into three triangles; the orange dot at right marks a concave vertex. Next are n=6 and n=8 at a shared scale with R=2. The outer circle passes through vertices; the inner touches the sides. Orange OA is R, vertical OM is r; M is the midpoint of AB. Last: AOM for n=8 at its own uniform scale, α=180°/8=22.5°, OA=R, OM=r, AM=s/2.

Two circles and one right triangle

Let O be the shared centre, A and B adjacent vertices, and M the midpoint of AB. Then OA=OB=R, OM=r, AB=s and OM⊥AB. Segment OM is an apothem; its length is the inradius.

Why a shared centre? Find it at the intersection of two adjacent angle bisectors. Equal sides and angles then give consecutive congruent triangles from that point to adjacent vertices. Distances to all vertices are therefore equal, as are the altitudes to the sides. One circle passes through the vertices and another touches the sides. The radius to a vertex is longer than the perpendicular to a side, so R>r.

Altitude OM halves the central angle: ∠AOM=180°/n. In right triangle AOM, the hypotenuse is R, adjacent leg r, and opposite leg s/2. Thus:

$s=2R\sin\left(\frac{180\text{°}}{n}\right),\qquad r=R\cos\left(\frac{180\text{°}}{n}\right).$

Here n is an integer at least 3, and the sine and cosine arguments are in degrees. Also, R=s/(2sin(180°/n)) and perimeter P=ns. WYDOT, Plane Geometry, E-4.

Lay out an octagon around its centre

With n=8 and R=2 m, the central angle is 45° and its half is 22.5°. Hence s=4sin22.5°≈1.531 m and r=2cos22.5°≈1.848 m.

Opposite vertices are 4 m apart; opposite sides are 2r≈3.696 m apart. Because eight is even, opposite vertices and sides pair symmetrically about the centre. An arbitrary odd n has no equivalent pair of “opposite sides”.

For the layout, mark eight directions at 45° intervals from O and set off R=2 m on each. Join adjacent points by segments, not arcs. The numbers are rounded to explain the model; actual layout precision depends on the tools.

Extra: a hexagon without a protractor

A regular hexagon has central angle 60°. The triangle between adjacent radii is equilateral, so the side equals R. On a given circle, step off successive chords of length R with a compass. Six equal steps close an exact hexagon in the ideal model. Euclid IV.15.

A numerical formula for arbitrary n and an exact construction using only unmarked straightedge and compass are different tasks. We have shown a specific possible construction; this lesson does not prove the general constructibility classification.