A polygon is a plane shape (two-dimensional) with straight sides. Examples include triangles, quadrilaterals, pentagons, hexagons and so on.

Regular

A “Regular Polygon” has:

  • all sides equal and
  • all angles equal.

Otherwise it is irregular.

Regular Pentagon Irregular Pentagon

Internal Angles

The internal angle of a regular polygon with “n” sides can be calculated using:

(n-2) × 180° / n

For example the interior angle of an octagon (8 sides) is:

(8-2) × 180° / 8 = 6×180°/8 = 135°

And for a square it is (4-2) × 180° / 4 = 2×180°/4 = 90°

External Angle

The External Angle and Internal Angle are measured from the same line, so they add up to 180°.

So the external angle is just 180° – Internal Angle

The internal angle of this octagon is 135°, so the external angle is 180°-135° = 45°

And a hexagon’s internal angle is 120°, so the external angle is 180°-120° = 60°

Diagonals

Polygons (except the triangle) have diagonals (lines from one corner to another, but not the sides).

The number of diagonals is equal to n(n – 3) / 2.

Examples:

  • a square has 4(4-3)/2 = 4×1/2 = 2 diagonals
  • an octagon has 8(8-3)/2 = 8×5/2 = 20 diagonals.

(Note: This works for regular and irregular polygons)

“Circumcircle, Incircle, Radius and Apothem … “

“Circumcircle, Incircle, Radius and Apothem … “

Sounds quite musical if you repeat it a few times, but they are just the names of the “outer” and “inner” circles (and each radius) that can be drawn on a regular polygon like this:

The “outside” circle is called a circumcircle, and it connects all vertices (corner points) of the polygon.

The “inside” circle is called an incircle and it just touches each side of the polygon at its midpoint.

The radius of the circumcircle is also the radius of the polygon.

The radius of the incircle is the apothem of the polygon.

Formulas

If we take just one “sector” of an “n“-sided regular polygon, and then cut it in half, we end up with a small triangle that has all the important information in it:

(Note: The angles are in radians, not degrees)

The small triangle is right-angled and so we can use sine, cosine and tangent to show how the side, radius, apothem and “n” are related:

sin(π/n) = (Side/2) / Radius Side = 2 × Radius × sin(π/n)
cos(π/n) = Apothem / Radius Apothem = Radius × cos(π/n)
tan(π/n) = (Side/2) / Apothem Side = 2 × Apothem × tan(π/n)

There are a lot more relationships like those (most of them just “re-arrangements”), but those will do for now.

Area

The area is now easy to work out … just add up the areas of all the little triangles!

The area of a triangle is half of the base times height, so:

Area of Small Triangle = ½ × Apothem × (Side/2)

And we know (from the “tan” formula above) that:

Side = 2 × Apothem × tan(π/n)

So: Area of Small Triangle = ½ × Apothem × (Apothem × tan(π/n))
= ½ × Apothem2 × tan(π/n)

And there are 2 such triangles per side, or 2n for the whole polygon:

Area of Polygon = n × Apothem2 × tan(π/n)

Quite a simple formula, really!

Other Area Formulas

If you don’t know the apothem, here is the same formula re-worked for radius and side:

Area of Polygon = ½ × n × Radius2 × sin(2 × π/n)

Area of Polygon = ¼ × n × Side2 / tan(π/n)

A Table of Values

We can use the formulas to make a table of the Side, Apothem and Area for several polygons, using a Radius of “1”:

Name Sides
(n)
Shape Interior Angle Radius Side Apothem Area
Triangle
(or Trigon)
3 60° 1 1.732…
(√3)
0.5 1.299…
(¾√3)
Quadrilateral
(or Tetragon)
4 90° 1 1.414…
(√2)
0.707…
(1/√2)
2
Pentagon 5 108° 1 1.176… 0.809… 2.378…
Hexagon 6 120° 1 1 0.866…
(½√3)
2.598…
((3/2)√3)
Heptagon
(or Septagon)
7 128.571° 1 0.868… 0.901… 2.736…
Octagon 8 135° 1 0.765… 0.924… 2.828…
(2√2)
Pentacontagon 50 172.8° 1 0.126… 0.998… 3.133…

Graph

And here is a graph of the table above, but with number of sides (“n”) from 3 to 30.

Notice that as “n” gets bigger, the Apothem is tending towards 1 (equal to the Radius) and that the Area is tending towards π = 3.1416…, just like a circle.

What is the Side length tending towards?

Side vs Radius comparison btwn Hexagon & Octagon:

Hexagon
2′ sides = 2′ radial
4′ sides = 4′ radial
6′ sides = 6′ radial
8′ sides = 8′ radial
10′ sides = 10′ radial
12′ sides = 12′ radial
16′ sides = 16′ radial

Octagon:
2′ sides = 2.7′ radial
4′ sides = 5.2′ (+/-?) radial
6′ sides = 7.9′ radial
8′ sides = 10.5′ radial
10′ sides = 13.1′ radial
12′ sides = 15.7(?) radial