Polygons

A polygon is a 2-dimensional many sided shape. If all the sides are the same length it is a regular polygon, if not it is an irregular polygon.

A polygon is usually thought of as a convex shape (roughly circular), but they don’t have to be, they can fold in on themselves and have intersecting lines. The only requirements are that they must have straight sides and form a closed 2D shape. So all these shapes are valid polygons:

Lets forget about these though and focus on convex polygons. The most visually satisfying are regular polygons. Here are the first nine:

An n-sided polygon has n interior angles, as shown below. If it is a regular polygon, each interior angle is the same, look at the regular polygons above.

Each interior angle has an exterior angle ‘partner’.

The sum of each interior angle and exterior angle is 180°, so:

e₁ + i₁ = 180°

e₂ + i₂ = 180°

Again, if it is a regular polygon, all the exterior angles are equal.

The sum of the exterior angles is 360°. Imagine walking round the polygon say anti-clockwise. Each time you reach a corner you have to turn anti-clockwise through the exterior angle. By the time you get back to the starting point, you have made one full turn, 360°, and turned through all of the exterior angles added together. Hence we can say for any polygon:

e₁ + e₂ + e₃ + …. + e_n = 360°

where e_n is the nth exterior angle and …. denotes all other exterior angles between 3 and n. It can be written more compactly as:

∑e = 360°

where the ∑ symbol means sum of. For a regular polygon each e is the same so:

e + e + e + …. + e = 360°

which can be written:

n × e = 360°

rearranging to make e the subject gives:

e = 360°/n

This immediately enables us to write down a formula for the interior angle, i, of a regular polygon starting from the fact that the sum of the interior and exterior angles is 180°:

i + e = 180°

rearranging to make i the subject:

i = 180° – e

substituting for e:

i = 180° – 360°/n

factorising by pulling out the common factor of 180°:

i = 180°(1-2/n)

pulling 1/n out of the bracket:

i = 180°×(n-2)/n

which is a formula that enables us to calculate the interior angle for any regular polygon. So for example the interior angle of a regular pentagon is:

i = 180°×(5-2)/5 = 36° × 3 = 108°

Hiding in this formula is the more general result for the sum of the interior angles (multiplying both sides by n):

n×i = ∑i = 180°×(n-2)

This result holds for any polygon, regular or irregular.

Both versions of this formula need to be learnt and applied for GCSE! It is much much easier to do this if you have a deep understanding of where the formula come from.

Here is a sample question (EdExcel Nov 2014 1MAO/1H):

Here is my worked solution.

There are other ways to get the answer, such as considering the polygon CKFED, or by calculating the external angle of the octagon. So long as you lay the answer out neatly, explaining what you are doing it doesn’t matter which approach you use.