Polygons and Interior Angles

What you will learn

• Name polygons by their number of sides.
• Find the sum of the interior angles of any polygon.
• Find the interior and exterior angles of a regular polygon.

Before you start: you should know that the angles of a triangle add to $180^\circ$ and those of a quadrilateral to $360^\circ$. See Triangles and Quadrilaterals and Their Properties.

The idea

A polygon is a flat shape with straight sides that closes up. Its names come from Greek: penta (five), hexa (six), hepta (seven), octa (eight), nona (nine), deca (ten) and dodeca (twelve). In a regular polygon all the sides are equal and all the angles are equal.

Why $(n-2) \times 180^\circ$?

Pick one corner and draw diagonals (straight lines between corners that are not neighbours) from it to every other corner. A polygon with $n$ sides is cut into $n - 2$ triangles, and each triangle has angles adding to $180^\circ$.

So the interior angles (the ones inside) add to $(n - 2) \times 180^\circ$. For a pentagon that is $3 \times 180 = 540^\circ$.

Regular polygons

In a regular polygon every interior angle is the same, so share the total equally: $\dfrac{(n-2) \times 180}{n}$.

Exterior angles

Extend one side at each corner. The angle between the extension and the next side is an exterior angle. As you walk round the shape you turn through one full turn, so the exterior angles of any polygon add to $360^\circ$. In a regular polygon each is $360 \div n$, and each interior angle is $180^\circ$ minus it.

Worked examples

Example 1: Sum of angles

Find the sum of the interior angles of a decagon.

1. A decagon has $n = 10$ sides.
2. $(10 - 2) \times 180 = 8 \times 180 = 1440$.

Answer: $1440^\circ$

Example 2: A regular nonagon

Find each interior angle of a regular nonagon.

1. Sum: $(9 - 2) \times 180 = 1260$.
2. Each angle: $1260 \div 9 = 140$.
3. Check with exterior angles: $360 \div 9 = 40$, and $180 - 40 = 140$.

Answer: $140^\circ$

Example 3: Counting sides

Each exterior angle of a regular polygon is $24^\circ$. How many sides does it have?

1. $n = 360 \div 24 = 15$.

Answer: $15$ sides

Tip: exterior angles are often the quickest route. They always total $360^\circ$, whatever the number of sides.

Why it matters

Tilers and designers need to know which polygons fit together without gaps. Regular hexagons do, because three angles of $120^\circ$ make exactly $360^\circ$, which is why honeycomb and bathroom tiles use them. Pentagons alone cannot tile a floor, since $108^\circ$ does not divide $360^\circ$.