Angle Sums and Properties of Polygons

Deriving the Sum of Angles in Polygons

Think about drawing a single diagonal line across a four-sided square; this instantly splits it into two triangles. This is the triangle partitioning principle. Any $n$-sided polygon can be split into triangles by drawing diagonals from a single vertex to all non-adjacent vertices.

The number of triangles created is always exactly two less than the total number of sides. This gives us the causal n-2 formula. Because each triangle contains $180^\circ$, we multiply the number of triangles by $180^\circ$ to derive the total sum.

Worked Example: Sum of Interior Angles

Derive and calculate the sum of the interior angles for a regular octagon.

Step 1: State the formula relating sides to triangles.

• $\text{Sum} = (n - 2) \times 180^\circ$

Step 2: Substitute the number of sides ($n = 8$ for an octagon).

• $\text{Sum} = (8 - 2) \times 180^\circ$
• $\text{Sum} = 6 \times 180^\circ$

Step 3: Calculate the final sum.

• $\text{Sum} = 1080^\circ$

Properties of Regular Polygons

You can sketch a wonky five-sided shape easily, but a perfect mathematical pentagon follows strict rules. A regular polygon is a shape where all sides are equal length and all interior angles are equal. Conversely, an irregular polygon has sides and/or angles that are not all equal.

Regular polygons have highly predictable symmetry properties. A regular polygon with $n$ sides will always have exactly $n$ lines of symmetry and an order of rotational symmetry equal to $n$. If the polygon has an even number of sides (like a hexagon or octagon), its opposite sides are always parallel.

Exterior Angles and the "Gateway" Method

Every interior angle has a corresponding exterior angle. This is the angle formed between one side of a polygon and the straight-line extension of the adjacent side. Note that the exterior angle is not the reflex angle outside the shape.

The sum of exterior angles for any convex polygon is always exactly $360^\circ$, regardless of how many sides it has. Because an interior angle and an exterior angle sit together on a straight line, they are supplementary angles that add up to $180^\circ$.

Worked Example: Calculating Individual Angles

Calculate the size of an individual exterior angle and interior angle for a regular decagon.

Step 1: Write the formula for the exterior angle.

$$\text{Exterior Angle} = \frac{360^\circ}{n}$$

Step 2: Substitute the number of sides ($n = 10$ for a decagon).

• $\text{Exterior Angle} = 360^\circ \div 10$

Step 3: Calculate the exterior angle.

• $\text{Exterior Angle} = 36^\circ$

Step 4: Use the supplementary subtraction rule to find the interior angle.

• $\text{Interior Angle} = 180^\circ - \text{Exterior Angle}$
• $\text{Interior Angle} = 180^\circ - 36^\circ$

Step 5: Calculate the final interior angle.

• $\text{Interior Angle} = 144^\circ$