Interior and exterior angles of polygons

For a polygon with $n$ sides:

Why $(n - 2)$? From one corner you can split any polygon into $n - 2$ triangles, each with angles adding up to 180°. A pentagon splits into 3 triangles: $3 \times 180° = 540°$.

In a regular polygon all sides and angles are equal, so:

Example: a regular 12-sided polygon has exterior angles of $360° \div 12 = 30°$, so each interior angle is $180° - 30° = 150°$.

Using the exterior angle is usually quicker than finding the interior sum and dividing.

Work backwards through the exterior angle.

Example: each interior angle of a regular polygon is 140°. The exterior angle is $180° - 140° = 40°$, so $n = 360° \div 40° = 9$ sides.

Can a regular polygon have interior angles of 125°? The exterior angle would be 55°, and $360 \div 55$ isn't a whole number, so no.

Find the interior sum, then subtract the angles you know.

Example: a pentagon has angles 100°, 110°, 95°, 120° and $x$. The interior sum is 540°, so $x = 540° - 425° = 115°$.

With algebra: a hexagon has angles $x, 2x, 2x, 2x, 2x, 3x$. Then $12x = 720$, so $x = 60°$.

A shape tessellates if copies fit together with no gaps. At each meeting point the angles must add up to 360°. Regular hexagons tessellate because $3 \times 120° = 360°$; regular pentagons don't, because 108° doesn't divide into 360° exactly.

• Using $n \times 180°$ instead of $(n - 2) \times 180°$.
• Dividing the interior sum by $n$ for an irregular polygon (only works for regular ones).
• Forgetting that exterior angles add up to 360° whatever the number of sides.

1. Find the sum of the interior angles of a 7-sided polygon.
2. Find each interior angle of a regular 20-sided polygon.
3. A regular polygon has exterior angles of 24°. How many sides does it have?
4. A quadrilateral has angles 85°, 95° and 110°. Find the fourth.

Answers: 1. $5 \times 180° = 900°$. 2. Exterior $18°$, interior $162°$. 3. $360 \div 24 = 15$. 4. $360° - 290° = 70°$.

More practice in our angles in polygons questions.