Polygons and their angle sums

Naming polygons, convex and non-convex polygons, the angle sum formula (n − 2) × 180°, interior and exterior angles of regular polygons.

Polygons

A polygon is a closed shape with straight sides: pentagon ($5$), hexagon ($6$), heptagon ($7$), octagon ($8$), nonagon ($9$), decagon ($10$), dodecagon ($12$). A regular polygon has all sides and all angles equal.

Angle sum

Angle sum of an n-sided polygon

$$S = (n - 2) \times 180^\circ$$

Angle sums grow by 180° per side

Drag the point to change the number of sides.

Example 1●

Find the size of each interior angle of a regular octagon.

$S = 6 \times 180 = 1080^\circ$

$1080 \div 8 = 135^\circ$

Example 2●●

Each interior angle of a regular polygon is $156^\circ$. How many sides does it have?

Each exterior angle is $180 - 156 = 24^\circ$.

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

Flashcards

Angle sum formula

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

Sum of exterior angles

$360^\circ$

Regular hexagon interior angle

$120^\circ$

Practice

Find each interior angle of a regular octagon.
A regular polygon has interior angles of $156^\circ$. How many sides?
Find the angle sum of a pentagon.
Find each interior angle of a regular hexagon.
Find each exterior angle of a regular decagon.
Find the angle sum of a dodecagon ($12$ sides).
A pentagon has angles $100^\circ, 105^\circ, 110^\circ, 115^\circ$ and $x^\circ$. Find $x$.
A polygon has angle sum $1260^\circ$. How many sides?