Geometry Reference
Polygons
Polygon angle problems usually ask us to connect the number of sides, the sum of the interior angles, one interior angle, or one exterior angle.
Fact Table
| Question | Use | Reminder |
|---|---|---|
| Sum of interior angles. | $(n-2)180^\circ$ | Works for any polygon with $n$ sides. |
| One interior angle of a regular polygon. | $\frac{(n-2)180^\circ}{n}$ | Regular means all sides and angles match. |
| Sum of exterior angles. | $360^\circ$ | Use one exterior angle at each vertex. |
| One exterior angle of a regular polygon. | $\frac{360^\circ}{n}$ | Interior and exterior angles are supplementary. |
| Number of sides from exterior angle. | $n=\frac{360^\circ}{\text{exterior angle}}$ | Often the fastest route. |
Content Formulas
Interior Sum
$$S=(n-2)180^\circ$$
Regular Interior Angle
$$I=\frac{(n-2)180^\circ}{n}$$
Exterior Sum
$$E_{\text{sum}}=360^\circ$$
Regular Exterior Angle
$$E=\frac{360^\circ}{n}$$
If one regular interior angle is given, the matching exterior angle is $180^\circ-I$. Then use $n=\frac{360^\circ}{E}$.
Classic Examples
Interior Angle Sum
Find the sum of the interior angles of a 12-sided polygon.
Geometry MoveUse the interior sum formula with $n=12$.
$$S=(n-2)180^\circ$$
$$=(12-2)180^\circ$$
$$=1800^\circ$$
Number of Sides
A regular polygon has each exterior angle equal to $24^\circ$. How many sides does it have?
Geometry MoveExterior angles of a polygon add to $360^\circ$.
$$n=\frac{360^\circ}{24^\circ}$$
$$=15$$
Interior Angle to Sides
Each interior angle of a regular polygon is $150^\circ$. Find the number of sides.
Geometry MoveConvert the interior angle to an exterior angle first.
$$E=180^\circ-150^\circ$$
$$=30^\circ$$
$$n=\frac{360^\circ}{30^\circ}$$
$$=12$$