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.
Solved Examples
Jump directly to a worked example by polygon angle task.
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.
Solution Steps
- Use the interior angle sum formula.
- Substitute $n=12$.
- State the sum: $1800^\circ$.
$$S=(n-2)180^\circ$$
$$=(12-2)180^\circ$$
$$=1800^\circ$$
Number of Sides
A regular polygon has each exterior angle equal to $24^\circ$. How many sides does it have?
Solution Steps
- Divide the exterior angle sum by one exterior angle.
- State the number of sides: $15$.
$$n=\frac{360^\circ}{24^\circ}$$
$$=15$$
Interior Angle to Sides
Each interior angle of a regular polygon is $150^\circ$. Find the number of sides.
Solution Steps
- Convert the interior angle to its supplementary exterior angle.
- State the exterior angle.
- Divide $360^\circ$ by the exterior angle.
- State the number of sides: $12$.
$$E=180^\circ-150^\circ$$
$$=30^\circ$$
$$n=\frac{360^\circ}{30^\circ}$$
$$=12$$