Polygons and Angle Sums
What Is a Polygon?
A polygon is a closed shape made of straight line segments — think triangles, quadrilaterals, pentagons, hexagons, and so on. The number of sides always equals the number of angles (vertices). A polygon is convex if every interior angle is less than 180^{\circ} (no "caving in"), and concave otherwise. We'll focus on convex polygons in this lesson.
The Interior Angle Sum Formula
You already know that a triangle's angles sum to 180^{\circ}. Larger polygons follow a pattern: every time you add a side, the angle sum increases by another 180^{\circ}. This is because any convex polygon with n sides can be split into n-2 non-overlapping triangles by drawing diagonals from a single vertex.
This gives the Interior Angle Sum Formula for a polygon with n sides:
S = (n-2)\times 180^{\circ}For example, a quadrilateral (n=4) has an angle sum of (4-2)\times 180^{\circ} = 360^{\circ}, and a hexagon (n=6) has an angle sum of (6-2)\times 180^{\circ} = 720^{\circ}.
Regular Polygons
A polygon is regular if all of its sides are equal in length and all of its angles are equal in measure (think of a stop sign — a regular octagon). Since a regular polygon's total interior angle sum is shared equally among its n angles, each interior angle measures:
\text{Each interior angle} = \frac{(n-2)\times 180^{\circ}}{n}Exterior Angles Always Sum to 360°
At each vertex of a convex polygon, the exterior angle is supplementary to the interior angle (they form a linear pair along the extended side). No matter how many sides the polygon has, the exterior angles of any convex polygon always add up to exactly 360^{\circ} — think of walking around the perimeter and turning at each corner; by the time you return to your starting point and direction, you've made one full 360^{\circ} turn.
For a regular polygon, each exterior angle therefore measures \frac{360^{\circ}}{n}.
Example
Find the measure of each interior angle and each exterior angle of a regular decagon (a 10-sided polygon).
- Interior angle sum: S = (10-2)\times 180^{\circ} = 8 \times 180^{\circ} = 1440^{\circ}
- Each interior angle (since it's regular, divide equally among 10 angles): \frac{1440^{\circ}}{10} = 144^{\circ}
- Each exterior angle is supplementary to the interior angle, or more directly: \frac{360^{\circ}}{10} = 36^{\circ}
- Check: 144^{\circ} + 36^{\circ} = 180^{\circ}, confirming the interior and exterior angles form a linear pair. ✓
Key Takeaways
- Interior angle sum of an n-gon: S=(n-2)\times 180^{\circ}.
- Regular polygons split that sum evenly: each interior angle is \frac{(n-2)\times 180^{\circ}}{n}.
- The exterior angles of any convex polygon always sum to 360^{\circ}, regardless of the number of sides.
- Each exterior angle of a regular n-gon measures \frac{360^{\circ}}{n}.