Interior angles are the angles inside a polygon, formed where two sides meet at each vertex. Knowing their total helps you solve for missing angles, classify shapes, and understand why different polygons fit together in patterns. The key idea is that any polygon can be divided into triangles, and each triangle has an angle sum of 180 degrees.
This makes polygon angle sums predictable instead of something to memorize one shape at a time.
To find the sum of the interior angles of an n-sided polygon, draw diagonals from one vertex to split the polygon into n - 2 triangles. Since each triangle contributes 180 degrees, the interior angle sum is (n - 2)180 degrees. If the polygon is regular, all its interior angles are equal, so each angle is the total sum divided by n.
These formulas are useful in geometry proofs, architecture, tiling patterns, and computer graphics.
Key Facts
- Interior angle sum of an n-sided polygon: S = (n - 2)180 degrees.
- A polygon with n sides can be divided into n - 2 triangles from one vertex.
- Each interior angle of a regular n-gon: A = ((n - 2)180 degrees) / n.
- A triangle has interior angle sum 180 degrees.
- A quadrilateral has interior angle sum (4 - 2)180 degrees = 360 degrees.
- Number of sides from a known interior angle sum: n = S / 180 degrees + 2.
Vocabulary
- Polygon
- A closed flat shape made of straight line segments.
- Interior angle
- An angle inside a polygon formed by two sides that meet at a vertex.
- Regular polygon
- A polygon with all sides equal in length and all interior angles equal in measure.
- Diagonal
- A line segment that connects two nonadjacent vertices of a polygon.
- Vertex
- A corner point where two sides of a polygon meet.
Common Mistakes to Avoid
- Using n times 180 degrees for the interior angle sum is wrong because a polygon with n sides divides into n - 2 triangles, not n triangles.
- Forgetting to divide by n for a regular polygon gives the total angle sum instead of the measure of one interior angle.
- Using the regular polygon formula on an irregular polygon is wrong because irregular polygons do not have equal interior angles.
- Counting triangles incorrectly from one vertex leads to the wrong sum because only diagonals to nonadjacent vertices form the n - 2 triangle pattern.
Practice Questions
- 1 Find the sum of the interior angles of a 9-sided polygon.
- 2 A regular polygon has 12 sides. Find the measure of each interior angle.
- 3 A polygon is divided from one vertex into 5 triangles. Explain how many sides the polygon has and how you know.