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Polygon Interior and Exterior Angles
Polygon Angle Formulas
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Polygons appear everywhere in geometry, design, engineering, and computer graphics, so understanding their interior and exterior angles is a basic skill with many uses. The angle patterns in polygons help students classify shapes, solve for unknown measures, and recognize regularity and symmetry. As the number of sides increases, the angle relationships follow simple formulas that make even large polygons manageable. Learning these formulas builds a strong bridge between arithmetic, algebra, and geometry.
Interior angles are the angles inside a polygon, while exterior angles are formed when one side is extended at a vertex. For any polygon with n sides, the sum of the interior angles is found by dividing the shape into triangles, which leads to the formula (n - 2) x 180 degrees. The sum of one exterior angle at each vertex is always 360 degrees, no matter how many sides the polygon has. In regular polygons, where all sides and angles are equal, each interior and exterior angle can be found by dividing these totals evenly.
Key Facts
- Sum of interior angles of an -sided polygon: degrees
- Sum of one exterior angle at each vertex of any polygon: 360 degrees
- Each exterior angle of a regular n-gon: 360/n degrees
- Each interior angle of a regular n-gon: [(n - 2) x 180]/n degrees
- Interior angle + adjacent exterior angle = 180 degrees
- Triangle: 180 degrees, quadrilateral: 360 degrees, pentagon: 540 degrees, hexagon: 720 degrees, heptagon: 900 degrees, octagon: 1080 degrees
Vocabulary
- Polygon
- A polygon is a closed flat figure made of straight line segments.
- Interior angle
- An interior angle is an angle formed inside a polygon by two adjacent sides.
- Exterior angle
- An exterior angle is an angle formed outside a polygon when one side is extended.
- Regular polygon
- A regular polygon has all sides equal in length and all interior angles equal in measure.
- Vertex
- A vertex is a corner point where two sides of a polygon meet.
Common Mistakes to Avoid
- Using n x 180 for the interior angle sum, which is wrong because a polygon can be divided into n - 2 triangles, not n triangles.
- Forgetting that exterior angles must be taken one at each vertex in the same direction, which is why their total is 360 degrees.
- Confusing the sum of all interior angles with one interior angle of a regular polygon, which leads to answers that are far too large.
- Mixing up interior and exterior angles in regular polygons, even though they are supplementary and must add to 180 degrees.
Practice Questions
- 1 Find the sum of the interior angles of a 9-sided polygon.
- 2 A regular hexagon has all exterior angles equal. Find the measure of one exterior angle and one interior angle.
- 3 Explain why the sum of the exterior angles of any polygon is always 360 degrees, even when the polygon has many sides.