Polygon Angle Sum Formulas Cheat Sheet

A printable reference covering polygon interior angle sums, exterior angle sums, regular polygon angles, and missing angle equations for grades 5-10.

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Polygon angle sum formulas help students find missing angles in triangles, quadrilaterals, and larger polygons. This cheat sheet gives the main rules for interior angles, exterior angles, and regular polygons in one place. Students need these formulas to solve geometry problems accurately and to recognize patterns as the number of sides changes. The most important formula is the interior angle sum, $S = 180(n - 2)$ , where $n$ is the number of sides. The exterior angles of any convex polygon always add to $360^\circ$ when one exterior angle is taken at each vertex. For a regular polygon, each interior angle is $\frac{180(n - 2)}{n}$ and each exterior angle is $\frac{360}{n}$ .

Key Facts

A polygon with $n$ sides can be divided into $n - 2$ triangles from one vertex.
The sum of the interior angles of an $n$ -gon is $S = 180(n - 2)$ degrees.
The sum of the exterior angles of any convex polygon is $360^\circ$ .
Each exterior angle of a regular $n$ -gon is $\frac{360}{n}$ degrees.
Each interior angle of a regular $n$ -gon is $\frac{180(n - 2)}{n}$ degrees.
An interior angle and its adjacent exterior angle form a straight line, so their measures add to $180^\circ$ .
A triangle has interior angle sum $180^\circ$ because $180(3 - 2) = 180^\circ$ .
A quadrilateral has interior angle sum $360^\circ$ because $180(4 - 2) = 360^\circ$ .

Vocabulary

Polygon: A closed two-dimensional figure made from straight line segments that meet only at their endpoints.
Interior angle: An angle formed inside a polygon by two adjacent sides.
Exterior angle: An angle formed outside a polygon by extending one side of the polygon.
Regular polygon: A polygon with all sides congruent and all interior angles congruent.
Convex polygon: A polygon in which all interior angles are less than $180^\circ$ and no sides bend inward.
n-gon: A polygon with $n$ sides, where $n$ represents any whole number greater than or equal to $3$ .

Common Mistakes to Avoid

Using $180n$ for the interior angle sum is wrong because a polygon with $n$ sides divides into $n - 2$ triangles, not $n$ triangles.
Forgetting that exterior angles add to $360^\circ$ is wrong because the exterior angle sum does not depend on the number of sides for a convex polygon.
Dividing the interior angle sum by $n$ for any polygon is wrong because $\frac{180(n - 2)}{n}$ gives each angle only when the polygon is regular.
Confusing one interior angle with the total interior angle sum is wrong because $S = 180(n - 2)$ gives the combined sum of all interior angles.
Adding an interior angle and its adjacent exterior angle to $360^\circ$ is wrong because they form a straight line, so they add to $180^\circ$ .

Practice Questions

1 Find the sum of the interior angles of a $9$ -gon.
2 A regular polygon has $12$ sides. Find the measure of each interior angle and each exterior angle.
3 The interior angles of a pentagon are $100^\circ$ , $110^\circ$ , $95^\circ$ , $120^\circ$ , and $x^\circ$ . Find $x$ .
4 Explain why the exterior angle sum of a convex polygon stays $360^\circ$ even when the polygon has more sides.

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