Polygons & Interior Angles Reference Cheat Sheet

A printable reference covering polygon names, interior angle sums, regular polygon angles, and exterior angles for grades 6-8.

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This cheat sheet covers how to identify polygons and calculate their interior and exterior angles. Students need these rules to solve geometry problems involving triangles, quadrilaterals, and larger polygons. It is especially useful for checking work, comparing polygon types, and remembering formulas for regular polygons. The most important idea is that a polygon with $n$ sides can be divided into $n - 2$ triangles. That leads to the interior angle sum formula $S = 180^\circ(n - 2)$ . For a regular polygon, each interior angle is $\frac{180^\circ(n - 2)}{n}$ , and each exterior angle is $\frac{360^\circ}{n}$ .

Key Facts

A polygon is a closed two-dimensional figure made of straight line segments.
The sum of the interior angles of an $n$ -gon is $S = 180^\circ(n - 2)$ .
A triangle has $3$ sides and an interior angle sum of $180^\circ$ .
A quadrilateral has $4$ sides and an interior angle sum of $360^\circ$ .
In any convex polygon, the sum of one exterior angle at each vertex is $360^\circ$ .
For a regular $n$ -gon, each interior angle is $\frac{180^\circ(n - 2)}{n}$ .
For a regular $n$ -gon, each exterior angle is $\frac{360^\circ}{n}$ .
At each vertex of a convex polygon, the interior angle and its adjacent exterior angle add to $180^\circ$ .

Vocabulary

Polygon: A polygon is a closed flat shape made only of straight sides.
Interior angle: An interior angle is an angle inside a polygon formed by two sides that meet at a vertex.
Exterior angle: An exterior angle is an angle formed outside a polygon by extending one side.
Regular polygon: A regular polygon has all sides congruent and all interior angles congruent.
Convex polygon: A convex polygon has all interior angles less than $180^\circ$ and no sides that cave inward.
Diagonal: A diagonal is a segment that connects two nonadjacent vertices of a polygon.

Common Mistakes to Avoid

Using $180^\circ n$ 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 is wrong because $180^\circ(n - 2)$ gives the total interior angle sum, not one angle.
Mixing up interior and exterior angles is wrong because each regular exterior angle is $\frac{360^\circ}{n}$ , while each regular interior angle is $\frac{180^\circ(n - 2)}{n}$ .
Assuming every polygon with the same number of sides is regular is wrong because a polygon can have equal side counts without equal sides or equal angles.
Counting sides incorrectly is wrong because the value of $n$ controls every formula, including $S = 180^\circ(n - 2)$ and $\frac{360^\circ}{n}$ .

Practice Questions

1 Find the sum of the interior angles of a polygon with $9$ sides.
2 Find each interior angle of a regular octagon.
3 A regular polygon has each exterior angle equal to $45^\circ$ . How many sides does it have?
4 Explain why the interior angle sum of a pentagon is greater than the interior angle sum of a quadrilateral.

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