Polygons & Quadrilaterals Cheat Sheet

A printable reference covering polygon angle sums, regular polygons, quadrilateral properties, perimeter, and area formulas for grades 5-8.

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Polygons and quadrilaterals are the building blocks of many geometry problems, diagrams, and real-world designs. This cheat sheet helps students quickly identify shapes, classify quadrilaterals, and choose the correct formula. It is especially useful when comparing properties like parallel sides, equal sides, equal angles, and symmetry. Students in grades 5-8 can use it as a compact reference for homework, review, and test preparation. The most important ideas include naming polygons by their number of sides, finding angle sums, and using perimeter and area formulas. For any polygon with $n$ sides, the interior angle sum is $(n - 2) \times 180^\circ$ . Quadrilaterals always have an interior angle sum of $360^\circ$ , but different types have different side and angle properties. Area formulas such as $A = lw$ , $A = \frac{1}{2}bh$ , and $A = \frac{1}{2}(b_1 + b_2)h$ connect shape properties to measurement.

Key Facts

A polygon is a closed two-dimensional figure made only of straight line segments.
The sum of the interior angles of an $n$ -sided polygon is $(n - 2) \times 180^\circ$ .
The sum of the exterior angles of any convex polygon is $360^\circ$ .
Each interior angle of a regular $n$ -gon is $\frac{(n - 2) \times 180^\circ}{n}$ .
Each exterior angle of a regular $n$ -gon is $\frac{360^\circ}{n}$ .
The perimeter of any polygon is the sum of all side lengths, so $P = s_1 + s_2 + s_3 + \cdots$ .
The area of a rectangle is $A = lw$ , and the area of a square is $A = s^2$ .
The area of a trapezoid is $A = \frac{1}{2}(b_1 + b_2)h$ , where $b_1$ and $b_2$ are the parallel bases.

Vocabulary

Polygon: A closed plane figure formed by three or more straight sides.
Regular polygon: A polygon with all sides congruent and all angles congruent.
Quadrilateral: A polygon with exactly four sides and an interior angle sum of $360^\circ$ .
Parallelogram: A quadrilateral with two pairs of opposite sides that are parallel.
Trapezoid: A quadrilateral with at least one pair of parallel sides.
Diagonal: A line segment that connects two nonadjacent vertices of a polygon.

Common Mistakes to Avoid

Using the exterior angle formula for one angle in a nonregular polygon is wrong because $\frac{360^\circ}{n}$ only gives each exterior angle when all exterior angles are equal.
Forgetting to subtract $2$ in the interior angle sum formula is wrong because a polygon with $n$ sides can be divided into $n - 2$ triangles, not $n$ triangles.
Confusing perimeter with area is wrong because perimeter measures distance around a figure, while area measures the space inside the figure.
Assuming every quadrilateral with one pair of parallel sides is a parallelogram is wrong because a parallelogram must have two pairs of parallel sides.
Using a slanted side as the height is wrong because height must be perpendicular to the base, not just any side length.

Practice Questions

1 Find the sum of the interior angles of a polygon with $9$ sides.
2 A regular hexagon has $6$ equal sides. Find the measure of each interior angle using $\frac{(n - 2) \times 180^\circ}{n}$ .
3 Find the area of a trapezoid with bases $8\text{ cm}$ and $14\text{ cm}$ and height $5\text{ cm}$ .
4 Explain why every square is a rectangle, but not every rectangle is a square.

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