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 sides can be divided into triangles. That leads to the interior angle sum formula . For a regular polygon, each interior angle is , and each exterior angle is .
Key Facts
- A polygon is a closed two-dimensional figure made of straight line segments.
- The sum of the interior angles of an -gon is .
- A triangle has sides and an interior angle sum of .
- A quadrilateral has sides and an interior angle sum of .
- In any convex polygon, the sum of one exterior angle at each vertex is .
- For a regular -gon, each interior angle is .
- For a regular -gon, each exterior angle is .
- At each vertex of a convex polygon, the interior angle and its adjacent exterior angle add to .
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 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 for the interior angle sum is wrong because a polygon with sides divides into triangles, not triangles.
- Forgetting to divide by for a regular polygon is wrong because gives the total interior angle sum, not one angle.
- Mixing up interior and exterior angles is wrong because each regular exterior angle is , while each regular interior angle is .
- 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 controls every formula, including and .
Practice Questions
- 1 Find the sum of the interior angles of a polygon with sides.
- 2 Find each interior angle of a regular octagon.
- 3 A regular polygon has each exterior angle equal to . 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.
Understanding Polygons & Interior Angles Reference
Polygon angle rules come from turning and partitioning, not from a list that must be memorized without meaning. Draw diagonals from one vertex to every nonadjacent vertex. The shape is split into triangular regions, provided the polygon is convex.
Each triangle contributes one hundred eighty degrees. This gives a reliable visual reason that larger polygons have larger interior angle totals.
A useful check is that adding one side creates one more triangle, so the total increases by one hundred eighty degrees. This pattern helps students build a table for pentagons, hexagons, heptagons, and beyond without starting over each time.
Regular polygons need special care because the word regular has two requirements. All sides must have equal length, and all interior angles must have equal measure. A shape with equal sides alone is not always regular.
A rhombus, for example, can have four equal sides while its angles are not all equal. When every interior angle is equal, the full interior total can be shared evenly among the vertices.
As the number of sides grows, each interior angle gets closer to one hundred eighty degrees. This makes sense because a many-sided regular polygon begins to look more like a circle, though it never becomes a circle because its boundary still has straight segments.
Exterior angles are easiest to understand as turns made while walking around a shape. Start along one side and keep moving in the same direction around the boundary. At every corner, turn enough to follow the next side.
After returning to the starting direction, the total turning is one complete rotation. This remains true even when the sides have different lengths or the interior angles have different measures, as long as one exterior turn is chosen at every vertex of a convex polygon.
Students often accidentally add both possible exterior angles at a corner. Use the small outside turning angle next to the interior angle, not the large reflex angle around the other side.
These ideas appear in tiling, design, construction, and computer graphics. Floor tiles fit perfectly around a point only when their angles make a full turn with no gap or overlap. Equilateral triangles, squares, and regular hexagons can do this by themselves.
Regular pentagons cannot tile a flat floor by themselves because their interior angles do not fit evenly into a full turn. In coordinate geometry, angle knowledge helps when drawing shapes accurately or checking whether a set of points could form a particular polygon. When solving a problem, first count the vertices carefully.
Then decide whether the problem asks for a total, one interior angle, or one exterior angle. Finally, check whether the polygon is regular, convex, or concave, since those details determine which shortcut is safe to use.