This cheat sheet covers how to find interior and exterior angles in polygons, including regular and irregular shapes. Students need these rules to solve geometry problems quickly and to recognize which formula fits each situation. Worked examples help connect the formulas to common classroom questions about triangles, quadrilaterals, pentagons, and other polygons.
The most important idea is that an -sided polygon can be split into triangles, so its interior angle sum is . For any convex polygon, one exterior angle at each vertex always adds to . In a regular polygon, all interior angles are equal and all exterior angles are equal, so division can be used to find one angle.
Key Facts
- The sum of the interior angles of an -sided polygon is .
- The measure of each interior angle of a regular -gon is .
- The sum of one exterior angle at each vertex of any convex polygon is always .
- The measure of each exterior angle of a regular -gon is .
- An interior angle and its adjacent exterior angle form a straight line, so their measures add to .
- If each exterior angle of a regular polygon is , then the number of sides is .
- A triangle has interior angle sum , and a quadrilateral has interior angle sum .
- For an irregular polygon, use the total interior angle sum first, then subtract the known angles to find a missing angle.
Vocabulary
- Polygon
- A polygon is a closed flat shape made from straight line segments.
- Interior angle
- An interior angle is an angle inside a polygon formed by two adjacent sides.
- Exterior angle
- An exterior angle is formed outside a polygon by extending one side and using the adjacent side.
- Regular polygon
- A regular polygon has all sides equal and all interior angles equal.
- Convex polygon
- A convex polygon has no interior angle greater than .
- Adjacent angles
- Adjacent angles share a side and a vertex, such as an interior angle and its matching exterior angle.
Common Mistakes to Avoid
- Using for the interior angle sum is wrong because a polygon with sides splits into triangles, so the correct formula is .
- Dividing the interior angle sum by for an irregular polygon is wrong because equal angles are only guaranteed in a regular polygon.
- Forgetting that exterior angles total is wrong because one exterior angle at each vertex makes a full turn around any convex polygon.
- Adding an interior angle and its adjacent exterior angle to is wrong because they form a straight line, so they add to .
- Confusing the number of sides with the number of triangles is wrong because an -sided polygon contains triangles when drawn from one vertex.
Practice Questions
- 1 Find the sum of the interior angles of a -sided polygon.
- 2 A regular polygon has sides. Find one interior angle and one exterior angle.
- 3 Each exterior angle of a regular polygon is . How many sides does the polygon have?
- 4 Explain why the exterior angles of a convex polygon add to even when the polygon is not regular.
Understanding Angles in Polygons Interior and Exterior Worked Examples
Drawing diagonals is more than a trick for getting an angle total. It explains where the rule comes from. Start at one vertex and draw diagonals to every non-adjacent vertex that can be reached inside the shape.
The polygon breaks into triangles with no overlaps. Each triangle contributes one hundred and eighty degrees. This method is especially helpful when a student forgets a formula, because the diagram rebuilds the idea.
It works cleanly for convex polygons, where every diagonal stays inside the boundary. Concave polygons need more care because some diagonals may lie outside the shape.
Exterior angles describe turning rather than the space inside a shape. Imagine walking around the edge of a polygon. At each corner, turn just enough to follow the next side.
After returning to the starting direction, the total turn is one complete rotation. This turning idea explains why the exterior angle rule is true even when the side lengths are different. The shape can look uneven, yet the total turning remains fixed.
Use one exterior angle at every vertex. Mixing an exterior angle at one corner with an interior angle at another corner produces an incorrect total.
For an irregular polygon, organise the information before doing arithmetic. First identify the number of sides by counting vertices, not by guessing from the drawing. Find the full interior total.
Then add the known interior angles and subtract that result from the full total. If a given angle is exterior, convert it to its adjacent interior angle before adding it to the list. For example, an exterior angle of seventy degrees gives an interior angle of one hundred and ten degrees.
A quick estimate helps catch mistakes. In a convex polygon, every interior angle must be greater than zero degrees and less than one hundred and eighty degrees.
Regular polygons give useful links between several pieces of information. Knowing one exterior angle can reveal the number of sides because equal turns fit around a full rotation. Knowing one interior angle can first give its exterior partner, then lead to the side count.
This is often easier than using a longer interior-angle calculation. A regular shape must have equal side lengths and equal angles.
A shape with equal-looking angles in a sketch is not automatically regular. In exam diagrams, markings or written statements provide the evidence.
Polygon angles appear in tiling patterns, road signs, roof frames, game design, and computer graphics. Designers often need shapes to meet around a point without gaps. Angle totals show whether a planned pattern will fit.
When learning worked examples, pay attention to the labels and the type of angle shown. Extend a side only when an exterior angle is needed. Do not treat the angle outside the shape on the opposite side as the adjacent exterior angle.
Finally, check whether the answer matches the shape. A missing angle in a nearly rectangular figure should be close to a right angle, not an unusually tiny or huge value.