📖Definition
A polygon is a closed two-dimensional shape made entirely of straight line segments. Polygons are classified by their number of sides, from triangles (3 sides) to quadrilaterals (4 sides) up to shapes with many more sides.
📐Formula
Here n is the number of sides. The formula works because any polygon can be split into (n − 2) triangles, each contributing 180° to the total.
📝Step-by-Step Guide
Count the number of sides
Determine n, the number of straight sides the polygon has.
Apply the angle sum formula
Substitute n into (n − 2) × 180° to find the total of all interior angles.
Divide for a regular polygon
If all sides and angles are equal, divide the total by n to find the measure of one interior angle.
Use the total to find a missing angle
If all but one angle are known, subtract their sum from the total to find the missing angle.
🧭Where the angle sum formula comes from
Any polygon can be divided into triangles by drawing diagonals from a single vertex. A polygon with n sides always splits into exactly (n − 2) triangles.
Since every triangle's angles sum to 180°, the total for the polygon is (n − 2) × 180°.
🧭Interior angles vs. exterior angles
While the interior angle sum grows with the number of sides, the exterior angles of any convex polygon always add up to exactly 360°, no matter how many sides it has.
This is a useful check: if a polygon's exterior angles do not sum to 360°, an angle has been measured or labeled incorrectly.
⚠️Common Mistakes to Avoid
- Using the wrong value of n, especially miscounting sides on complex shapes
- Confusing the interior angle sum formula with the fact that exterior angles always sum to 360°
- Assuming every polygon has equal angles without checking whether it is regular
- Forgetting to divide by n when asked for a single interior angle of a regular polygon
- Mixing up the number of sides with the number of triangles used to derive the formula
✏️Practice Problems
Find the sum of interior angles in a hexagon (6 sides)
💡 Hint: Use (n − 2) × 180° with n = 6.
Answer: 720°
Find the sum of interior angles in a quadrilateral
💡 Hint: A quadrilateral has 4 sides.
Answer: 360°
Find the measure of one interior angle of a regular pentagon (5 sides)
💡 Hint: Find the total sum, then divide by 5.
Answer: 108°
A quadrilateral has three angles of 90°, 80°, and 100°. Find the fourth angle.
💡 Hint: All four angles sum to 360°.
Answer: 90°
A regular polygon has interior angles of 144°. How many sides does it have?
💡 Hint: Set (n − 2) × 180° / n equal to 144° and solve for n.
Answer: 10 sides
Find the sum of interior angles in a decagon (10 sides)
💡 Hint: Use (n − 2) × 180° with n = 10.
Answer: 1440°
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❓Polygons FAQ
What makes a polygon "regular"?
A regular polygon has all sides equal in length and all interior angles equal in measure. A square is a regular quadrilateral; a general rectangle is not regular unless it is a square.
Does the angle sum formula work for any polygon?
It works for any simple polygon, meaning the sides do not cross themselves, whether the shape is regular or irregular, convex or concave.
How many sides does a polygon need to have interior angles summing to over 1000°?
Solving (n − 2) × 180° > 1000° gives n > 7.55, so an octagon (8 sides), with an angle sum of 1080°, is the smallest one that qualifies.