Geometry
Interior Angles of Polygons: Why Triangles Always Add Up to 180°
Ask someone to draw a triangle and measure its three angles, and the total will land on 180°. Stretch it long and thin, squash it almost flat, redraw it sideways — the individual angles change, the sum does not. That stubbornness is not a coincidence or a rounding trick. It is the single fact that makes it possible to say anything useful about the interior angles of a pentagon, a hexagon, or a twenty-sided shape you would never attempt by hand.
Better still, the reason behind it fits in your head in about thirty seconds. Once you can see it, the polygon formula stops being a line to memorise and becomes something you can rebuild from a blank page. That matters more than it sounds. Formulas you can rebuild are formulas you can trust under pressure, whether you are checking homework, estimating a roof angle, or trying to remember why a regular octagon has such a tidy-looking corner.
Why Every Triangle Adds Up to 180°
Take any triangle and label its three angles A, B and C. Now draw a line through the apex — the vertex holding angle A — parallel to the side opposite it. That line creates two new angles, one on each side of A. Because the line is parallel to the base, the left-hand one equals angle B and the right-hand one equals angle C. These are alternate angles, the same idea often taught as "Z angles".
So along the top line, three angles sit side by side: B, then A, then C. They run in a straight line, and a straight line measures 180°. Therefore A + B + C = 180°.
Every interior angle formula for polygons is this one result, applied over and over.
Notice what the argument never used: side lengths, whether the triangle is isosceles or scalene, how tall or flat it looks. Only the behaviour of parallel lines mattered. That is why the rule holds for the long, ugly, hand-drawn triangle at the bottom of a page just as firmly as for a neat equilateral one.
You can also see it by tearing the three corners off a paper triangle and lining them up. The three tips meet in a straight line. The proof above is the precise version of that little experiment.
Turning Any Polygon into a Fan of Triangles
Quadrilaterals, pentagons and their larger relatives look like a different problem, but they are only triangles glued together. Find the glue lines and the angle sums fall out for free.
The method, step by step:
- Pick one vertex of the polygon.
- Draw a diagonal from it to every other vertex that is not already joined to it by a side, skipping its two immediate neighbours.
- Count the triangles you have drawn.
- Multiply that number by 180°.
Try it on a hexagon. From a single vertex you can draw three diagonals, and they cut the hexagon into four triangles. Four triangles means 4 × 180° = 720°, the total of all six interior angles added together.
The count follows a pattern. An n-sided polygon gives you n − 3 diagonals from one vertex, and each new diagonal adds a triangle, so you finish with n − 2 triangles. That is where the familiar formula comes from — not from a textbook decree, but from slicing.
A Quadrilateral and a Pentagon by Hand
A quadrilateral needs just one diagonal. That splits it into two triangles, so its interior angles total 2 × 180° = 360°. This is why every rectangle has four right angles adding to 360°, but also why a lopsided, irregular four-sided plot of land follows the same total.
A pentagon gives three triangles, because 5 − 2 = 3. The total is 3 × 180° = 540°. If you measured a regular pentagon, each of its five equal angles would be 540° ÷ 5 = 108°. The fan method explains both the total and the regular case in one move.
The Interior Angle Formula
Put the two ideas together and you have it:
Sum of interior angles = (n − 2) × 180°, where n is the number of sides.
- Triangle (3 sides): 180°
- Quadrilateral (4 sides): 360°
- Pentagon (5 sides): 540°
- Hexagon (6 sides): 720°
- Octagon (8 sides): 1,080°
- Decagon (10 sides): 1,440°
- Icosagon (20 sides): 3,240°
Each extra side adds another 180°, because each extra side buys you another triangle in the fan. That makes a useful sanity check: if your answer for a hexagon comes out smaller than your answer for a pentagon, something has gone wrong along the way.
Regular Polygons: Splitting the Total Evenly
A regular polygon has equal sides and equal angles, so the total divides evenly between the vertices. Each interior angle is (n − 2) × 180° ÷ n.
For a regular pentagon, that is 540° ÷ 5 = 108°. For a regular hexagon, it is 720° ÷ 6 = 120°. An octagon gives 1,080° ÷ 8 = 135°, and a decagon gives 1,440° ÷ 10 = 144°. As the number of sides grows, each individual angle gets closer and closer to 180°, which is why a regular 100-sided polygon looks almost like a circle.
This division only works for regular polygons. An irregular hexagon still has interior angles totalling 720°, but the six angles can be wildly different — one might be 200° if the shape is concave, while another is tiny.
Finding a Missing Angle
The formula is most useful when one angle is unknown. Suppose a quadrilateral has interior angles of 85°, 95° and 110°. The three known angles add to 290°. Since every quadrilateral totals 360°, the missing angle is 360° − 290° = 70°.
The same approach works for a pentagon. If four of its angles are 100°, 110°, 120° and 130°, those add to 460°. A pentagon must total 540°, so the fifth angle is 540° − 460° = 80°. Always start by finding the required total for that many sides, then subtract what you know.
Exterior Angles: The Other Way Around
There is a second, often faster route to the same results. If you extend one side of a convex polygon at each vertex, the angle between the side and the extension is an exterior angle. For any convex polygon, the exterior angles add to 360°, no matter how many sides it has.
For a regular polygon, each exterior angle is 360° ÷ n. Because an interior angle and its exterior angle sit on a straight line, they add to 180°. That gives another formula for each interior angle of a regular polygon:
Interior angle = 180° − (360° ÷ n)
Check it with a hexagon: 360° ÷ 6 = 60°, and 180° − 60° = 120°. It matches the earlier answer. This exterior-angle view is handy when you want one angle quickly, and it reinforces why the interior angles grow as n grows: the exterior angle must shrink so the total stays fixed at 360°.
Common Mistakes and Quick Checks
- Using n instead of n − 2. A triangle has 3 sides, but the formula uses 1 triangle, not 3.
- Forgetting that the formula gives the sum, not each angle. Divide by n only for a regular polygon.
- Mixing up interior and exterior angles. Interior + exterior = 180° at any vertex of a convex polygon.
- Assuming every angle must be less than 180°. In a concave polygon, one interior angle can be greater than 180°.
- Checking the trend: each added side increases the interior-angle sum by 180°. If your totals decrease, you have made an arithmetic slip.
The triangle is not just the simplest polygon. It is the unit of measurement for all the rest.
Once the 180° fact is secure, polygons stop being a list of numbers to memorise. A quadrilateral is two triangles, a pentagon is three, a hexagon is four, and any n-sided shape is n − 2. The formula is not something handed down from above; it is a triangle, repeated.
