ShapesPolygons and their angles
A polygon is a closed flat shape with straight sides. Triangles, quadrilaterals, pentagons and hexagons are all polygons. A circle is not, because its edge is curved. This page gives the names, the angle rules and a table for polygons of 3 to 12 sides.
Names by number of sides
| Sides | Name | Sum of interior angles | Each interior angle | Each exterior angle | Diagonals |
|---|---|---|---|---|---|
| 3 | Triangle | 180° | 60° | 120° | 0 |
| 4 | Quadrilateral | 360° | 90° | 90° | 2 |
| 5 | Pentagon | 540° | 108° | 72° | 5 |
| 6 | Hexagon | 720° | 120° | 60° | 9 |
| 7 | Heptagon | 900° | 128.57° | 51.43° | 14 |
| 8 | Octagon | 1080° | 135° | 45° | 20 |
| 9 | Nonagon | 1260° | 140° | 40° | 27 |
| 10 | Decagon | 1440° | 144° | 36° | 35 |
| 11 | Hendecagon | 1620° | 147.27° | 32.73° | 44 |
| 12 | Dodecagon | 1800° | 150° | 30° | 54 |
The last three columns are for regular polygons, whose sides and angles are all equal.
The angle sum rule
Sum of interior angles = (n − 2) × 180°
The reason is simple. From one corner, draw lines to all the other corners. The polygon is cut into n − 2 triangles, and each triangle has angles adding to 180°. A quadrilateral makes 2 triangles, so 360°. A pentagon makes 3, so 540°.
Exterior angles always add to 360°
Walk round any polygon. At each corner you turn through the exterior angle. By the time you are back at the start, facing the way you began, you have made one full turn: 360°. This is true however many sides there are. For a regular polygon, each exterior angle is 360° ÷ n.
At every corner, the interior and exterior angles add to 180°, because they lie on a straight line.
Working backwards
Each interior angle of a regular polygon is 150°. How many sides? The exterior angle is 180° − 150° = 30°, and 360° ÷ 30° = 12. It has 12 sides. Working through the exterior angle is nearly always the quickest route.
Can a regular polygon have an interior angle of 100°? The exterior angle would be 80°, and 360 ÷ 80 = 4.5, which is not a whole number. So no.
Diagonals
A diagonal joins two corners that are not next to each other. From each corner you can draw n − 3 diagonals, and counting each one once gives
Number of diagonals = n(n − 3) ÷ 2
A triangle has none, a quadrilateral 2, a pentagon 5 and a hexagon 9.
Convex and concave
In a convex polygon every interior angle is less than 180° and every diagonal lies inside. A concave polygon has at least one angle greater than 180°, a corner that points inwards. The angle sum rule holds for both.
Which regular polygons tile a floor?
Only three regular polygons fit together on their own with no gaps: the equilateral triangle, the square and the regular hexagon. Their angles (60°, 90°, 120°) divide exactly into 360°. That is why bathroom tiles are squares or hexagons and why bees build hexagonal cells.
For the area of a regular polygon, see the regular polygon page.