What is a polygon? Sides, corners and diagonals
A polygon is a closed shape with straight sides. A triangle has 3 sides, a quadrilateral 4, a pentagon 5, a hexagon 6, a heptagon 7 and an octagon 8. The number of sides always equals the number of corners (vertices).
A diagonal is a line that joins two corners that are not next to each other. A side is not a diagonal. From one corner of an n-sided polygon you can draw (n − 3) diagonals, because you cannot join a corner to itself or to its two neighbours.
The total number of diagonals in the whole polygon is n × (n − 3) ÷ 2. We divide by 2 because each diagonal is counted from both ends.
Convex and non-convex polygons
In a convex polygon every angle inside is smaller than 180°. If you pick any two points inside, the straight line between them stays inside. A rubber band around it touches every side.
In a non-convex polygon (also called concave) at least one inside angle is bigger than 180°. It looks like it has a dent, like a dart or an arrow head. Some diagonals of such a polygon lie outside the shape.
The angle-sum rule below is for convex polygons, and it also works for a dented polygon if you split it carefully into triangles from the dent corner.
Angle sum of a convex polygon
Pick one corner and draw all diagonals from it. They cut the polygon into (n − 2) triangles. The corners of these triangles together fill exactly the corners of the polygon. Every triangle has an angle sum of 180°. So:
Angle sum = (n − 2) × 180°
Check: triangle (n = 3) gives 180°. Quadrilateral (n = 4) gives 360°. Pentagon gives 540°. Hexagon gives 720°. Each extra side adds one more triangle, which adds 180°.
Each angle of a regular polygon
A regular polygon has all sides equal and all angles equal. So one angle = (n − 2) × 180° ÷ n. For a regular hexagon, 720° ÷ 6 = 120°. For a regular octagon, 1080° ÷ 8 = 135°.
The outside (exterior) angle at each corner is 180° minus the inside angle. The outside angles of any convex polygon always add up to 360°, so each outside angle of a regular polygon is 360° ÷ n.
Try it
Draw a pentagon on paper. Pick one corner, draw its 2 diagonals, and colour the 3 triangles. Now cut out the three corners of every triangle and place them side by side. Do they make a straight line each time (180°)? Do the same with a hexagon and count the 4 triangles.
Key formulas and definitions
- Diagonals from one corner = n − 3
- Total diagonals = n(n − 3) ÷ 2
- Triangles made from one corner = n − 2
- Angle sum of a convex polygon = (n − 2) × 180°
- Each angle of a regular polygon = (n − 2) × 180° ÷ n
- Sum of exterior angles of a convex polygon = 360°
Worked examples
1. How many diagonals can be drawn from one corner of a hexagon, and how many triangles do they make?
n = 6. Diagonals from one corner = 6 − 3 = 3. Triangles = 6 − 2 = 4.
2. Find the sum of the interior angles of a heptagon (7 sides).
Sum = (7 − 2) × 180° = 5 × 180° = 900°.
3. Find one angle of a regular octagon.
Sum = (8 − 2) × 180° = 1080°. One angle = 1080° ÷ 8 = 135°.
4. Four angles of a pentagon are 100°, 110°, 120° and 90°. Find the fifth angle.
Pentagon sum = 540°. The four angles add to 100 + 110 + 120 + 90 = 420°. Fifth angle = 540° − 420° = 120°.
5. The angle sum of a polygon is 1440°. How many sides does it have?
(n − 2) × 180° = 1440°, so n − 2 = 8 and n = 10. It is a decagon.
6. Each angle of a regular polygon is 150°. Find n and the number of diagonals in the whole polygon.
The outside angle is 180° − 150° = 30°. n = 360° ÷ 30° = 12. Total diagonals = 12 × 9 ÷ 2 = 54.
Common mistakes
- Using n × 180° instead of (n − 2) × 180°. Always subtract 2 first.
- Counting the sides as diagonals. A diagonal joins corners that are not neighbours.
- Dividing the angle sum by n for a polygon that is not regular. The angles are only equal for a regular polygon.
- Forgetting that the diagonals from one corner number n − 3, not n − 1.