Sum of Interior Angles of a Convex Polygon

These QuickNotes give you the most important points to remember for the ESSLCE. They are based on the MoE Grade 9 Mathematics Textbook, Unit 6, Chapter 6.1.

~6 min read

Summary

A polygon is a simple closed plane figure made of three or more line segments.
A polygon is convex when every interior angle is less than 180°, and concave when at least one interior angle is more than 180°.
Diagonals from one vertex divide an n-sided polygon into n − 2 triangles.
The sum of the interior angles of an n-sided polygon is (n − 2) × 180°.

Key Words

Polygon: a simple closed plane figure formed by three or more line segments joined end to end.
Side: one of the line segments that forms the polygon.
Vertex: the common end point where two sides meet. The plural is vertices.
Interior angle: an angle formed inside the polygon at a vertex.
Convex polygon: a polygon in which every interior angle is less than 180°.
Concave polygon: a polygon that has at least one interior angle of more than 180°.
Diagonal: a line segment that joins two vertices which are not next to each other.
n-gon: the general name for a polygon with n sides. For example, a 16-sided polygon is a 16-gon.

What Is a Polygon?

A polygon is a simple closed plane figure formed by three or more line segments joined end to end.
No two segments that follow each other are allowed to lie on the same straight line.
The segments are called sides, the points where two sides meet are called vertices, and the angles formed inside are called interior angles.
The triangle, with three sides, is the simplest polygon. After it come the quadrilateral with four sides and the pentagon with five sides.
A polygon always has the same number of sides and vertices. A hexagon has 6 sides and 6 vertices.

Convex and Concave Polygons

A polygon is convex when the measure of each interior angle is less than 180°. All its vertices point outwards.
A polygon is concave when at least one interior angle measures more than 180°. Such a polygon has at least one vertex that points inwards, like a dent.
Here is a quick test: extend each side of the polygon into a full line. If none of these lines cuts through the polygon, it is convex. If at least one line cuts through it, the polygon is concave.

Dividing a Polygon into Triangles

You already know that the interior angles of any triangle add up to 180°. We use this fact to find the angle sum of every other polygon.
Pick one vertex of the polygon and draw all the diagonals from that vertex. The diagonals cut the polygon into triangles that do not overlap.
A quadrilateral needs one diagonal and gives 2 triangles, so its interior angles add up to 2 × 180° = 360°.
A pentagon gives 3 triangles, so its interior angles add up to 3 × 180° = 540°.
The pattern is simple: an n-sided polygon always gives n − 2 triangles.

The Formula for the Sum of Interior Angles

Since an n-sided polygon splits into n − 2 triangles, and each triangle gives 180°, we get the formula below.
Sum of interior angles of an n-sided polygon = (n − 2) × 180°.
This tells us that every extra side adds one more triangle, so the angle sum grows by 180° each time.
You can use the formula in both directions: from the number of sides to the sum, or from the sum back to the number of sides.

Name of polygon	Sides (n)	Triangles (n − 2)	Sum of interior angles
Triangle	3	1	180°
Quadrilateral	4	2	360°
Pentagon	5	3	540°
Hexagon	6	4	720°
Heptagon	7	5	900°
Octagon	8	6	1,080°
Nonagon	9	7	1,260°
Decagon	10	8	1,440°
n-gon	n	n − 2	(n − 2) × 180°

Worked example: find the sum of the interior angles of a hexagon.

A hexagon has 6 sides, so use n = 6 in the formula.

Sum = (6 − 2) × 180° = 4 × 180° = 720°.

Worked example: the sum of the interior angles of a polygon is 1,080°. How many sides does it have?

Set up the equation: (n − 2) × 180° = 1,080°.

Divide both sides by 180°: n − 2 = 1,080 ÷ 180 = 6.

So n = 8. The polygon is an octagon.

Common Mistakes to Avoid

Writing n × 180° instead of (n − 2) × 180°. The number of triangles is always 2 less than the number of sides.
Forgetting to add 2 back when you work backwards. If the sum is 1,080°, then 1,080 ÷ 180 = 6 gives the number of triangles, not the number of sides. The polygon has 6 + 2 = 8 sides.
Calling a polygon concave just because it looks thin or stretched. The only test is the interior angles: concave means at least one angle is more than 180°.
Mixing up the angle sum of a polygon with the measure of one angle. The formula in this chapter gives the total of all interior angles, not each one.

Easy Ways to Remember

Think “two less”: the number of triangles inside a polygon is always two less than the number of sides.
Every new side adds exactly one more triangle, so the angle sum grows by 180° for each extra side.
Anchor yourself with the three sums you already know: triangle 180°, quadrilateral 360°, pentagon 540°.
Convex sounds like “curved out”: all vertices point outwards. Concave has a “cave”: one vertex sinks inwards.

Quiz

Tap an answer to check it.

1. What is the name of a polygon with ten sides?

2. Which one of the following is true about a concave polygon?

3. Into how many triangles can a heptagon be divided by diagonals drawn from one vertex?

4. What is the sum of the interior angles of a 12-sided polygon?

5. The sum of the interior angles of a polygon is 1,440°. How many sides does it have?

Remember: An n-sided polygon splits into n − 2 triangles, so its interior angles always add up to (n − 2) × 180°. A polygon is convex when every interior angle is less than 180° and concave when at least one angle is more than 180°.