Geometry · Triangles & polygons · 2.3

Polygon angle sum.

Any polygon with n sides can be split into (n − 2) triangles from one corner, so its interior angles add to (n − 2) × 180°. A hexagon (6 sides): (6 − 2) × 180 = 720°.

01 / THE IDEA

See it work.

The figure

See the shape.

6 sides

Watch the method

Polygon angle sum

Interior sum = (n − 2) × 180°.

1 / 2
1

Polygon angle sum

Split the polygon into (n − 2) triangles; each contributes 180°.

(n−2)×180(n-2)\times 180
2

Substitute n

An 6-sided polygon makes 4 triangles.

(6−2)×180=720∘(6-2)\times 180 = 720^\circ
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Find the measure.

Type the number — units optional.

streak 0
best 0

…

— / WHY IT WORKS

The idea underneath.

Split into triangles

Drawing diagonals from one vertex makes n − 2 triangles, each 180°.

Why it works

Count the triangles, times 180.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Interior angle sum of a 10-gon?

1

Polygon angle sum

Split the polygon into (n − 2) triangles; each contributes 180°. (n-2)× 180

2

Substitute n

An 10-sided polygon makes 8 triangles. (10-2)× 180 = 1440°

EX 2medium

Interior angle sum of a 9-gon?

1

Polygon angle sum

Split the polygon into (n − 2) triangles; each contributes 180°. (n-2)× 180

2

Substitute n

An 9-sided polygon makes 7 triangles. (9-2)× 180 = 1260°

EX 3hard

Interior angle sum of a 10-gon?

1

Polygon angle sum

Split the polygon into (n − 2) triangles; each contributes 180°. (n-2)× 180

2

Substitute n

An 10-sided polygon makes 8 triangles. (10-2)× 180 = 1440°

04 / COMMON TRAPS

Where students slip.

Trap

Using n × 180.

Fix

It’s (n − 2) × 180 — two fewer triangles than sides.

Trap

Forgetting to subtract 2.

Fix

A polygon makes n − 2 triangles, not n.

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (4)

C1

Interior angle sum of a 7-gon?

Need a hint?
  • Count the sides as n.
  • Use (n − 2) × 180.
C2

Interior angle sum of a 10-gon?

Need a hint?
  • Count the sides as n.
  • Use (n − 2) × 180.
C3

Interior angle sum of a 6-gon?

Need a hint?
  • Count the sides as n.
  • Use (n − 2) × 180.
C4

Interior angle sum of a 3-gon?

Need a hint?
  • Count the sides as n.
  • Use (n − 2) × 180.
Apply

Apply (4)

A1

Interior angle sum of a 5-gon?

Need a hint?
  • Count the sides as n.
  • Use (n − 2) × 180.
A2

Interior angle sum of a 10-gon?

Need a hint?
  • Count the sides as n.
  • Use (n − 2) × 180.
A3

Interior angle sum of a 6-gon?

Need a hint?
  • Count the sides as n.
  • Use (n − 2) × 180.
A4

Interior angle sum of a 10-gon?

Need a hint?
  • Count the sides as n.
  • Use (n − 2) × 180.
Extension

Extension (4)

E1

Interior angle sum of a 7-gon?

Need a hint?
  • Count the sides as n.
  • Use (n − 2) × 180.
E2

Interior angle sum of a 6-gon?

Need a hint?
  • Count the sides as n.
  • Use (n − 2) × 180.
E3

Interior angle sum of a 6-gon?

Need a hint?
  • Count the sides as n.
  • Use (n − 2) × 180.
E4

Interior angle sum of a 6-gon?

Need a hint?
  • Count the sides as n.
  • Use (n − 2) × 180.