Geometry

Regular polygons

Add sides. Investigate the angles.

Predict, test and explain

Step 1 of 3

Predict

What is the interior angle sum of a hexagon?

interior sum=(5−2)×180∘=540∘\text{interior sum} = (5-2)\times180^\circ = 540^\circ
108∘108^\circ

Scroll the diagram sideways to see every label.

each interior=108∘\text{each interior} = 108^\circeach exterior=72∘\text{each exterior} = 72^\circ

InvestigateAdd one side. What happens to the total interior angle?

Explore and compare
Can you break the rule?

The exterior angles of a regular polygon add to one full turn.

Always, sometimes or never? Test it in the simulation.

Hint

Show the exterior angles and change the number of sides.

Check the reasoning

Always. Walking around a convex regular polygon turns you through one complete rotation.

n×360∘n=360∘n\times\frac{360^\circ}{n}=360^\circ

One counterexample disproves “always”. Examples alone do not prove it.

Make a regular polygon with this exterior angle.

45∘45^\circ

Try it in the simulation. Change one thing at a time.

Hint

The exterior angles add to one full turn. Turn on exterior angles.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Find each exterior angle and each interior angle of a regular octagon.

Hint

The eight equal exterior angles total one full turn; each interior angle adds to a straight angle with its exterior angle.

Reveal answer

Each exterior angle is 45 degrees and each interior angle is 135 degrees.

exterior=360∘8=45∘,interior=180∘−45∘=135∘\text{exterior}=\frac{360^\circ}{8}=45^\circ,\qquad\text{interior}=180^\circ-45^\circ=135^\circ

Build an explanation

Hint 1

Split the polygon into triangles from one corner.

Hint 2

An n-sided polygon splits into n − 2 triangles.

Hint 3

Its interior sum is (n − 2) × 180°; for a regular polygon, divide by n.

Worked example

This example uses fixed values, separate from the diagram controls.

Find one interior and one exterior angle of a regular pentagon.

  1. The interior sum is (5 − 2) × 180° = 540°.
  2. Each interior angle is 540°/5 = 108°.
  3. Each exterior angle is 180° − 108°.

Answer: Interior 108°; exterior 72°.

Connect this idea

Watch out: Dividing the interior sum equally gives individual angles only when all interior angles are equal.