E. Spatial Sense · Spatial Sense: Angles, Tessellations, Transformations, Scale, Pythagoras and Composite Measurement

Interior and Exterior Angles of Polygons

E2.2 A1About 50 minutes

Learning goals

I can explain why the interior angles of a polygon with n sides add to (n − 2) × 180°.
I can find each interior and exterior angle of a regular polygon.
I can find a missing angle in any polygon.
I can work backwards from an angle to find how many sides a regular polygon has.

Words to know

polygon
A closed flat shape made of straight sides.

interior angle
An angle inside a polygon, at one of its vertices.

exterior angle
The angle between one side of a polygon and the next side extended in a straight line. An interior angle and its exterior angle add to 180°.

regular polygon
A polygon with all sides equal and all angles equal, like a square or a stop sign shape.

convex polygon
A polygon with no dents: every interior angle is less than 180°.

n-gon
A polygon with n sides. A 12-gon has 12 sides (also called a dodecagon).

Remember from Grade 6

In Grade 6 you learned that the interior angles of a triangle add to 180° and of a quadrilateral add to 360° (a diagonal splits it into 2 triangles). You also learned that an interior angle and its exterior angle add to 180°, and that if you walk around a convex polygon, the exterior angles add to one full turn: 360°. In Grade 8 you'll turn these facts into rules that work for any polygon, and use them to solve problems.

Let's start with a stop sign

A stop sign in Canada is a regular octagon: 8 equal sides and 8 equal angles. How big is each corner? You could measure it with a protractor, but a mathematician can work it out without measuring at all. By the end of this lesson, you will too.

Draw every diagonal from one vertex. A pentagon splits into 3 triangles and a hexagon into 4 triangles. The angles of the triangles fill the polygon's corners exactly.

The rule for the sum of the interior angles

From one vertex, draw diagonals to every other vertex you can. A polygon with n sides always splits into n − 2 triangles. (The two sides that touch your starting vertex don't make new triangles.) Each triangle has 180°, so:

Sum of the interior angles = (n − 2) × 180°

Regular polygons

In a regular polygon all the angles are equal, so:

Each interior angle = (n − 2) × 180° ÷ n
Each exterior angle = 360° ÷ n (the exterior angles of a convex polygon always add to 360°)
Check: interior + exterior = 180°.

Extend each side of a regular pentagon. Each exterior angle is 72°, and each interior angle and its exterior angle make a straight line: 108° + 72° = 180°. Walking around the pentagon, you turn 72° five times: one full turn of 360°.

Polygon	Sides (n)	Triangles (n − 2)	Sum of interior angles	Each angle (if regular)	Each exterior angle (if regular)
Triangle	3	1	180°	60°	120°
Quadrilateral	4	2	360°	90°	90°
Pentagon	5	3	540°	108°	72°
Hexagon	6	4	720°	120°	60°
Octagon	8	6	1 080°	135°	45°
Decagon	10	8	1 440°	144°	36°
Dodecagon (12-gon)	12	10	1 800°	150°	30°

Each time you add a side, the sum grows by 180°, because there is one more triangle inside.

Worked example

Example 1: The stop sign

Find the size of each interior angle of a regular octagon.

n = 8, so the octagon splits into 8 − 2 = 6 triangles.
Sum of the interior angles: 6 × 180° = 1 080°.
The 8 angles are equal: 1 080° ÷ 8 = 135°.
Check with the exterior angle: 360° ÷ 8 = 45°, and 180° − 45° = 135° ✓

Each corner of a stop sign is 135°.

Worked example

Example 2: A missing angle

A hexagon has angles of 110°, 130°, 125°, 140°, 95° and one unknown angle x. Find x.

Sum of the interior angles of a hexagon: (6 − 2) × 180° = 720°.
Add the known angles: 110° + 130° + 125° + 140° + 95° = 600°.
x = 720° − 600° = 120°.

x = 120° (reason: the interior angles of a hexagon add to 720°).

Worked example

Example 3: Working backwards

Each interior angle of a regular polygon is 144°. How many sides does it have?

Each exterior angle is 180° − 144° = 36°.
The exterior angles add to 360°, and they are all equal: 360° ÷ 36° = 10.
So there are 10 exterior angles, which means 10 vertices and 10 sides.

It has 10 sides (a regular decagon).

Worked example

Example 4: Polygons and parallel lines together

Trapezoid ABCD has AB parallel to DC. Angle A = 65° and angle B = 110°. AD and BC are the slanted sides. Find angles D and C.

Side AD is a transversal of the parallel sides AB and DC. Angles A and D are co-interior, so they add to 180°: D = 180° − 65° = 115°.
Side BC is another transversal. Angles B and C are co-interior: C = 180° − 110° = 70°.
Check: 65° + 110° + 70° + 115° = 360° ✓ (a quadrilateral).

Angle D = 115° and angle C = 70°.

Think like a mathematician

The sum rule (n − 2) × 180° works for every polygon, even ones with a dent.
The each angle rule only works for regular polygons, where all the angles are equal.
When working backwards, use the exterior angle. Dividing 360° by the exterior angle is much quicker than solving with the interior angle.
Reasonableness check: the interior angles of a regular polygon are always less than 180°, and they get closer to 180° as the number of sides grows.

Hands-on activity

Walk a polygon

You need: some space outdoors (a yard, a park or a snowy field), or a large room, and a partner.

Choose a regular polygon, like a pentagon. Work out its exterior angle: 360° ÷ 5 = 72°.
Walk 5 steps forward, then turn left by about 72° (a bit less than a right-angle turn). Repeat 5 times.
Did you end up back where you started, facing the same way? You made one full turn in total: 5 × 72° = 360°.
In snow, look at your footprints from above (or have your partner trace your path with chalk on a driveway). What shape did you make?
Now try a hexagon (turn 60° each time) and an octagon (turn 45°).

What to look for: the path closes up because the turns (the exterior angles) add to 360°. Small turning errors are normal. The more sides, the smaller each turn, and the more the path looks like a circle.

Math talk: "How much would you turn at each corner to walk a regular 20-gon? What would the interior angle be?" (18°, so 162°.)

Just for fun

What did the math teacher say when her parrot flew out the window?

"Polygon!"

Your turn

What is the sum of the interior angles of a heptagon (7 sides)?

What is the size of each interior angle of a regular hexagon?

What is the size of each exterior angle of a regular pentagon?

A quadrilateral has angles of 85°, 95°, 100° and x. Find x.

A pentagon has angles of 100°, 110°, 120°, 90° and x. Find x.

Each exterior angle of a regular polygon is 30°. How many sides does it have?

Can a regular polygon have interior angles of 100°? Explain.

More practice: warm-up

A triangle has angles of 50° and 60°. Find the third angle.

How many triangles do you get when you draw all the diagonals from one vertex of an octagon?

What is the sum of the interior angles of a pentagon?

What do the exterior angles of any convex polygon add to?

An interior angle is 130°. What is the exterior angle beside it?

More practice: on your own

Find each interior angle of a regular nonagon (9 sides).

A hexagon has four angles of 125° each. The other two angles are equal. Find them.

Each interior angle of a regular polygon is 156°. How many sides does it have?

The interior angles of a polygon add to 1 980°. How many sides does it have?

One angle of a parallelogram is 58°. Find the other three angles.

Challenge

The angles of a pentagon are x, x + 20°, x + 40°, x + 60° and x + 80°. Find all five angles.

Each interior angle of a regular polygon is 5 times its exterior angle. How many sides does it have?

Sam says he drew a polygon whose interior angles add to 1 000°. Explain why that's impossible.

Check your understanding

Quiz

What is the sum of the interior angles of a hexagon?

What is the size of each interior angle of a regular octagon?

The exterior angles of a convex polygon (one at each vertex) add to:

Each exterior angle of a regular polygon is 40°. How many sides does it have?

A quadrilateral has angles of 70°, 80° and 90°. What is the fourth angle?

Which expression gives the sum of the interior angles of a polygon with n sides?

For parents and teachers

Big idea: The Ontario curriculum (E2.2) asks Grade 8 students to solve problems involving angle properties of polygons. Grade 6 covered triangles and quadrilaterals. Now your child generalizes: any polygon with n sides has interior angles that add to (n − 2) × 180°.

Common difficulties:

Using n × 180° instead of (n − 2) × 180°. Have your child draw the diagonals from one vertex and count the triangles.
Dividing by n for a polygon that is not regular. The "each angle" rule needs all angles equal.
Working backwards with the interior angle. Show them the exterior-angle shortcut: 360° ÷ exterior angle = number of sides.

How to help: Look for regular polygons around you: a stop sign (octagon), hexagonal nuts and bolts, a soccer ball (pentagons and hexagons), floor tiles. Ask your child to work out the angle at each corner. The walking activity is a fun way to feel exterior angles. This lesson also builds A1 skills: generalizing a pattern into a rule and explaining why it works.