KS3 · Maths

Exterior angles and regular polygons

Walk all the way round a shape, turning at each corner. How much have you turned by the time you're back at the start?

Maths · Geometry

Drag corner B and watch the turns

Exterior at A 90°. Exterior at B 38°. Exterior at C 109°. Exterior at D 48°. Exterior at E 75°. Interior at B 142°. Interior at C 71°. Classification: an irregular pentagon. Relationship: However you drag B, the five exterior angles add up to 360°. And at B and at C, the interior angle and exterior angle make a straight line: 180°.

Exterior at A90°Exterior at B38°Exterior at C109°Exterior at D48°Exterior at E75°Interior at B142°Interior at C71°drag B — two angles trade, the total doesn't

an irregular pentagon

However you drag B, the five exterior angles add up to 360°. And at B and at C, the interior angle and exterior angle make a straight line: 180°.

At each corner, one edge has been stretched out into a straight line (the bold coloured lines). The exterior angle is the angle between that stretched edge and the next edge. Drag B along the top edge: add up the five exterior readings as you go. Readings are rounded to whole degrees, so your total can be a degree out.

Maths · Angles

Which one is the exterior angle?

Look at one corner of a polygon. One of its edges has been stretched out past the corner as a straight line.

Which of these do you think the exterior angle at that corner is?

The angle outside the polygon, between the stretched-out edge and the next edge.All of the space outside the corner, the whole way round from one edge to the other.The angle inside the polygon where the two edges meet.It depends which edge you stretch: the other edge would give a different size.

How sure are you?

I'm sureI think soI'm guessing

The angle outside the polygon, between the stretched-out edge and the next edge.

This matches the evidence

Yes. The stretched edge and the next edge make a small angle on the outside, and that is the exterior angle. It sits on a straight line with the interior angle, so the two make 180°.

All of the space outside the corner, the whole way round from one edge to the other.

This is a common misconception

That big outside angle is the reflex angle, and the exterior angle is not all of it. Only the part between the stretched-out edge and the next edge counts. The rest is the interior angle's partner on the straight line.

The angle inside the polygon where the two edges meet.

This is a common misconception

That one is the interior angle, formed inside the polygon by two of its edges. The exterior angle is on the outside, and the two of them add up to 180° because the stretched edge makes a straight line.

It depends which edge you stretch: the other edge would give a different size.

This is a common misconception

Either edge can be stretched out, and both give an exterior angle of the same size. Choosing one direction round the shape (all clockwise, say) just keeps you to one exterior angle per corner.

Exterior angle = the angle on the outside between a stretched-out edge and the next edge. It is never the whole reflex angle, and it makes 180° with the interior angle.

Missing angle in an irregular polygon

Problem

A pentagon has interior angles of 100°, 95°, 103° and 88°. Find the fifth interior angle.

Maths · Regular polygons

Could a regular polygon have this angle?

Divide 360 by each exterior angle. Is the answer a whole number of sides? Decide first, then read why.

Still to sort

A regular polygon has this exterior angle (0)

360 ÷ the angle is a whole number.

Where the line is: The division comes out exactly, with nothing left over.

No regular polygon has this exterior angle (0)

360 ÷ the angle is not a whole number.

Where the line is: The division leaves a decimal, and there is no such thing as 7.2 sides.

7 of 7 still to sort.

In a regular polygon every exterior angle is the same size. Share one full turn of 360° between the corners and each exterior angle is 360° ÷ the number of sides. Flip it round: the number of sides is 360° ÷ the exterior angle. And you can't have a polygon with a fraction of a side.

Maths · Algebra

The 360° fact proves something you already know

A triangle's three interior angles are a, b and c. Use the exterior angles to show that a + b + c = 180°. Step through it and watch what each line does.

GoalShow that the interior angles of a triangle add up to 180°

(180 − a) + (180 − b) + (180 − c) = 360

start

Each exterior angle is 180° minus its interior angle, and the three exterior angles of the triangle total 360°.

Step 1 of 4

Each exterior angle is 180° minus its interior angle, and the three exterior angles of the triangle total 360°.

Watch out: The brackets matter. The minus sign in 540 − (a + b + c) takes away a, b and c together.

Maths · Regular polygons

Where does this working go wrong?

A regular hexagon and a regular octagon sit side by side and share one edge. At one end of that shared edge, the hexagon's corner, the octagon's corner and an empty gap fit together around the point. Find the size of the gap.

A student's working — which line goes wrong?

Angles around a point add up to 360°, so gap = 360° − hexagon corner − octagon corner.
Hexagon corner: 360° ÷ 6 = 60°, and 180° − 60° = 120°.
Octagon corner: 360° ÷ 8 = 45°.
Gap = 360° − (120° + 45°) = 195°.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Walk all the way round any polygon and you turn exactly once.

What you need to know

An interior angle is inside the polygon, between two edges. An exterior angle is outside, between a stretched-out edge and the next edge.
An exterior angle and its interior angle make a straight line, so they add up to 180°: exterior = 180° − interior.
The exterior angles of any polygon add up to 360°, however many sides it has. Walk round the edge and you finish exactly one full turn.
In a regular polygon every exterior angle is equal, so each is 360° ÷ the number of sides, and the number of sides is 360° ÷ the exterior angle.
A regular polygon can only have a given exterior angle if 360 divided by it is a whole number.

The big picture

The exterior angles of any polygon add up to 360°. Walk round the edge, turning through each exterior angle, and by the time you're back at the start you've turned exactly once. Zoom out until the polygon is a dot and the exterior angles sit round that one point. Each exterior angle makes a straight line with its interior angle (180° together), and in a regular polygon each one is 360° ÷ the number of sides.

Key points

1Exterior angle: the angle outside a polygon between a stretched-out edge and the next edge. It is not the whole reflex angle outside the corner.

2Stretch either edge at a corner and you get an exterior angle of the same size.

3Interior + exterior = 180° at the same corner.

4Exterior angles of any polygon total 360°: one full turn.

5Missing angle: change the given interior angles to exterior angles, subtract their total from 360°, then change back if you need the interior angle.

6Regular polygon: exterior angle = 360° ÷ n, interior angle = 180° − exterior angle, and n = 360° ÷ exterior angle.

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Worked example

Problem

Each exterior angle of a regular polygon is 24°. (a) How many sides does it have? (b) What is each interior angle?

⚠ Watch out

Using 360 ÷ n as the interior angle. For a regular polygon, 360 ÷ n is each exterior angle. For 5 sides, 360 ÷ 5 = 72° is the exterior angle, and the interior angle is 180° − 72° = 108°.

🧠

Memory hook

One lap round the edge is one full turn. Share the 360° between the corners and you get 360 ÷ sides.

✓

Check yourself

Priya says a regular polygon can have exterior angles of 14°. Is she right? Check: 360 ÷ 14 is about 25.7, not a whole number, so no.

Flashcards

(13)

What is an interior angle of a polygon?

An angle inside the polygon, formed by two of its edges.

What is an exterior angle?

The angle on the outside between a stretched-out edge and the next edge. It is not the whole reflex angle outside the corner.

Which edge do you stretch out to make an exterior angle at a corner?

Either one. Both give an exterior angle of the same size.

At one corner, what do the interior angle and exterior angle add up to, and why?

180°, because the stretched-out edge makes a straight line. They are supplementary.

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

360°, however many sides it has.

Why do the exterior angles make one full turn?

Walk round the polygon turning through each exterior angle and you are back at the start, having turned exactly once.

How do you find each exterior angle of a regular polygon with n sides?

360° ÷ n. A regular pentagon: 360 ÷ 5 = 72°.

Regular polygon, exterior angle known: how do you find the number of sides?

360° ÷ the exterior angle. 45° gives 360 ÷ 45 = 8 sides.

What do you take away from 180° to get an interior angle of a regular polygon?

Its exterior angle. An exterior angle of 36° gives an interior angle of 144°.

How can you tell whether a regular polygon can have a given exterior angle?

360 ÷ the angle must be a whole number. 40° gives 9 sides, but 50° gives 7.2, so no regular polygon has 50°.

How do you find a missing angle in an irregular polygon using exterior angles?

Change each given interior angle into an exterior angle, subtract their total from 360°, then change back if you need the interior angle.

An interior angle of a regular polygon is 144°. How many sides has it?

Exterior angle = 180° − 144° = 36°, then 360 ÷ 36 = 10 sides. (The interior-sum route, 180(n − 2) = 144n, also gives n = 10.)

Two regular polygons share a corner point and an edge. How do you find the gap around that point?

360° minus the two interior angles. Or add the two exterior angles: a hexagon and an octagon give 60° + 45° = 105°.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Which one is the exterior angle?

Could a regular polygon have this angle?

The 360° fact proves something you already know

Where does this working go wrong?

More KS3 Maths topics