Good Learning Co.
SavannahYear 8
← Maths
MathsAngles, congruence and similarity30 min★★★ difficulty

Angle sums in triangles and polygons

You only really need one fact — the angles in a triangle add to 180° — because every polygon can be cut into triangles.

Lesson overview

What you'll learn in this lesson

Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon

Key learning points

  • • Why a triangle is 180°
  • • Any polygon
  • • Regular polygons
  • • Writing reasons

This lesson at a glance

  • 30 minutes
  • 19 parts to scroll through
  • 4 quick checks
  • Marked quiz at the end
  • Gentle pace: short sittings with pauses

Words to know

deriveanglestrianglededucepolygon

Scroll down — the lesson carries on below

Part 1 of 19 · Discover5%
1

Watch & discover

Part 1 of 19

Watch first

Angle sums in triangles and polygons

An English teaching video on this topic from SHFT - Your Learning. Watch it, then scroll down for the lesson.

In a nutshell

Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon

2

What you already know

Part 2 of 19

Before we start

What you already know

You know angles on a straight line sum to 180° and around a point to 360°.

3

Key words

Part 3 of 19

Key words

Words you'll need today

interior angle
An angle inside the polygon at a vertex.
exterior angle
The angle between a side extended and the next side.
regular polygon
A polygon with all sides and all angles equal.
4

Learning cycle

Part 4 of 19

Learning cycle 1 of 2

Part 1 — Deriving the rules

A short piece of teaching, then a check to make sure it has landed.

5

Explore the idea

Part 5 of 19

Learn

Why a triangle is 180°

Draw a line through one vertex parallel to the opposite side. The two outer angles are alternate angles equal to the other two angles of the triangle, and together with the third they lie on a straight line — 180°. This is a proof, not a measurement, which is why it holds for every triangle ever drawn.

6

Explore the idea

Part 6 of 19

Learn

Any polygon

Split a polygon from one vertex into triangles: a quadrilateral gives 2, a pentagon 3, a hexagon 4. So the interior angle sum is (n − 2) × 180°. A pentagon is 540°, a hexagon 720°, a decagon 1,440°. You never need to memorise these once you can count the triangles.

7

Quick check

Part 7 of 19

Quick check

Interior angle sum of a pentagon?

8

Quick check

Part 8 of 19

Quick check

Each interior angle of a regular octagon?

9

Reset break

Part 9 of 19

Pause

That's sitting 1 of 2 done

Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.

Stop here for now
10

Learning cycle

Part 10 of 19

Learning cycle 2 of 2

Part 2 — Using them in problems

A short piece of teaching, then a check to make sure it has landed.

11

Explore the idea

Part 11 of 19

Learn

Regular polygons

In a regular polygon, each interior angle is (n − 2) × 180° ÷ n. For a regular hexagon that is 720 ÷ 6 = 120°. Exterior angles of any polygon always sum to 360°, so each exterior angle of a regular n-gon is 360 ÷ n — often the faster route: 360 ÷ 6 = 60°, and 180 − 60 = 120°.

12

Explore the idea

Part 12 of 19

Learn

Writing reasons

Exam-style angle questions award marks for reasons, not just numbers. Use the proper phrases: 'angles in a triangle sum to 180°', 'base angles of an isosceles triangle are equal', 'alternate angles are equal', 'exterior angles of a polygon sum to 360°'. One reason per step is enough.

13

Quick check

Part 13 of 19

Quick check

Exterior angles of any polygon sum to...

14

Quick check

Part 14 of 19

Quick check

A triangle has angles 47° and 68°. The third is...

15

Common mix-ups

Part 15 of 19

Common mix-ups

Things people often get wrong

  • Using 360° for the angle sum of every polygon.
  • Dividing the interior sum by the wrong number of sides.
  • Giving an answer with no reason in a 'give reasons' question.
16

Challenge round

Part 16 of 19

Game · Sort it

Which of these are true?

Drag each card into the right column. Tap a card first if dragging is fiddly.

True

Not true

17

Challenge round

Part 17 of 19

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

The two outer ____ are alternate angles equal to the other two angles of the triangle, and together with the third they lie on a straight line — 180°.

This is a proof, not a measurement, which is why it holds for every ____ ever drawn.

Split a ____ from one vertex into triangles: a quadrilateral gives 2, a pentagon 3, a hexagon 4.

18

Mastery quiz

Part 18 of 19

Marked quiz

End of lesson quiz: Angle sums in triangles and polygons

6 questions, marked with the reasoning shown. No timer.

  1. 1. Interior angle sum of a hexagon?

  2. 2. Each exterior angle of a regular decagon?

  3. 3. A regular polygon has exterior angles of 24°. How many sides?

  4. 4. Angles in a quadrilateral sum to...

  5. 5. Isosceles triangle, apex 40°. Each base angle?

  6. 6. Which is a valid reason?

19

Lesson round-up

Part 19 of 19

Lesson round-up

Ready when you are

Quiz score

Not sat

Games

Not played

Points this lesson

0

Best run

0 in a row

Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?

Ask Luna

Do the work

Your written work for this lesson

This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.

Article

The cost of a school trip

A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.

If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.

The school decides to charge £20 and to use anything left over for a second trip later in the year.

Written for Good Learning Co.

Reading questions

  1. Work out the total cost of the trip if 50 pupils go.
  2. Show why the price per pupil falls when more pupils go, using numbers.
  3. At £20 each, how many pupils must go before the trip makes money? Show your working.

  • Working out

    Working out

    Do the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.

    10–15 questions

  • Worked example, explained

    Writing

    Copy one question out and write underneath, in words, why each step works.

    1 question, 4–5 sentences

  • Problem in words

    Working out

    Write your own real-life problem that uses today's method, then solve it.

    1 problem + solution

Saved as you type.

Part 1 of 19 · Watch & discover

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