Good Learning Co.
SavannahYear 8
← Maths
MathsShape, probability and data30 min★★★ difficulty

Angles in triangles and polygons

Polygon angle facts are not things to memorise separately: every one of them can be derived from the triangle, which is why deriving beats remembering.

Part of your national curriculum
  • Geometry and measures: Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon

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 totals 180°
  • • Angle sum of any polygon
  • • Regular polygons
  • • Reasoning, not just answers

This lesson at a glance

  • 30 minutes
  • 17 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 17 · Discover6%
1

Watch & discover

Part 1 of 17

Watch first

Angles in triangles and polygons

An English teaching video on this topic from mathantics. 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

Learning cycle

Part 2 of 17

Learning cycle 1 of 2

Part 1 · Why a triangle totals 180°

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

3

Explore the idea

Part 3 of 17

Learn

Why a triangle totals 180°

Draw a line through one vertex parallel to the opposite side. The two outer angles are alternate angles equal to the base angles, and together with the third angle they form a straight line of 180°. That single argument is the foundation for everything else in this lesson.

4

Explore the idea

Part 4 of 17

Learn

Angle sum of any polygon

A polygon with n sides can be split into n − 2 triangles from one vertex, so the interior angles total 180(n − 2)°. A hexagon gives 720° and an octagon 1,080°. The exterior angles of any convex polygon always total 360°, however many sides it has.

5

Quick check

Part 5 of 17

Quick check

The interior angles of a hexagon total

6

Quick check

Part 6 of 17

Quick check

The exterior angles of any convex polygon total

7

Learning cycle

Part 7 of 17

Learning cycle 2 of 2

Part 2 · Regular polygons

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

8

Reset break

Part 8 of 17

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
9

Explore the idea

Part 9 of 17

Learn

Regular polygons

In a regular polygon every exterior angle is 360 ÷ n, and each interior angle is 180 minus that. Working through the exterior angle is almost always quicker: if each interior angle is 156°, the exterior is 24° and n = 360 ÷ 24 = 15 sides.

10

Explore the idea

Part 10 of 17

Learn

Reasoning, not just answers

Full marks require the reason with the value: 'x = 68° because alternate angles between parallel lines are equal'. A chain of reasons should read like a short argument, each line justified by a named fact.

11

Quick check

Part 11 of 17

Quick check

Each exterior angle of a regular decagon is

12

Quick check

Part 12 of 17

Quick check

A full-mark angle answer includes

13

Explore the idea

Part 13 of 17

Worked example

Worked answer: each interior angle of a regular polygon is 156°. How many sides? (3 marks)

Interior and exterior angles at a vertex lie on a straight line, so the exterior angle is 180 − 156 = 24° (1). The exterior angles of any polygon total 360°, so n = 360 ÷ 24 (1), giving 15 sides (1). Dividing 156 into 360 is a frequent error and gives a non-integer, which should itself signal the mistake.

14

Challenge round

Part 14 of 17

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

15

Challenge round

Part 15 of 17

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

The two outer ____ are alternate angles equal to the base angles, and together with the third angle they form a straight line of 180°.

A ____ with n sides can be split into n − 2 triangles from one vertex, so the interior angles total 180(n − 2)°.

16

Mastery quiz

Part 16 of 17

Marked quiz

End of lesson quiz: Angles in triangles and polygons

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

  1. 1. The interior angles of a hexagon total

  2. 2. The exterior angles of any convex polygon total

  3. 3. Each exterior angle of a regular decagon is

  4. 4. A full-mark angle answer includes

17

Lesson round-up

Part 17 of 17

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 17 · Watch & discover

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

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