MathsNumber, algebra and geometry: connected foundations30 min★★ difficulty

Chapter 8: Triangles and polygons

Every polygon angle sum can be derived by dividing the shape into triangles.

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

• Triangle sum
• Polygon formula
• Regular polygons

This lesson at a glance

30 minutes
14 parts to scroll through
3 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 14 · Discover7%

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Part 1 of 14

Watch first

Chapter 8: 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

Learning cycle

Part 2 of 14

Learning cycle 1 of 2

Part 1 · Triangle sum

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

Explore the idea

Part 3 of 14

Learn

Triangle sum

The interior angles of a triangle total 180°. Exterior angles of any polygon total 360° when one is taken at each vertex.

Explore the idea

Part 4 of 14

Learn

Polygon formula

An n-sided polygon divides into n−2 triangles from one vertex, so its interior angle sum is (n−2)×180°. A hexagon totals 720°.

Quick check

Part 5 of 14

Quick check

What is the interior angle sum of a pentagon?

Reset break

Part 6 of 14

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

Quick check

Part 7 of 14

Quick check

A triangle has angles 48° and 67°. Find the third.

Learning cycle

Part 8 of 14

Learning cycle 2 of 2

Part 2 · Regular polygons

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

Explore the idea

Part 9 of 14

Learn

Regular polygons

In a regular polygon all interior angles are equal, so divide the total by n. A regular hexagon has interior angle 720÷6=120°.

Quick check

Part 10 of 14

Quick check

Each interior angle of a regular hexagon is:

Challenge round

Part 11 of 14

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

Challenge round

Part 12 of 14

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

The interior ____ of a triangle total 180°.

Exterior angles of any ____ total 360° when one is taken at each vertex.

Mastery quiz

Part 13 of 14

Marked quiz

End of lesson quiz: Chapter 8: Triangles and polygons

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

1. What is the interior angle sum of a pentagon?
2. A triangle has angles 48° and 67°. Find the third.
3. Each interior angle of a regular hexagon is:

Lesson round-up

Part 14 of 14

Lesson round-up

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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

Work out the total cost of the trip if 50 pupils go.
Show why the price per pupil falls when more pupils go, using numbers.
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

Write your answer, notes or working here

Saved as you type.

Photos of paper work, diagrams or working outHow many minutes did the written work take?

Part 1 of 14 · 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