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SavannahYear 8
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MathsYear 9 Maths: standard form, modelling and Pythagoras30 min★★★ difficulty

Pythagoras' theorem

In any right-angled triangle, the three sides are locked together by one relationship. Once you know two of them, the third is fixed — which is why builders, navigators and game designers all rely on it.

Lesson overview

What you'll learn in this lesson

Use Pythagoras' theorem to solve problems involving right-angled triangles

Key learning points

  • • What the theorem says
  • • Finding the longest side
  • • Finding a shorter side
  • • Using it in real problems

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

pythagorastheoremproblemsinvolvingright-angled

Scroll down — the lesson carries on below

Part 1 of 19 · Discover5%
1

Watch & discover

Part 1 of 19

Watch first

Pythagoras' theorem

An English teaching video on this topic from Cognito. Watch it, then scroll down for the lesson.

In a nutshell

Use Pythagoras' theorem to solve problems involving right-angled triangles

2

What you already know

Part 2 of 19

Before we start

What you already know

You should already be able to square and square-root numbers and rearrange a simple equation.

Recap: Modelling with expressions and formulae
3

Key words

Part 3 of 19

Key words

Words you'll need today

Hypotenuse
The longest side, always opposite the right angle.
Square
Multiply a number by itself.
Square root
The number that was squared to give this value.
Right-angled triangle
A triangle containing an angle of 90°.
4

Learning cycle

Part 4 of 19

Learning cycle 1 of 2

Finding the hypotenuse

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

5

Explore the idea

Part 5 of 19

Learn

What the theorem says

For a right-angled triangle with shorter sides a and b and hypotenuse c, a² + b² = c². The squares are areas: the square drawn on the longest side has exactly the same area as the two smaller squares combined. Identifying the hypotenuse first is the safest habit, because it is always opposite the right angle and always the longest side.

6

Explore the idea

Part 6 of 19

Learn

Finding the longest side

If the two shorter sides are 6 cm and 8 cm, then c² = 36 + 64 = 100, so c = 10 cm. Square, add, then square-root. Leave the answer exact when a question asks for it, for example √52, or round sensibly to one or two decimal places for a measurement. Always check the answer is longer than either short side, otherwise something has gone wrong.

7

Quick check

Part 7 of 19

Quick check

The hypotenuse is always...

8

Quick check

Part 8 of 19

Quick check

Shorter sides 9 and 12. The hypotenuse is...

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

Finding a shorter side and applying it

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

11

Explore the idea

Part 11 of 19

Learn

Finding a shorter side

When the hypotenuse is known you subtract instead. If c = 13 and a = 5, then b² = 169 − 25 = 144, so b = 12. Subtracting the wrong way round gives a negative square, which is the clearest signal that the hypotenuse has been misidentified. Writing the equation as c² − a² = b² before substituting keeps the order right.

12

Explore the idea

Part 12 of 19

Learn

Using it in real problems

Ladders against walls, diagonals of rectangles, distances between coordinates and the shortest route across a field are all Pythagoras questions in disguise. Sketch the triangle, label the right angle, and mark which side is unknown. For coordinates, the horizontal and vertical gaps are the two shorter sides, so the distance between (1, 2) and (4, 6) is √(3² + 4²) = 5.

13

Quick check

Part 13 of 19

Quick check

Hypotenuse 25, one short side 7. The other is...

14

Quick check

Part 14 of 19

Quick check

Distance between (0, 0) and (5, 12) is...

15

Common mix-ups

Part 15 of 19

Common mix-ups

Things people often get wrong

  • People often add the sides instead of the squares. It is a² + b², not a + b.
  • People often forget to square-root at the end and leave the answer as the squared value.
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.

For a ____ triangle with shorter sides a and b and hypotenuse c, a² + b² = c².

Ladders against walls, diagonals of rectangles, distances between coordinates and the shortest route across a field are all ____ questions in disguise.

18

Mastery quiz

Part 18 of 19

Marked quiz

End of lesson quiz: Pythagoras' theorem

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

  1. 1. Pythagoras' theorem applies to...

  2. 2. Sides 5 and 12: hypotenuse =

  3. 3. Hypotenuse 10, short side 6: other side =

  4. 4. A triangle has sides 3, 4 and 6. Is it right-angled?

  5. 5. A ladder 5 m long reaches 4 m up a wall. Its foot is...

  6. 6. Distance between (2, 3) and (6, 6) is...

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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