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
- 25 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 25
Visual introductionPicture this
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.
In a nutshell
Use Pythagoras' theorem to solve problems involving right-angled triangles
What you already know
Part 2 of 25
Before we start
What you already know
You should already be able to square and square-root numbers and rearrange a simple equation.
Key words
Part 3 of 25
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°.
Reset break
Part 4 of 25
Pause
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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.
Learning cycle
Part 5 of 25
Learning cycle 1 of 2
Finding the hypotenuse
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 6 of 25
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.
Explore the idea
Part 7 of 25
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.
Reset break
Part 8 of 25
Pause
That's sitting 2 of 6 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.
Quick check
Part 9 of 25
Quick check
Part 10 of 25
Learning cycle
Part 11 of 25
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.
Reset break
Part 12 of 25
Pause
That's sitting 3 of 6 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.
Explore the idea
Part 13 of 25
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.
Explore the idea
Part 14 of 25
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.
Quick check
Part 15 of 25
Reset break
Part 16 of 25
Pause
That's sitting 4 of 6 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.
Quick check
Part 17 of 25
Common mix-ups
Part 18 of 25
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.
Challenge round
Part 19 of 25
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
Reset break
Part 20 of 25
Pause
That's sitting 5 of 6 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.
Challenge round
Part 21 of 25
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.
Challenge round
Part 22 of 25
Game · Recall cards
The hypotenuse is always...
Card 1 of 4
Exit quiz
Part 23 of 25
Exit quiz
Show what you've learned
6 questions, marked together at the end. Nothing is timed.
1. Pythagoras' theorem applies to...
2. Sides 5 and 12: hypotenuse =
3. Hypotenuse 10, short side 6: other side =
4. A triangle has sides 3, 4 and 6. Is it right-angled?
5. A ladder 5 m long reaches 4 m up a wall. Its foot is...
6. Distance between (2, 3) and (6, 6) is...
Mastery quiz
Part 24 of 25
Marked quiz
End of lesson quiz: Pythagoras' theorem
4 questions, marked with the reasoning shown. No timer.
1. The hypotenuse is always...
2. Shorter sides 9 and 12. The hypotenuse is...
3. Hypotenuse 25, one short side 7. The other is...
4. Distance between (0, 0) and (5, 12) is...
Lesson round-up
Part 25 of 25
Lesson round-up
Ready when you are
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Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?
Ask LunaPart 1 of 25 · Watch & discover