Draw a square on each side of a right-angled triangle. The two smaller squares together have exactly the same area as the biggest: a² + b² = c², where c is the hypotenuse, the side opposite the right angle.
Builders check a corner is square with a 3-4-5 triangle: 3 m along one wall, 4 m along the other, and if the diagonal is exactly 5 m, the corner is a right angle. People have used this for over three thousand years.
In a right-angled triangle, the longest side is opposite the right angle. It is called the hypotenuse, c. The two sides that meet at the right angle are a and b.
Pythagoras' theorem says that the squares on the two shorter sides add up to the square on the hypotenuse:
a2+b2=c2
To find the hypotenuse: square the two shorter sides, add, then take the square root.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Sides 3 and 4
Is the hypotenuse shorter or longer than going round the corner, 3 + 4?
TryTry it on the board and compare the result with your prediction.
Squares on the sides
Count the unit squares in each square, or read the numbers inside them.
Try it yourselfYour edits are saved in this browser.
Make a = 3 and b = 4. Do the two smaller squares add up to the biggest? Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Square, add, root
Here is the method.
Worked exampleBefore each step, say what you think comes next.
c is opposite the right angleFind the hypotenuse first.
a2+b2=62+82=36+64=100Square the two shorter sides and add.
c=100=10 cmTake the square root.
6<10<6+8Check: longer than each shorter side, shorter than both together.
Now try one: sides 12 cm and 16 cm.
Read Leah's working. Which is shorter: across or round the corner?
A right-angled triangle has sides 12 cm and 16 cm at the right angle. How long is the hypotenuse, in cm?
Leah finds the hypotenuse. Click the line where her method goes wrong.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
Set a = 5 and b = 12, then read the biggest square and c.
Try it yourselfYour edits are saved in this browser.
Make a right-angled triangle with sides 5 and 12 at the right angle. How long is the hypotenuse?
Use the board. This ticks when you get there.
The sides at the right angle are 9 m and 12 m. How long is the hypotenuse, in m?
The sides at the right angle are both 1 m. How long is the hypotenuse, in m, to two decimal places?
PractiseA few on your own, to check you have it.
The hypotenuse
The sides at the right angle are 8 cm and 15 cm. Hypotenuse, in cm?
The sides at the right angle are 2 cm and 3 cm. Hypotenuse, in cm, to two decimal places?
A park is a rectangle 30 m by 40 m. How long is the path straight across, corner to corner, in m?
Review: work out 144.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A hypotenuse of 10
Then: is 6-8-10 a new triangle, or 3-4-5 made bigger?
Try it yourselfYour edits are saved in this browser.
Need a hint?
c² = 100. Which two square numbers add to 100?
Another hint
36 + 64.
Make a right-angled triangle with whole-number sides whose hypotenuse is exactly 10.
Use the board. This ticks when you get there.
For teachers
Students build right-angled triangles with squares on each side, see that the two smaller squares exactly make the biggest, and use a² + b² = c² to find the hypotenuse (AC9M8M06, node 1). About 40 minutes.
I can find the hypotenuse of a right-angled triangle.
I can explain Pythagoras' theorem with squares on the sides.
I can check that an answer is sensible (longer than each short side, shorter than both together).
Before class
Do the Run board yourself first. Students need squares and square roots (Number Foundations) and right-angled triangles (Classifying triangles).
The one idea: the areas of the squares on the two shorter sides add to the area of the square on the hypotenuse. The common mistake is adding the sides (Leah's mistake). The sense check (between the longer short side and the sum) catches most slips.
This is the curriculum's Year 8 content; it is placed here, after circles, because it needs Year 7 squares and roots and triangle names.
Running the lesson
Predict: take a vote; students who chose 7 added; 25 forgot the root.
Run: count the small squares in the 9 and 16 and see the 25 together.
Investigate: write 'square, add, root' on the board.
Modify: for √2, ask between which whole numbers it lies before computing.
Make: ask whether 6-8-10 is new or a scaled 3-4-5.
The hard part (on purpose)
Keeping the order square, add, root. Ask students to say the three steps before they touch a calculator, and to check the answer against the two shorter sides.
When students get stuck
A student adds the sides (Leah's mistake). Ask: Is walking across shorter or longer than going round the corner? Fix: Square both, add, then root.
A student stops at c² (25 for 3 and 4). Ask: Is 25 a length or an area? Fix: It is the big square's area: root it for the side.
A student cannot tell which side is the hypotenuse. Ask: Where is the right angle? Which side does not touch it? Fix: The hypotenuse is opposite the right angle.
Stretch and support
Stretch: Find three more whole-number triangles (Pythagorean triples) that are not multiples of 3-4-5.
Support: Use squared paper for the squares; keep a list of squares to 20² visible.
Where it leads
Next: Finding a side (node 2), including a shorter side.
Where it leads: distance between points on a graph, trigonometry (Year 10), and 3D diagonals.
What each step is for
sides 3 and 4: Committing to a hypotenuse for legs 3 and 4 shows whether they add the sides (7), add the squares without rooting (25), or expect 5.
squares on the sides: Building a 3-4-5 triangle with squares on its sides shows the two small squares' areas add exactly to the big one.
square, add, root: Square both shorter sides, add, then root; Leah adds the sides, which is always too long. The hypotenuse is opposite the right angle.
one change at a time: One change at a time: another whole-number triangle on the board, a new pair of sides, a hypotenuse that is not a whole number, then why the hypotenuse is always the longest.
the hypotenuse: Whole-number and decimal hypotenuses and a real context, mixed; adding the sides is the planted slip; the review keeps square roots fresh.
a hypotenuse of 10: Working backwards from c = 10 means finding two squares that add to 100: a search, not a formula.
Now you can
Find the hypotenuse of a right-angled triangle.
Explain Pythagoras' theorem with squares on the sides.
Check that an answer is sensible (longer than each short side, shorter than both together).