Year 10 · Mathematics · Measurement
Pythagoras' theorem
Use Pythagoras' theorem to find unknown side lengths in right-angled triangles and to decide whether a triangle is right-angled.
Awaiting educator review. This material was drafted by our content team with AI assistance and is waiting for review by a qualified educator. Please check it suits your child before using it. Created 10 October 2026.
These lessons were written around Australian Curriculum strands and matched to New Zealand year levels by age (NZ Year 1 is the first year of school). They are not yet mapped to The New Zealand Curriculum, so check them against your own learning programme.
Learning goals
Learners will
- Identify the hypotenuse of a right-angled triangle.
- Use Pythagoras' theorem to find the hypotenuse or a shorter side.
- Use the theorem to test whether a triangle contains a right angle.
Success looks like
- I can point to the hypotenuse in any right-angled triangle.
- I can find the hypotenuse given the two shorter sides.
- I can find a shorter side given the hypotenuse and the other side.
- I can solve a worded problem, such as a ladder against a wall.
The big idea
In a right-angled triangle, the side opposite the right angle is called the hypotenuse. It is always the longest side. We usually call it c and call the other two sides a and b.
Pythagoras' theorem says: c^2 = a^2 + b^2. In words, if you draw a square on each side of a right-angled triangle, the area of the square on the hypotenuse equals the total area of the other two squares.
Finding the hypotenuse: square the two shorter sides, add the results, then take the square root. For sides 6 and 8: 36 + 64 = 100, and sqrt(100) = 10.
Finding a shorter side: rearrange to a^2 = c^2 − b^2. Square the hypotenuse, subtract the square of the known shorter side, then take the square root. A quick check: the answer must be shorter than the hypotenuse.
Testing for a right angle: if the square of the longest side equals the sum of the squares of the other two sides, the triangle is right-angled. If not, it is not.
Some sets of whole numbers fit the theorem exactly, such as 3, 4, 5 and 5, 12, 13. Multiples of these also work, such as 6, 8, 10. Builders have long used a 3-4-5 triangle to check that corners are square.
Worked examples
Finding the hypotenuse
A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the hypotenuse.
- c^2 = 6^2 + 8^2 = 36 + 64 = 100.
- c = sqrt(100) = 10.
Answer: 10 cm
Finding a shorter side
A right-angled triangle has a hypotenuse of 13 m and one shorter side of 5 m. Find the other side.
- a^2 = c^2 − b^2 = 13^2 − 5^2 = 169 − 25 = 144.
- a = sqrt(144) = 12.
- Check: 12 is shorter than the hypotenuse of 13, which makes sense.
Answer: 12 m
Is it right-angled?
A triangle has sides 7 cm, 24 cm and 25 cm. Is it right-angled?
- The longest side is 25: 25^2 = 625.
- The other two: 7^2 + 24^2 = 49 + 576 = 625.
- The two results are equal, so the theorem holds.
Answer: Yes, it is right-angled (the right angle is opposite the 25 cm side).
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
A right-angled triangle has shorter sides of 3 cm and 4 cm. How long is the hypotenuse?
5 cm — 3^2 + 4^2 = 9 + 16 = 25, and sqrt(25) = 5 cm.
Which side of a right-angled triangle is the hypotenuse?
The side opposite the right angle — The hypotenuse is opposite the right angle and is always the longest side, whichever way the triangle is drawn.
Which set of side lengths makes a right-angled triangle?
8, 15, 17 — 8^2 + 15^2 = 64 + 225 = 289 = 17^2. The other sets fail: 36 + 49 = 85 ≠ 81, 25 + 100 = 125 ≠ 144, 16 + 25 = 41 ≠ 36.
A right-angled triangle has a hypotenuse of 10 cm and one shorter side of 6 cm. How long is the other side?
8 cm — 10^2 − 6^2 = 100 − 36 = 64, and sqrt(64) = 8 cm.
A 5 m ladder leans against a vertical wall. Its foot is 3 m from the base of the wall on level ground. How high up the wall does it reach?
4 m — The ladder is the hypotenuse. 5^2 − 3^2 = 25 − 9 = 16, and sqrt(16) = 4 m.
A right-angled triangle has shorter sides of 9 cm and 12 cm. Find the length of the hypotenuse in cm.
15 cm — 9^2 + 12^2 = 81 + 144 = 225, and sqrt(225) = 15 cm.
A right-angled triangle has a hypotenuse of 26 m and a shorter side of 10 m. Find the other side in metres.
24 m — 26^2 − 10^2 = 676 − 100 = 576, and sqrt(576) = 24 m.
A right-angled triangle has shorter sides of 5 cm and 7 cm. Find the hypotenuse in cm, correct to one decimal place.
8.6 cm — 5^2 + 7^2 = 25 + 49 = 74, and sqrt(74) ≈ 8.602, which rounds to 8.6 cm.
Watch out for
- Adding the squares when finding a shorter side. If the hypotenuse is known, you subtract. A check: a shorter side must be less than the hypotenuse.
- Forgetting to take the square root at the end, giving c^2 instead of c.
- Thinking the hypotenuse is always the side drawn at the bottom or on the slant. Find the right angle first; the hypotenuse is directly opposite it.
- Using Pythagoras' theorem on triangles that are not right-angled. It only applies when there is a 90° angle.
Tips for parents
- Ask your learner to label the hypotenuse with a c before doing any calculation. This one habit prevents most errors.
- Encourage a quick reasonableness check: the hypotenuse must be longer than either shorter side but shorter than the two added together.
- Measure the diagonal of a rectangular table, book or TV screen together and compare it with the value from Pythagoras' theorem.
Go further
- Use string and a tape measure to mark out a 3 m, 4 m, 5 m triangle in the backyard and check that the corner is square.
- Find as many Pythagorean triads as you can with all sides less than 50. Which ones are not just multiples of another triad?
- Investigate the shortest walking route diagonally across a rectangular park compared with walking along two edges.