A right-angled triangle's legs are 16 cm and 12 cm: 16^2 + 12^2 = □ + □ = □, so the hypotenuse = □ cm. Fill in each blank

256, 144, 400, 20. Square each first, 256 and 144, which add to 400; 400 = 20 × 20, so the hypotenuse is 20 cm.

Pythagoras' theorem · Ages 13–14

Finding the hypotenuse

The hypotenuse faces the right angle; c² = a² + b², then the root.

How to do it

Method

The hypotenuse faces the right angle; c^2 = a^2 + b^2, then the root

6² + 8² = 100, 100 = 10

Another way

A multiple of a triple

6, 8 = 2 × (3, 4), so the hypotenuse = 2 × 5 = 10

In full

In a right-angled triangle the side opposite the right angle is the longest: the hypotenuse; the other two are the legs. However the triangle is turned, find the right angle first, and the side facing it is the hypotenuse. Pythagoras' theorem: the squares of the two legs add up to the square of the hypotenuse, a² + b² = c². To find the hypotenuse, square each leg, add, then take the square root. Legs of 6 cm and 8 cm: 6² + 8² = 36 + 64 = 100, so the hypotenuse = 100 = 10 cm. When the root doesn't come out whole, leave it as □ (say 41 cm) or take a calculator's approximation (41 ≈ 6.4). Careful: the hypotenuse is not the legs added, 6 + 8 = 14, far longer than 10. Or use a known Pythagorean triple: 6 and 8 are 2 times 3 and 4, so the hypotenuse is 2 times 5, which is 10.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
A right-angled triangle's legs are 16 cm and 12 cm: 16² + 12² = + = , so the hypotenuse = cm.
Fill in each blank
Show the answer and working
Answer: 256, 144, 400, 20
Square each first, 256 and 144, which add to 400; 400 = 20 × 20, so the hypotenuse is 20 cm.
Example 2
Triangle ABC in the picture is right-angled. Which side is the hypotenuse?
- A. It has none
- B. AB
- C. CA
- D. BC
Choose the right answer
Show the answer and working
Answer: C. CA
Find the right angle (the one with the little square); the side facing it, not touching that corner, is the hypotenuse, the longest side.
Example 3
In the right-angled triangle, the hypotenuse ? = cm.
Show the answer and working
Answer: 8
2² + 2² = 4 + 4 = 8, not a square number, so the root doesn't come out whole: the hypotenuse is 8 cm.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.A right-angled triangle's legs are 36 cm and 48 cm: 36² + 48² = + = , so the hypotenuse = cm.
Fill in each blank
Answer:
2.Triangle ABC in the picture is right-angled. Which side is the hypotenuse?
- A. It has none
- B. CA
- C. BC
- D. AB
Choose the right answer
Answer:
3.A right-angled triangle has legs of 10 cm and 5 cm. How long is its hypotenuse, in cm, to 1 decimal place?
Answer:
4.A right-angled triangle with legs of 12 cm and 16 cm has a hypotenuse of 20 cm.
True or false?
Answer:
5.A right-angled triangle's legs are 0.8 m and 0.6 m. How long is its hypotenuse, in m?
Answer:

All the answers

1. 1296, 2304, 3600, 60
2. D. AB
3. 11.2 cm
4. True
5. 1 m

Free worksheet: Finding the hypotenuseWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

Start playingPlay and learn in the secondary world, Star Galaxy: animated methods, monster battles, a review of every mistake