Find the hypotenuse
The hypotenuse is opposite the right angle. Its square equals the sum of the squares of the shorter sides
\[c^2=a^2+b^2\]
- Identify before calculating The hypotenuse need not be sloping in the drawing
- Convert before squaring Use the same length unit for both known sides
- Take the positive root The sum gives a squared length, not the side itself \(\displaystyle c=\sqrt{a^2+b^2}\)
- Round at the end Keep an exact square root until the requested final approximation
Worked example
A rectangular sign is 210 cm wide and 2.8 m high. Find its diagonal in metres
\[c^2=a^2+b^2\]
- \(\displaystyle 210\text{ cm}=2.1\text{ m}\)Use metres for both sides
- \(\displaystyle c^2=2.1^2+2.8^2\)The diagonal is opposite a right angle
- \(\displaystyle c^2=4.41+7.84=12.25\)Square each length and add
- \(\displaystyle c=\sqrt{12.25}=3.5\text{ m}\)Take the positive square root
- \(\displaystyle 2.8<3.5<4.9\)The diagonal exceeds the longer side and is less than the sum
Common mistakes
Adding side lengths
✕\(\displaystyle c=a+b\)✓\(\displaystyle c=\sqrt{a^2+b^2}\)
Add the squares of the shorter lengths, then take the positive square rootStopping at a squared length
✕\(\displaystyle \begin{aligned}x^2&=185\\x&=185\text{ cm}\end{aligned}\)✓\(\displaystyle \begin{aligned}x^2&=185\\x&=\sqrt{185}\text{ cm}\end{aligned}\)
Take the positive square root to change the squared length into a lengthDifficulty levels
introFind a hypotenuse with a whole-number answer
standardFind a hypotenuse and round as requested
challengeConvert mixed units before finding the hypotenuse
examChoose a rounded hypotenuse length
Questions about this topic
How do I find the hypotenuse?+
Find the right angle. The side opposite it is the hypotenuse, whatever way the triangle is drawn
When should I round?+
Round the final length only when requested. Keep the exact square root until that step
Why take the positive square root?+
A length is positive. The negative root does not represent a side length
What if there is no right angle?+
Pythagoras' theorem cannot be applied without establishing a right triangle
Does the diagonal include fixing allowances?+
No. Add any stated allowances separately after calculating the corner-to-corner span
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 9: Pythagoras and trigonometry
In this chapter
9A Finding the hypotenuse: Find a side of a right-angled triangleDetermine whether three sides form a right-angled triangle9B Finding a shorter side: Find a side of a right-angled triangle9C Pythagoras' theorem in 2D and 3D: Find a side of a right-angled trianglePythagoras in three dimensionsName sides and state trigonometric ratios9D Trigonometric ratios: Evaluate a trigonometric ratio9D Trigonometric ratios: Choose a trigonometric ratio9E Unknown side in the numerator: Find an unknown side of a right-angled triangle9E Unknown side in the numerator: Solve a practical right-angled-triangle problem9E Unknown side in the numerator: Evaluate a trigonometric ratio9E Unknown side in the numerator: Choose a trigonometric ratio9F Unknown side in the denominator: Find an unknown side of a right-angled triangle9F Unknown side in the denominator: Solve a practical right-angled-triangle problem9F Unknown side in the denominator: Choose a trigonometric ratioFind an angle of a right-angled triangle from two sides9G Finding angles with trigonometry: Solve a practical right-angled-triangle problem9G Finding angles with trigonometry: Evaluate a trigonometric ratio, and find an angle from its valueAngles of elevation and depressionRead and state three-figure bearingsSolve a navigation problem with bearingsTrigonometry in three dimensions