Find a side with Pythagoras' theorem
Use Pythagoras' theorem only for a right-angled triangle. The hypotenuse is opposite the right angle and is the longest side
\[c^2=a^2+b^2\]
- Choose the operation Add squares to find the hypotenuse. Subtract the square of the known shorter side to find the other shorter side \(\displaystyle \begin{aligned}c^2&=a^2+b^2\\a^2&=c^2-b^2\end{aligned}\)
- Match the units Convert the given lengths before squaring \(\displaystyle 1\text{ cm}=10\text{ mm}\)
- Return to a length Take the positive square root because a side length is positive \(\displaystyle \begin{aligned}x^2&=N\\x&=\sqrt{N}\end{aligned}\)
- Use exact or rounded form An unsimplified square root is an exact answer. For a decimal answer, round once at the end \(\displaystyle \sqrt{83}\text{ cm}\approx9.11\text{ cm}\)
Worked example
Find the other shorter side of this right-angled triangle in metres. The hypotenuse is labelled c and the known shorter side is labelled b
\[\begin{aligned}c&=5.8\text{ m}\\b&=400\text{ cm}\end{aligned}\]
- \(\displaystyle 400\text{ cm}=4\text{ m}\)Express both known lengths in metres
- \(\displaystyle a^2+4^2=5.8^2\)The sum of the squares of the shorter sides equals the square of the hypotenuse
- \(\displaystyle a^2+16=33.64\)Evaluate the two known squares
- \(\displaystyle a^2+16-16=33.64-16\)Subtract the square of the known shorter side from both sides
- \(\displaystyle a^2=17.64\)Simplify both sides
- \(\displaystyle a=\sqrt{17.64}\)Take the positive square root to obtain a length
- \(\displaystyle a=4.2\text{ m}\)This length is exact
- \(\displaystyle 4.2\text{ m}<5.8\text{ m}\)The missing shorter side is shorter than the hypotenuse
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 rootAdding squares for a shorter side
✕\(\displaystyle a^2=c^2+b^2\)✓\(\displaystyle a^2=c^2-b^2\)
Subtract the square of the known shorter side from the square of the hypotenuseStopping 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 either missing side with whole-number answers
standardFind either missing side and round as requested
challengeConvert length units before calculating, or complete an exact square-root length
examChoose a rounded or exact side 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 do I subtract?+
Subtract when the hypotenuse is known and a shorter side is missing. Subtract the square of the known shorter side from the square of the hypotenuse
When should I round?+
Round the final length only when requested. Keep the exact square root until that step
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 3: Pythagoras and trigonometry
In this chapter
3A Finding the hypotenuse: Find a side of a right-angled triangle3B Finding a shorter side: Find a side of a right-angled triangleApply Pythagoras' theorem in two-dimensional contextsPythagoras in three dimensions (optional extension)Name sides and state trigonometric ratiosChoose a trigonometric ratio3F Unknown side in the numerator: Evaluate a trigonometric ratio3F Unknown side in the numerator: Find an unknown side of a right-angled triangle3G Unknown side in the denominator: Find a side in the denominator3H Finding angles with trigonometry: Find an angle from a trigonometric ratioFind an angle of a right-angled triangle from two sidesAngles of elevation and depressionSolve a practical right-angled-triangle problem (optional practice)Read and state three-figure bearingsSolve a navigation problem with bearings