Finding a Shorter Side (Pythagoras) –

Finding a shorter side is the second key skill in Pythagoras' theorem questions. Instead of adding the squares of the two shorter sides, you subtract — and getting the direction of the subtraction right is where many students lose marks.

What Is Finding a Shorter Side?

When you know the hypotenuse and one of the shorter sides of a right-angled triangle, you can find the remaining shorter side by rearranging Pythagoras' theorem. The standard formula c² = a² + b² is rearranged so that the unknown shorter side is the subject.

The crucial difference from finding the hypotenuse is that you subtract instead of add. You always subtract the square of the known shorter side from the square of the hypotenuse. If you subtract the wrong way round, you get a negative number under the square root, which is impossible for a real length.

After subtracting, you take the square root to find the missing side. You should always check your answer by substituting back into the original formula to verify that a² + b² = c².

Key Formulas

Step-by-Step Method

1. Identify the hypotenuse (c) — the longest side, opposite the right angle.
2. Label the known shorter side (b) and the unknown shorter side (a).
3. Rearrange the formula: a² = c² − b².
4. Substitute the known values and subtract.
5. Take the square root to find a.
6. Check: does a² + b² = c²?

Worked Example 1 — Foundation Level

Question: A right-angled triangle has a hypotenuse of 13 cm and one side of 5 cm. Find the other side.

Working: a² = c² − b² a² = 13² − 5² a² = 169 − 25 a² = 144 a = √144

Answer: a = 12 cm. Check: 5² + 12² = 25 + 144 = 169 = 13². Correct.

Worked Example 2 — Higher Level

Question: A right-angled triangle has a hypotenuse of 20 cm and one leg of 11 cm. Find the other leg to 1 decimal place.

Working: a² = 20² − 11² a² = 400 − 121 a² = 279 a = √279

Answer: a = 16.7 cm (1 d.p.). Check: 11² + 16.7² = 121 + 278.89 = 399.89 ≈ 400 = 20². Correct (minor rounding difference).

Worked Example 3 — Exam Style

Question: A ladder is 6.5 m long and leans against a vertical wall. The foot of the ladder is 2.5 m from the base of the wall. How high up the wall does the ladder reach?

Working: The ladder is the hypotenuse. The distance from the wall is one shorter side. The height is the other. h² = 6.5² − 2.5² h² = 42.25 − 6.25 h² = 36 h = √36

Answer: The ladder reaches 6 m up the wall.

Common Mistakes

• Subtracting the wrong way round. Always subtract the smaller square from the larger: a² = c² − b². If you do b² − c² you get a negative, which is impossible.
• Adding instead of subtracting. When finding a shorter side, you subtract. When finding the hypotenuse, you add. Mixing these up is the most common Pythagoras error.
• Misidentifying the hypotenuse. If the question gives you two sides and does not specify which is the hypotenuse, the hypotenuse is always the longest one. If you label a shorter side as the hypotenuse, the subtraction may give a wrong result.

Exam Tips

• Write the rearranged formula a² = c² − b² before substituting — this earns a method mark.
• Always check your answer: substitute back into a² + b² and confirm it equals c².
• If the answer is not a whole number, the question will tell you how to round. Keep the full calculator value until the final step.

Practice Questions

Q1 (Foundation): A right-angled triangle has a hypotenuse of 17 cm and one side of 8 cm. Find the other side.

Q2 (Foundation): A rope stretches from the top of a 12 m pole to a point on the ground 9 m from the base. How long is the rope?

Q3 (Higher): A right-angled triangle has a hypotenuse of 25 cm and one side of 7 cm. Find the other side.

Practise finding a shorter side questions with instant feedback — completely free on GCSEMathsAI.

Related Topics

• Pythagoras' Theorem
• Finding the Hypotenuse
• Trigonometry SOHCAHTOA

Summary

• To find a shorter side, rearrange Pythagoras' theorem to a² = c² − b².
• Always subtract the square of the known shorter side from the square of the hypotenuse — never the other way round.
• Take the square root after subtracting to find the missing side length.
• Check your answer by verifying a² + b² = c² with all three sides.