How to Use Pythagoras' Theorem

The method, step by step

1. 01

   Confirm the triangle has a right angle

   a² + b² = c² (only for right-angled triangles)

   The theorem is not a general fact about triangles, and applying it to one without a right angle produces a confidently wrong answer. Look for the small square in the corner, or for wording that guarantees it — a wall meeting the ground, a ladder, a diagonal across a rectangle.

2. 02

   Identify the hypotenuse

   hypotenuse = the side opposite the right angle = the longest side

   Everything else depends on this. The hypotenuse is always opposite the right angle and always the longest side, whichever way the triangle has been drawn or rotated on the page. Labelling it before writing anything down prevents the error that produces every impossible answer in this topic.

3. 03

   Decide whether to add or subtract

   finding the hypotenuse: c² = a² + b² finding a shorter side: a² = c² - b²

   Adding always gives the longest side, so subtracting is what you need when the hypotenuse is already known. This single decision accounts for most wrong answers here, and it is settled by asking one question: is the missing side the longest one or not?

4. 04

   Substitute and square

   c² = 5² + 12² = 25 + 144 = 169

   Square each side length before adding. Adding first and squaring afterwards gives 17² = 289, which is wrong, and the error is easy to make when working quickly. Keeping the squares on their own line makes it visible.

5. 05

   Take the square root and give the units

   c = √169 = 13 cm

   The equation gives c², not c, and stopping at 169 is a lost mark rather than a wrong answer. Units come from the question and are carried through unchanged, since the square root of an area-like quantity returns to a length.

6. 06

   Sense-check against the hypotenuse

   13 > 12 > 5 ✓ (the hypotenuse must be the longest side)

   A hypotenuse shorter than one of the other sides means the add-or-subtract decision went the wrong way. This check takes a moment and catches the topic's most common error, which is the one no amount of accurate arithmetic will find.

01

Finding a shorter side

When the hypotenuse is known and one of the other sides is missing, the theorem is rearranged rather than replaced. For a right-angled triangle with hypotenuse 10 cm and one shorter side of 6 cm, the missing side satisfies a² = 10² − 6² = 100 − 36 = 64, so a = 8 cm.

The check is immediate: 8 is shorter than the hypotenuse of 10, as it must be. A student who subtracted in the wrong order would have reached √(36 − 100), which is not a real number — the arithmetic itself refuses, which is a useful warning.

02

When the answer is not a whole number

Most real triangles do not give tidy answers. With shorter sides of 7 cm and 9 cm, c² = 49 + 81 = 130, so c = √130 ≈ 11.4 cm to one decimal place. The exact answer is √130, and whether to leave it in surd form or round is decided by the question.

The triples that do come out whole are worth recognising, because they appear repeatedly in exam questions and spotting one saves the calculation entirely: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and any multiple of these such as 6-8-10 or 9-12-15.

• 3, 4, 5 — since 9 + 16 = 25
• 5, 12, 13 — since 25 + 144 = 169
• 8, 15, 17 — since 64 + 225 = 289
• 7, 24, 25 — since 49 + 576 = 625

03

Using it backwards to test for a right angle

The theorem works in reverse. If the squares of the two shorter sides add to the square of the longest, the triangle must be right-angled; if they do not, it is not. This is the converse, and questions ask for it in the form "show that this triangle is right-angled".

For sides 8, 15 and 17: 8² + 15² = 64 + 225 = 289, and 17² = 289, so the triangle is right-angled. For sides 5, 6 and 8: 5² + 6² = 25 + 36 = 61, but 8² = 64, so it is not. The conclusion has to be written in words to earn the mark.

04

How we teach this

We ask students to label the hypotenuse on the diagram before they write a single number. Nearly every error in this topic traces back to that label being assumed rather than checked, particularly when a triangle is drawn rotated or sits inside a larger shape.

The final sense-check — is the hypotenuse the longest side? — is taught as part of the method rather than as advice. It costs nothing and it is the only step that catches a wrong add-or-subtract decision, which accurate arithmetic never will.