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Study GuideYear 10Level 2 · Standard

Free Year 10 Pythagoras' Theorem Explained

Explains Pythagoras' theorem for right-angled triangles, how to find the hypotenuse, how to find a shorter side, and how to test whether a triangle is right-angled. Worked examples include exact answers, rounded answers and a practical ladder problem.

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Year
Year 10
Subject
Maths
Topic
Pythagoras' Theorem
Difficulty
Level 2 · Standard
Estimated time
20 minutes
Curriculum
New Zealand Curriculum
Answers
Not applicable
Format
PDF (A4) + print

Students will practise

  • understanding the relationship c[sup 2] = a[sup 2] + b[sup 2] in a right-angled triangle
  • identifying the hypotenuse
  • finding the hypotenuse and a shorter side
  • testing whether a triangle is right-angled

Curriculum: New Zealand Curriculum. We show specific outcome codes only where they have been verified against the official curriculum document.

What's next?

Completed: Pythagoras' Theorem Explained

  1. 1Year 10 Pythagoras' Theorem Worksheet — Level 1
  2. 2Year 10 Pythagoras' Theorem Worksheet — Level 2
  3. 3Year 10 Pythagoras' Theorem Worksheet — Level 3
  4. 4Trigonometry Cheat Sheet: SOH CAH TOA
Study guide

How to use this study guide

  1. Read it together first, pausing at each worked example to try the step before reading the answer.
  2. Attempt the "Check yourself" questions at the end without looking back.
  3. Then practise with a worksheet from the pathway above and finish with the topic test.

Common questions

Who is this study guide for?

Year 10 students (typically ages 14–15) working on pythagoras' theorem. It is pitched at level 2 · standard.

Are the answers included?

This is a study guide, so there is no separate answer sheet; the 'Check yourself' questions include answers.

How long does it take?

About 20 minutes. Short, regular sessions work best: two or three a week beats one long one.

Do I need to sign up to download?

No. Click Download Free PDF and it opens immediately. It is free for personal, classroom and homeschool use.

What should we do next?

Try Year 10 Pythagoras' Theorem Worksheet — Level 1.

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Success TutoringWorksheetsYear 10 · Mathematics · Pythagoras' Theorem

Pythagoras' Theorem Explained

What you need to know

In a right-angled triangle, the longest side is called the hypotenuse. It is always opposite the right angle. Pythagoras' theorem says that the square of the hypotenuse equals the sum of the squares of the other two sides: c[sup 2] = a[sup 2] + b[sup 2], where c is the hypotenuse and a and b are the two shorter sides. The theorem only works for right-angled triangles.

Finding the hypotenuse (shorter sides 6 and 8)

  1. c2 = 62 + 82
  2. c2 = 36 + 64 = 100
  3. c = √100 = 10

Finding a shorter side (hypotenuse 13, one side 5)

  1. Rearrange: b2 = c2 − a2.
  2. b2 = 132 − 52 = 169 − 25 = 144
  3. b = √144 = 12

When the answer is not a whole number (shorter sides 4 and 7)

  1. c2 = 42 + 72 = 16 + 49 = 65
  2. c = √65. This is the exact answer.
  3. As a decimal, c ≈ 8.06 (to 2 decimal places). Leave the answer as √65 unless the question asks you to round.
Common mistake: adding the squares when finding a shorter side. If you are given the hypotenuse, you must subtract. A quick sanity check: the hypotenuse must come out as the longest side.

Is the triangle right-angled?. The theorem works in reverse too. If the longest side squared equals the sum of the other two sides squared, the triangle is right-angled. If not, it is not.

Sides 7, 24 and 25

  1. Longest side is 25: 252 = 625.
  2. 72 + 242 = 49 + 576 = 625.
  3. The two results match, so the triangle is right-angled.
Pythagorean tripleCheck
3, 4, 59 + 16 = 25
5, 12, 1325 + 144 = 169
8, 15, 1764 + 225 = 289
7, 24, 2549 + 576 = 625
6, 8, 10 (3, 4, 5 doubled)36 + 64 = 100
Recognising the common triples saves time: any multiple of a triple (such as 9, 12, 15) is also a right-angled triangle.

A 5 m ladder leans against a wall with its foot 3 m from the wall. How high up the wall does it reach?

  1. The ladder is the hypotenuse (5 m), the ground distance is one shorter side (3 m) and the height is the other shorter side (h).
  2. h2 = 52 − 32 = 25 − 9 = 16
  3. h = √16 = 4 m

Check yourself

  1. Shorter sides 9 and 12. Find the hypotenuse.
  2. Hypotenuse 10, one shorter side 6. Find the other side.
  3. Shorter sides 5 and 5. Find the hypotenuse to 2 decimal places.
  4. Is a triangle with sides 6, 8 and 11 right-angled?
  5. Hypotenuse 17, one shorter side 8. Find the other side.
  6. Shorter sides 2 and 3. Find the hypotenuse, exact and to 2 decimal places.

Answers: 1. 15 2. 8 3. √50 ≈ 7.07 4. No (36 + 64 = 100, but 112 = 121) 5. 15 6. √13 ≈ 3.61

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Created by
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Last reviewed
1 October 2026
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