--- title: "Pythagoras' Theorem" description: "Apply Pythagoras’ Theorem (a² + b² = c²) to calculate unknown side lengths in right-angled triangles, including in real-world and coordinate-geometry contexts" canonical: https://lightmysky.com/learn/mathematics/pythagoras-theorem-mt_1VmTUxBrNd source: https://lightmysky.com/learn/mathematics/pythagoras-theorem-mt_1VmTUxBrNd.md retrieved: 2026-10-19 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Pythagoras' Theorem Apply Pythagoras’ Theorem (a² + b² = c²) to calculate unknown side lengths in right-angled triangles, including in real-world and coordinate-geometry contexts Subject: Mathematics · Area: Geometry · Ages 13 to 14 Page: https://lightmysky.com/learn/mathematics/pythagoras-theorem-mt_1VmTUxBrNd ## Ready when they can - Calculate the hypotenuse of a right-angled triangle given the two shorter sides - Find a shorter side given the hypotenuse and the other short side - Use Pythagoras’ Theorem to find the distance between two points on a coordinate grid ## Lesson: Pythagoras' Theorem A right-angled triangle has one corner that is exactly 90 degrees, like the corner of a book. The two sides that form that corner are called the legs. The third side sits opposite the right angle. It is called the hypotenuse, and it is always the longest side of the triangle. Pythagoras' Theorem links all three sides. Call the legs a and b, and call the hypotenuse c. Then a² + b² = c². Squaring means multiplying a number by itself, so 5² is 25. In words: square each leg, add the squares, and the total is the square of the hypotenuse. This rule only works on right-angled triangles. *(drawing: The legs a and b form the right angle. Square them, add the squares, and you get c², the square of the hypotenuse.)* **Example.** A ladder leans against a wall and meets the ground at a right angle. Its foot is 9 m from the wall and its top reaches 12 m up. The ground and the wall are the legs. Square them: 9² = 81 and 12² = 144. Add the squares: 81 + 144 = 225. Take the square root: c = 15. The ladder is 15 m long. To find a missing leg, turn the rule around: a² = c² − b². Say the hypotenuse is 25 cm and one leg is 7 cm. Square both: 25² = 625 and 7² = 49. Subtract the smaller square from the bigger one: 625 − 49 = 576. The square root of 576 is 24, so the missing leg is 24 cm. Always square the sides first. Using the plain lengths, like 25 − 7 = 18, gives a number that does not fit the triangle. **Example.** The same rule measures distances on a grid. A treasure chest sits at (1, 2) and a player stands at (13, 18). Find the gaps by subtracting the coordinates: 13 − 1 = 12 across, and 18 − 2 = 16 up. Those gaps are the legs of a right triangle, and the straight-line distance is its hypotenuse. Square and add: 12² + 16² = 144 + 256 = 400. The square root of 400 is 20, so they are 20 units apart. **Recap.** In a right-angled triangle, a² + b² = c², where c is the hypotenuse opposite the right angle. Add the squares of the legs to find the hypotenuse, subtract the squares to find a missing leg, and use the same rule for the distance between two points on a grid. *Adapted from OpenStax Algebra and Trigonometry (CC-BY 4.0), openstax.org* · [license](https://creativecommons.org/licenses/by/4.0/) ## Practice 16 questions on this page, each with its working shown. ## Needs first - [Simple formulae](https://lightmysky.com/learn/mathematics/simple-formulae-mt_3qrCtdoVAU) - [Integer powers and roots](https://lightmysky.com/learn/mathematics/integer-powers-and-roots-mt_hCVPYlF-7Y) - [First Quadrant Coordinates](https://lightmysky.com/learn/mathematics/first-quadrant-coordinates-mt_jBQS-CicNn) ## Opens up - [Pythagoras and Trigonometry in Three Dimensions](https://lightmysky.com/learn/mathematics/pythagoras-and-trigonometry-in-three-dimensions-mt_7yKxYqQHB2) - [Trigonometry basics](https://lightmysky.com/learn/mathematics/trigonometry-basics-mt_KB_Czd7RQH) - [The Pythagorean and Quotient Identities](https://lightmysky.com/learn/mathematics/the-pythagorean-and-quotient-identities-mt_NXBq37vxQh) - [Volume and Surface Area of Cones, Spheres and Pyramids](https://lightmysky.com/learn/mathematics/volume-and-surface-area-of-cones-spheres-and-pyramids-mt_s4K5YMBSzG) - [Parallel and Perpendicular Lines in Coordinates](https://lightmysky.com/learn/mathematics/parallel-and-perpendicular-lines-in-coordinates-mt_xNEmm139Ka)