--- title: "Choosing a Triangle Method" description: "Decide from what a question gives you which tool a triangle needs: Pythagoras, right-angle ratios, the sine rule or the cosine rule." canonical: https://lightmysky.com/learn/mathematics/choosing-a-triangle-method-mt_nubZzZjVvn source: https://lightmysky.com/learn/mathematics/choosing-a-triangle-method-mt_nubZzZjVvn.md retrieved: 2026-11-12 --- > **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`. # Choosing a Triangle Method Decide from what a question gives you which tool a triangle needs: Pythagoras, right-angle ratios, the sine rule or the cosine rule. Subject: Mathematics · Area: Geometry · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/choosing-a-triangle-method-mt_nubZzZjVvn ## Ready when they can - Sort triangle questions by what is given before calculating anything - Say why the chosen rule fits the given information best - Spot a diagram that has to be split into two triangles solved in order ## Lesson: Picking the right triangle rule Sort the question before you solve it. A right-angled triangle with two known sides: Pythagoras is the most direct. A side paired with its opposite angle, plus one more known side or angle: the sine rule starts. Two sides trapping their angle, or all three sides: the cosine rule uses them directly, and it also covers right triangles, since a 90 degree included angle reduces it to Pythagoras. **Example.** You know two angles and one side. Find the third angle from 180 degrees, and you now own a side with its opposite angle. That pair is the fingerprint of the sine rule, so use it. **Example.** You know two sides with the angle trapped between them, and you want the third side. No side pairs with an opposite angle, so the sine rule cannot start. The cosine rule uses the two sides and the trapped angle directly, so it is the most direct fit for this task. **Tip.** Some diagrams split into two right-angled triangles solved in order. Triangle one has hypotenuse 10 cm and one leg 6 cm, so Pythagoras gives the missing leg: root of 100 minus 36 is 8 cm. Triangle two shares that 8 cm leg and has another leg of 6 cm with the right angle between them, so its hypotenuse is the root of 64 plus 36, which is 10 cm. Solve the first, carry the shared side across, then solve the second. **Recap.** Name what is given, match it to one rule, and split shared-side diagrams into two ordered steps. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Pythagoras and Trigonometry in Three Dimensions](https://lightmysky.com/learn/mathematics/pythagoras-and-trigonometry-in-three-dimensions-mt_7yKxYqQHB2) - [The Cosine Rule](https://lightmysky.com/learn/mathematics/the-cosine-rule-mt_yy73gK5CEP) ## Opens up - [Bearing Problems with the Sine and Cosine Rules](https://lightmysky.com/learn/mathematics/bearing-problems-with-the-sine-and-cosine-rules-mt_YQFdU4aiFP) - [Arc Length, Sector Area and Segments](https://lightmysky.com/learn/mathematics/arc-length-sector-area-and-segments-mt_z9LIwmfKYJ)