--- title: "Triangle Area with Half ab sin C" description: "Find the area of any triangle from two sides and the angle between them, and run the same rule backwards to find an angle from a known area." canonical: https://lightmysky.com/learn/mathematics/triangle-area-with-half-ab-sin-c-mt_PiwizT4D2b source: https://lightmysky.com/learn/mathematics/triangle-area-with-half-ab-sin-c-mt_PiwizT4D2b.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`. # Triangle Area with Half ab sin C Find the area of any triangle from two sides and the angle between them, and run the same rule backwards to find an angle from a known area. Subject: Mathematics · Area: Geometry · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/triangle-area-with-half-ab-sin-c-mt_PiwizT4D2b ## Ready when they can - Use the two sides either side of the given angle, not any two sides - Find an angle when the area and two sides are known - Split a quadrilateral into two triangles to find its area ## Lesson: Half, times, times sine Half base times height always works for area, but you need the height. When nobody hands you a height, build one. Take a triangle with sides a and b and the angle C between them. Use side a as the base. Drop a perpendicular from the far corner to the base, making a right triangle whose hypotenuse is b and whose height is h. In that right triangle, h equals b sin C. Substitute into the base-height formula and the area is one half times a times b times sin C. The rule is that a and b must touch angle C: grab a far side and the answer breaks. *(drawing: Drop a perpendicular from the far corner to the base, then read the height from the right triangle.)* When angle C is obtuse, the perpendicular meets the extension of the base instead of the base itself. The auxiliary triangle still has hypotenuse b and an acute angle of 180 minus C, and sin of 180 minus C equals sin C, so the height is unchanged. At exactly 90 degrees the perpendicular is side b itself, which is the right-angle case of the same formula. The formula also runs backwards. When you know the area and two sides, solve for the sine of the angle between them: sin C equals 2 times area over a times b, then take the inverse sine. Sine only reaches 1, so the area can never exceed half of a times b, and that maximum happens only at 90 degrees. When sin C is exactly 1 both candidates coincide at 90 degrees, counted once. For any smaller sine, both C and 180 minus C are possible, check which one fits your triangle. **Example.** A four-sided shape needs one extra move. Draw a diagonal that stays inside the shape to split the quadrilateral into two triangles, find each triangle's area with the same formula, then add the two areas together. A split giving 12 and 7 makes 19 for the whole shape, and the same adding works whatever each half measures. In a concave quadrilateral one diagonal can fall outside, so always choose the internal one. **Recap.** Multiply the two touching sides by the sine between them, halve it, and split quadrilaterals before adding. ## Practice 16 questions on this page, each with its working shown. ## Needs first - [The Cosine Rule](https://lightmysky.com/learn/mathematics/the-cosine-rule-mt_yy73gK5CEP) ## Opens up - [Pythagoras and Trigonometry in Three Dimensions](https://lightmysky.com/learn/mathematics/pythagoras-and-trigonometry-in-three-dimensions-mt_7yKxYqQHB2) - [Determinants and What They Measure](https://lightmysky.com/learn/mathematics/determinants-and-what-they-measure-mt_fWr2D_1BdM) - [The Cross Product and Oriented Area](https://lightmysky.com/learn/mathematics/the-cross-product-and-oriented-area-mt_QEBM_Ms-SK) - [Arc Length, Sector Area and Segments](https://lightmysky.com/learn/mathematics/arc-length-sector-area-and-segments-mt_z9LIwmfKYJ)