---
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: 2026-09-12
---

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# 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 fails when nobody hands you a height. If instead you know two sides with the angle squeezed between them, use area equals one half times a times b times sin C. The strict rule is that a and b must be the two sides that actually touch angle C: grab a far side and the answer breaks.

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. Watch out, because two angles can share one sine value, so check which one fits your triangle.

**Example.** A four-sided shape needs one extra move. Draw a diagonal 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 14 and 9 makes 23 for the whole shape, and the same adding works whatever each half measures.

**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)