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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
Multiply the two touching sides by the sine between them, halve it, and split quadrilaterals before adding.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.