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Trigonometry

Area of a triangle from two sides and an angle

The area of any triangle from two sides and the angle between them, no height needed.

A=12absin⁡C

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

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CABb = 8a = 5c ≈ 4.44
  • Law of cosines: c = √(a² + b² − 2ab cos C) ≈ 4.4405
  • Angles: A ≈ 34.26°, B ≈ 115.7°, C = 30° (total 180°)
  • Area = ½ab sin C ≈ 10

What the symbols mean

A
the area of the triangle (A)
a
one side
b
another side
C
the angle between sides a and b

When to use it

Use it whenever you know two sides and the included angle, for a field, a roof panel or a triangle on a survey, without having to find a perpendicular height. Working backwards, it finds the angle that gives a certain area; there are usually two (an acute one and an obtuse one with the same sine). With three sides, use Heron's formula.

Worked example

A triangular garden has sides of 5 m and 8 m with an angle of 30° between them. What is its area?

  1. Start from the formula.

    A=12⁢a⁢b⁢sin⁡(C)

  2. Substitute a = 5, b = 8, C = 30° and work it out.

    A=12·5·8⁢sin⁡(30°)
    =0.5·5·8·0.5
    =2.5·8·0.5
    =20·0.5
    =10

Answer Area = 10 m²

Why it works

The usual formula is ½ × base × height. Take a as the base; the height from the opposite corner is bsin⁡C, which gives 12absin⁡C.

Common mistakes

  • Using an angle that is not between the two sides.
  • Forgetting the ½.
  • Leaving the calculator in radians: sin 30 in radians is negative, which would give a negative area.