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General Formula for Area of a Triangle

With the help of sine, we can generalize the formula for the area of a triangle to work for any triangle. The formula looks like this:

Formula

General Formula for Area of a Triangle

T = 1 2 b c sin A (1) A = sin 1 ( 2T b c), (2) where b and c are the sides meeting in angle A, and where T is the area of the triangle ABC.

Rule

Uses

You can use this generalized formula for the area of a triangle to

  • Find the area of a triangle where you know two sides and the angle between them. Formula (1).

  • Find an angle in a triangle where you know the area and the two sides forming the angle. Formula (2).

Example 1

Find the area of a triangle with sides b = 30 and c = 15 that meet in an angle A = 135°

Insert the numbers into a calculator and compute:

T = 1 2 30 15 sin 135° 159.9.

Example 2

Find the angle A in a triangle ABC, where A is formed by the two sides b = 10 and c = 5, and the area of the triangle is T = 12.5

You can insert this information into Formula (2) in CAS:

A = sin 1 (2 12.5 10 5 ) = 30°.

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