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

\begin{align} T & = \frac{1}{2} \cdot b \cdot c \cdot \sin ∠ A & (1) \\ ∠ A & = \sin^{- 1} (\frac{2 T}{b \cdot c}) & (2) \end{align}

where

b

and

c

are the sides meeting in angle

∠ A

, and where

T

is the area of the triangle

△ A B C

Rule

Uses

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

Find the area of a triangle when you know two sides and the angle between them—Formula (1).
Find an angle in a triangle when 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 at the angle $∠ A = 135 °$

Insert the numbers into a calculator and compute:

T = \frac{1}{2} \cdot 30 \cdot 15 \cdot \sin 135 ° \approx 159.9

Example 2

Find the angle $∠ A$ in a triangle $△ A B C$ , 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) from above:

A = \sin^{- 1} (\frac{2 \cdot 12.5}{10 \cdot 5}) = 30 °

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