Non-right Triangles with Law of Cosines: Learn It 3

Using Heron’s Formula to Find the Area of a Triangle

We already learned how to find the area of an oblique triangle when we know two sides and an angle. We also know the formula to find the area of a triangle using the base and the height. When we know the three sides, however, we can use Heron’s formula instead of finding the height. Heron of Alexandria was a geometer who lived during the first century A.D. He discovered a formula for finding the area of oblique triangles when three sides are known.

Heron’s formula

Heron’s formula finds the area of oblique triangles in which sides [latex]a,b[/latex], and [latex]c[/latex] are known.

[latex]\text{Area}=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}[/latex]

where [latex]s=\frac{\left(a+b+c\right)}{2}[/latex] is one half of the perimeter of the triangle, sometimes called the semi-perimeter.

Find the area of the triangle using Heron’s formula.A triangle with angles A, B, and C and opposite sides a, b, and c, respectively. Side a = 10, side b - 15, and side c = 7.

Use Heron’s formula to find the area of a triangle with sides of lengths [latex]a=29.7\text{ft},b=42.3\text{ft}[/latex], and [latex]c=38.4\text{ft}[/latex].

A Chicago city developer wants to construct a building consisting of artist’s lofts on a triangular lot bordered by Rush Street, Wabash Avenue, and Pearson Street. The frontage along Rush Street is approximately 62.4 meters, along Wabash Avenue it is approximately 43.5 meters, and along Pearson Street it is approximately 34.1 meters. How many square meters are available to the developer?A triangle formed by sides Rush Street, N. Wabash Ave, and E. Pearson Street with lengths 62.4, 43.5, and 34.1, respectively.

Find the area of a triangle given [latex]a=4.38\text{ft},b=3.79\text{ft,}[/latex] and [latex]c=5.22\text{ft}\text{.}[/latex]