Heron’s Formula
Heron’s Formula is a clever method for calculating the area of a triangle. It does not require the triangle’s height to compute the area; instead, it requires the lengths of the three sides which are easier to find. In the formula, the sides of the triangle are labeled as [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex]. It is standard in geometry to denote sides in lower case letters as you can see here. In addition, we will need the semi-perimeter of the triangle which is simply the sum of the three sides divided by [latex]2[/latex]. The semi-perimeter is a required value in the formula.
Example 1: Find the area of the triangle below using Heron’s Formula.
Just a few housekeeping notes. We label the vertices (corners) of the triangles with uppercase letters. In this case, we use A, B, and C. The side opposite vertex A (or angle A) is side [latex]a[/latex]. The side opposite vertex B is side [latex]b[/latex]. And finally, the side opposite vertex C is side [latex]c[/latex]. Note that for the side lengths, we denote them with lower case letters.
With that said, we are able to map the values of the sides of the triangle.
Since,
[latex]a=5[/latex] feet
[latex]b=13[/latex] feet
[latex]c=12[/latex] feet
The semi-perimeter of the triangle is
We have all the necessary information to calculate the area of the triangle using Heron’s formula.
Therefore, the area of triangle [latex]\triangle[/latex] ABC is [latex]30[/latex] square feet.
Example 2: Find the area of the triangle below using Heron’s Formula.
In our previous lesson, we learned how to find the area of a triangle using its base and height. We may not need to use Heron’s formula because it appears that side [latex]\overline{BC}[/latex] forms a right angle with side [latex]\overline{CA}[/latex] which implies that [latex]\overline{BC}[/latex] is the height and [latex]\overline{CA}[/latex] is the base of triangle [latex]\triangle{ABC}[/latex]. Let’s calculate the area of the triangle using the standard formula.
Since,
base = [latex]8[/latex]
height = [latex]6[/latex]
Therefore, the area of [latex]\triangle{ABC}[/latex] is
Using the standard formula, we calculated the area of the triangle as [latex]24[/latex] square inches.
What we want now is to show that we can get the same area using Heron’s Formula. From the triangle,
[latex]a=6[/latex]
[latex]b=8[/latex]
[latex]c=10[/latex]
Calculating our semi-perimeter, we get
Let’s put it all together to solve for the area of the triangle using Heron’s formula.
Therefore, the area of the triangle is [latex]24[/latex] square inches. This answer is an exact match to the solution using the standard formula (1/2 base times height). Great!
You might also be interested in: