# Heron’s Formula to Calculate Area of Triangle

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 $a$, $b$, and $c$. 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 $2$. 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 $a$. The side opposite vertex B is side $b$. And finally, the side opposite vertex C is side $c$. 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,

$a=5$ feet

$b=13$ feet

$c=12$ 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 $\triangle$ ABC is $30$ 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 $\overline{BC}$ forms a right angle with side $\overline{CA}$ which implies that $\overline{BC}$ is the height and $\overline{CA}$ is the base of triangle $\triangle{ABC}$. Let’s calculate the area of the triangle using the standard formula.

Since,

base = $8$

height = $6$

Therefore, the area of $\triangle{ABC}$ is

Using the standard formula, we calculated the area of the triangle as $24$ square inches.

What we want now is to show that we can get the same area using Heron’s Formula. From the triangle,

$a=6$

$b=8$

$c=10$

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 $24$ square inches. This answer is an exact match to the solution using the standard formula (1/2 base times height). Great!

You may also be interested in these related math lessons or tutorials:

Heron’s Formula Practice Problems with Answers

Area of a Triangle

Area of a Circle