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 $a$ , $b$ , 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.

triangle ABC has side lengths 5 ft, 12 ft, and 13 ft

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.

triangle ABC with where side a =9 feet, side b= 12 feet, and side c= 8 feet

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.

triangle ACB has side lengths 6, 8 and 10

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 might also be interested in:

Heron’s Formula Practice Problems with Answers

Area of a Triangle

Area of a Circle