Finding the Area of a Triangle Given All Three Side Lengths

In this tutorial, we will learn how to find the area of a triangle when all three side lengths are known. This method uses Heron’s Formula, which is a reliable way to calculate the area of any triangle.

Formula

The area of a triangle with sides \(a\), \(b\), and \(c\) is given by:

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

Explanation of Terms:

  • \(a, b, c\): The lengths of the three sides of the triangle.
  • \(s\): The semi-perimeter of the triangle, calculated as:
    \[
    s = \frac{a + b + c}{2}
    \]

The following is the illustration of a triangle with the lengths of sides marked.

Area of a Triangle Formula using Heron's Formula

Examples

Example 1: Find the area of a triangle with sides 5, 6, and 7.

Given: \(a = 5\), \(b = 6\), \(c = 7\)

First, calculate the semi-perimeter:

\[ s = \frac{a + b + c}{2} = \frac{5 + 6 + 7}{2} = 9 \]

Next, substitute into Heron’s Formula:

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

\[ \text{Area} = \sqrt{9(9-5)(9-6)(9-7)} \]

Now simplify:

\[ \text{Area} = \sqrt{9 \times 4 \times 3 \times 2} = \sqrt{216} \]

Finally, calculate the result:

\[ \text{Area} = 14.7 \text{ sq. units (approximately)} \]

Example 2: Find the area of a triangle with sides 8, 15, and 17.

Given: \(a = 8\), \(b = 15\), \(c = 17\)

First, calculate the semi-perimeter:

\[ s = \frac{a + b + c}{2} = \frac{8 + 15 + 17}{2} = 20 \]

Next, substitute into Heron’s Formula:

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

\[ \text{Area} = \sqrt{20(20-8)(20-15)(20-17)} \]

Now simplify:

\[ \text{Area} = \sqrt{20 \times 12 \times 5 \times 3} = \sqrt{3600} \]

Finally, calculate the result:

\[ \text{Area} = 60 \text{ sq. units} \]

With this step-by-step process, you can easily find the area of any triangle given its three side lengths!