Calculating the Area of a Triangle with 3 Sides

Introduction

When it comes to geometry, triangles are one of the most fundamental shapes. They are formed by three sides and three angles. Calculating the area of a triangle is an essential skill that is often required in various mathematical and real-world applications. In this blog post, we will explore different ways to calculate the area of a triangle when only the lengths of the three sides are known.

The Formula

Before we dive into the different methods of calculating the area of a triangle, let’s first understand the formula that is commonly used.

The formula for finding the area of a triangle with three known side lengths (a, b, and c) is known as Heron’s formula:

A = √(s(s-a)(s-b)(s-c))

Where A represents the area of the triangle and s represents the semi-perimeter of the triangle, which is calculated by adding all three sides and dividing by 2:

s = (a + b + c) / 2

Method 1: Using Heron’s Formula

As mentioned earlier, Heron’s formula is a widely used method to calculate the area of a triangle when only the lengths of the three sides are known. Let’s go through an example to understand how it works:

Consider a triangle with side lengths a = 5, b = 6, and c = 7. We can calculate the semi-perimeter (s) using the formula mentioned above:

s = (5 + 6 + 7) / 2 = 9

Now, we can plug in the values of a, b, c, and s into Heron’s formula:

A = √(9(9-5)(9-6)(9-7))

Simplifying further:

A = √(9 * 4 * 3 * 2) = √(216) ≈ 14.7

Therefore, the area of the triangle is approximately 14.7 square units.

Method 2: Using Trigonometry

Another method to calculate the area of a triangle with three known side lengths is by using trigonometric functions. This method is particularly useful when you know at least one angle of the triangle. Let’s consider an example to illustrate this method:

Suppose we have a triangle with side lengths a = 8, b = 10, and c = 12, and we also know that one of the angles, angle A, is 45 degrees.

First, we need to find the height of the triangle using trigonometry. We can use the sine function to calculate the height (h) of the triangle:

h = b * sin(A)

Substituting the values:

h = 10 * sin(45) ≈ 7.07

Now, we can calculate the area of the triangle by multiplying the base (b) and height (h) and dividing by 2:

A = (b * h) / 2 = (10 * 7.07) / 2 ≈ 35.35

Therefore, the area of the triangle is approximately 35.35 square units.

Conclusion

Calculating the area of a triangle with three known side lengths is an important skill in geometry. In this blog post, we explored two methods to calculate the area of a triangle using Heron’s formula and trigonometry. Both methods have their own advantages and can be used depending on the available information about the triangle. By understanding these methods, you can confidently calculate the area of a triangle even when only the lengths of the sides are known.