Area of a Triangle with 3 Sides

A triangle is a fundamental geometric shape. It is used commonly in mathematics, engineering, and other real-world applications. When you know the height, you can easily calculate the area of a triangle. However, when you only know the three sides of the triangle, you need a different approach. In this article, we will get into how to find the area of a triangle with three sides using Heron’s formula. Let’s dive in!

1.0What is the Area of a Triangle with 3 Sides?

A triangle has three sides and three angles. If you know the three sides, the area can be calculated without needing the height or any of the angles. This method is particularly useful for calculating the area of scalene triangles, a form of triangle with unequal sides.

For a triangle with sides a, b, and c, the traditional area formula is 

Area = ½ x base x height

However, if you don’t know the height, this formula becomes useless. This is where a triangle area using Heron’s formula comes into play. It allows you to calculate the area of the triangle directly from the side lengths. 

2.0Step-by-Step Calculation Using Heron’s Formula

Named after the ancient Greek mathematician, Hero of Alexandria, Heron’s formula is widely used for calculating the area of a triangle. It states,

$Area=\sqrt{s(s-a)(s-b)(s-c)}$

Where:

• a, b, and c are the lengths of the three sides of the triangle
• s is the semi-perimeter of the triangle

$s=\frac{a+b+c}{2}$

Step 1: Calculate the semi-perimeter (s)

Add all three sides and divide by 2.

s = (a + b + c)/2​

Step 2: Substitute into Heron’s Formula

Once s is known, plug the values into the formula:

$Area=\sqrt{s(s-a)(s-b)(s-c)}$

Step 3: Simplify and Compute the Area

Perform the multiplications inside the square root, then take the square root to find the area.

3.0Types of Triangles When Using Heron’s Formula

Heron’s formula is very versatile and can be used for all types of triangles.

4.0Why Heron’s Formula is Useful

1. No Need for Height: It is difficult to find the height when the triangles have different sides. 
2. Versatile: It works for every type of triangle, including acute, obtuse, and right-angled.
3. Direct Calculation: When it comes to the real world, it simplifies calculation when you need to find the area of a triangle when the sides are known.

5.0Solved Problems

Problem 1: Area of a Scalene Triangle

Given: Triangle with sides 9 cm, 12 cm, and 15 cm. Find the area.

Solution:

Calculate semi-perimeter:

s = (9 + 12 + 15)/2

= 36/2 = 18 cm

Apply Heron’s formula:

$Area=\sqrt{s(s-a)(s-b)(s-c)}$

$=\sqrt{18(18-9)(18-12)(18-15)}$

$=\sqrt{18\times 9\times 6\times 3)}$

$=\sqrt{2916}$

= 54 cm²

Answer: 54 cm²

Problem 2: Triangle Area Using Heron’s Formula for Unequal Sides

Given: A Triangle with sides 8 cm, 10 cm, and 6 cm.

Solution:

Semi-perimeter:

s = (8 + 10 + 6)/2

= 24/2 = 12 cm

Heron’s formula:

$Area=\sqrt{12(12-8)(12-10)(12-6)}$

$=\sqrt{12\times 4\times 2\times 6}$

$=\sqrt{576}$

= 24 cm²

Problem 3: Equilateral Triangle Example

Given: All sides are 5 cm.

Solution:

Semi-perimeter:

s = (5 + 5 + 5)/2

= 15/2 = 7.5 cm

Heron’s formula:

$Area=\sqrt{7.5(7.5-5)(7.5-5)(7.5-5)}$

$=\sqrt{7.5\times 2.5\times 2.5\times 2.5}$

$=\sqrt{117.1875}$

= 10.83 cm²

Answer: 10.83 cm²

6.0Sample Questions on Area of a Triangle

Q1. How can I find the area of a triangle when the sides are known?

Use Heron’s formula. First, calculate the semi-perimeter 

$s=\frac{a+b+c}{2}$

Then, apply, $Area=\sqrt{s(s-a)(s-b)(s-c)}$