Area of a Triangle Using Heron’s Formula

Heron's formula is a method for calculating the area of a triangle when you know the lengths of all three sides. This is a very useful formula because it does not require knowing the height of the triangle, which is often not easy to obtain. Heron was a Greek engineer and mathematician who developed this formula. In this lesson, we will explore Heron's formula in depth using simple language, visual examples, and step-by-step guides to help you understand how to find the area of a triangle using Heron's formula.

Understanding Heron's formula

Before diving into the actual formula, it is important to know what a triangle is. A triangle is a polygon with three sides and three vertices. It is one of the basic shapes in geometry. The total of all the angles in a triangle is 180 degrees.

Definition of Heron formula

Heron's formula states that the area of a triangle with sides of length a, b, and c is:

    a = √(s(sa)(sb)(sc))

where s is the semi-perimeter of the triangle, which is calculated as:

    S = (a + b + c) / 2

Now, let's explain step-by-step how to use this formula.

A step-by-step guide to using Heron's formula

Step 1: Identify the sides of the triangle

First, you must know the length of the three sides of the triangle. Suppose we have a triangle with sides a, b and c. These can be any positive numbers.

Step 2: Calculate the semi-perimeter

Find the semi-perimeter s by adding the three sides together and dividing by 2. This value helps balance the equation in Heron's formula.

For example, if a = 5, b = 6, and c = 7, then the semi-perimeter will be calculated as follows:

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

Step 3: Plug in the formula

Once you find the value of s, you can substitute it into Heron's formula along with the side lengths to find the area of the triangle.

Use our example:

    A = √(9(9-5)(9-6)(9-7))
      = √(9 × 4 × 3 × 2)
      = √(216)
      = √(6 × 6 × 6)
      = 6√6

So, the area of a triangle with sides 5, 6, and 7 is approximately 14.7 square units.

Visual example

Exploring different examples

Example 1: Equilateral triangle

Consider an equilateral triangle where all sides are equal. Let each side be a = 3.

Step 1: Calculate the semi-perimeter

    S = (3 + 3 + 3) / 2
      = 4.5

Step 2: Use Heron's formula

    A = √(4.5(4.5-3)(4.5-3)(4.5-3))
      = √(4.5 × 1.5 × 1.5 × 1.5)
      = √(15.1875)
      ≈ 3.897

Therefore, the area of an equilateral triangle with a side of 3 units is approximately 3.897 square units.

Example 2: Isosceles triangle

In an isosceles triangle two sides are equal. Let's say the equal sides are 5 units and the base is 4 units.

Step 1: Calculate the semi-perimeter

    S = (5 + 5 + 4) / 2
      = 7

Step 2: Use Heron's formula

    A = √(7(7-5)(7-5)(7-4))
      = √(7 × 2 × 2 × 3)
      = √(84)
      ≈ 9.165

Thus, the area of an isosceles triangle is approximately 9.165 square units.

Properties of triangles related to Heron's formula

Understanding more about triangles will help you make better use of Heron's formula and give you a deeper insight into the nature of triangles.

Relation to right triangles

In the case of right triangles, Heron's formula can still be used, but there is also a simpler method to calculate the area:

    Area = 1/2 × base × height

But, if you only know the lengths of the sides, Heron's formula is more appropriate. Just make sure you know which side lengths correspond to the opposite and adjacent sides of the right angle, and which is the hypotenuse.

Heron's formula for scalene triangles

For scalene triangles, where all sides have different lengths, Heron's formula provides a suitable way to calculate the area without the need for additional information such as height. This versatility underscores why Heron's is often used and taught in academics.

More examples for practice

To master Heron's formula, it is useful to practice with different examples. Here are some additional exercises to try:

Example 3: Triangle with sides 7, 8, 9

Step 1: Calculate the semi-perimeter

    S = (7 + 8 + 9) / 2
      = 12

Step 2: Use Heron's formula

    A = √(12(12-7)(12-8)(12-9))
      = √(12 × 5 × 4 × 3)
      = √(720)
      ≈ 26.833

Therefore, the area of the triangle is approximately 26.833 square units.

Example 4: Triangle with sides 13, 14, 15

Step 1: Calculate the semi-perimeter

    S = (13 + 14 + 15) / 2
      = 21

Step 2: Use Heron's formula

    A = √(21(21-13)(21-14)(21-15))
      = √(21 × 8 × 7 × 6)
      = √(7056)
      ≈ 84

Thus, the area of the triangle is 84 square units.

Conclusion

Heron's formula is a great way to find the area of a triangle when you know the lengths of the sides. With practice, you can easily remember the steps and apply them to any triangle when only the lengths of the sides are given. This formula highlights the beauty of mathematics, providing a way to creatively solve problems with the details available. Keep practicing different examples to gain proficiency and familiarity with this valuable mathematical tool!

To further understand and master Heron's formula, try creating your own visual representations of triangles with different side lengths, and calculate their areas using what you've learned here.

Comments