10.1.1. Introduction to Heron's formula

We know that the area of a triangle when its height is given, is $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ . Now suppose that we know the lengths of the sides of a scalene triangle and not the height. Can you still find its area? For instance, you have a triangular park whose sides are 40 m, 32 m, and 24 m. How will you calculate its area? Definitely if you want to apply the formula, you will have to calculate its height. But we do not have a clue to calculate the height. Try doing so. If you are not able to get it, then go to the next section.

The area of a triangle can typically be found using the formula that involves its base and height. However, if we only know the sides of the triangle (like with a scalene triangle), calculating the height may not be straightforward or possible. This introduces a problem for calculating the area based solely on side lengths, which is where Heron's formula comes into play. It allows us to find the triangle's area without needing to know its height.

Imagine a triangular park where you want to put in flower beds. You know the lengths of the triangle's sides but not how high it is from the base to the top point where the flowers would grow. If you only have the side lengths, you might feel stuck. Heron's formula acts like a special tool that lets you calculate the area even when you don’t know how tall the triangle is.

Heron was born in about 10AD possibly in Alexandria in Egypt. He worked in applied mathematics. His works on mathematical and physical subjects are so numerous and varied that he is considered to be an encyclopedic writer in these fields. His geometrical works deal largely with problems on mensuration written in three books.

Heron of Alexandria was a notable figure in the history of mathematics. His contributions laid foundational work in applied mathematics, including geometry. His exploration of different geometrical shapes and their areas demonstrates his extensive knowledge. His book focused on measurements of various shapes, but notably, it contains the formula for the area of a triangle, leading to his prominent legacy in mathematics.

Think of Heron like a pioneer explorer in a vast land of geometry. Just as explorers mapped new territories, Heron mapped out mathematical principles that help us navigate the complexities of shapes, starting with triangles. His writings are like guides we still use today for our own explorations in geometry.

The formula given by Heron about the area of a triangle, is also known as Hero’s formula. It is stated as: $\text{Area of a triangle} = \sqrt{s(s-a)(s-b)(s-c)}$ where a, b and c are the sides of the triangle, and $s = \frac{a+b+c}{2}$, the semi-perimeter of the triangle.

Heron's formula allows us to calculate the area of any triangle when we know the lengths of all three sides. We first find the semi-perimeter (s) by adding the lengths of the three sides and dividing by 2. Once we have s, we can plug it into the formula using the three sides to find the area. This is particularly useful for triangles where finding the height is complicated.

Imagine you are an architect needing to figure out how much space a triangular rooftop will cover. You have the measurements of the side lengths but no idea of the height. Heron’s formula is like a secret recipe that uses just those side lengths to calculate the area, ensuring you know how much roofing material you'll need without any guesswork.

This formula is helpful where it is not possible to find the height of the triangle easily. Let us apply it to calculate the area of the triangular park ABC, mentioned above (see Fig. 10.2). Let us take a = 40 m, b = 24 m, c = 32 m, so that we have s = $\frac{40 + 24 + 32}{2} = 48$ m. Thus, we can apply Heron's formula to find the area.

When applying Heron's formula to the triangular park, we start by determining the semi-perimeter using the side lengths of the triangle. After calculating s, we then substitute it and the sides into the formula, allowing us to find the area in an effective manner without needing to calculate the height. This process exemplifies the utility of Heron’s formula in practical situations.

Think of the triangular park as a pizza slice. You know the lengths of the crust and the sides (like your measuring tape), but you can’t measure how high it rises in the middle. Heron’s formula lets you calculate the area of that pizza slice just from knowing those crust lengths, helping you decide how many slices you can serve to your friends at a picnic!

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