10.1. Area of a Triangle — by Heron’s Formula

We know that the area of triangle when its height is given, is 1/2 × base × 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.

To find the area of a triangle, usually, we use the formula: Area = 1/2 × base × height. However, this requires knowing the height, which isn't always available. In cases where only the lengths of the sides of a triangle are known (like in our example with sides 40 m, 32 m, and 24 m), we need a different approach. This is where Heron’s formula comes into play, allowing us to calculate the area without needing the height.

Imagine you are measuring a triangular plot of land to build a garden, but the height of the plot is overgrown with bushes, making it difficult to measure. Instead of clearing the bushes, you can use the lengths of the sides to calculate the area using Heron’s formula.

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.

Heron of Alexandria was a significant figure in the development of mathematics, particularly in geometry. He wrote extensively on various subjects, and his work contributed to many mathematical principles used today, including the formula for calculating the area of a triangle based on its sides. This historical context helps us appreciate the origins of Heron’s formula, which is still relevant in contemporary mathematics.

Think of Heron as a pioneer of his time, like an inventor today who creates new technology. Just as modern inventors build on the ideas of the past, mathematicians like us leverage Heron’s groundbreaking work to solve problems more efficiently today.

The formula given by Heron about the area of a triangle is stated as: Area of a triangle = √s(s - a)(s - b)(s - c), where a, b, and c are the sides of the triangle, and s = semi-perimeter, i.e., half the perimeter of the triangle = (a + b + c) / 2.

Heron's formula for calculating the area of a triangle involves first determining the semi-perimeter (s) of the triangle. The semi-perimeter is calculated by adding the lengths of all three sides and dividing by 2. Once you have the semi-perimeter, you can substitute it and the sides (a, b, c) into Heron's formula to find the area. This formula is particularly useful because it allows the calculation of the area without requiring the height of the triangle.

Picture making a cake without needing to measure how thick it will be. Instead, if you know the lengths of your ingredients (the sides), you can calculate how much batter you'll need (the area) using Heron’s formula.

Let us apply it to calculate the area of the triangular park ABC. Let us take a = 40 m, b = 24 m, c = 32 m, so that we have s = (40 + 24 + 32) / 2 = 48 m. s - a = (48 - 40) m = 8 m, s - b = (48 - 24) m = 24 m, s - c = (48 - 32) m = 16 m. Therefore, area of the park ABC = √(48)(8)(24)(16) m² = 384 m².

To calculate the area of triangle ABC using Heron's formula, we first find the semi-perimeter (s) as 48 m. Next, we calculate each difference (s - a, s - b, s - c) to get 8 m, 24 m, and 16 m respectively. Plugging these values into Heron’s formula, we compute the area, concluding that the area of the triangular park is 384 m². This example illustrates the practical application of Heron's formula in determining areas without measuring heights.

Consider you are planning a new playing field in a park, and you have a blueprint showing the lengths of the sides. By using Heron's formula, you can quickly figure out how much grass and other materials you will need to cover that area without having complicated measurements for height.

We can check that the area of the park is 1/2 × 32 × 24 m² = 384 m². We find that the area we have got is the same as we found by using Heron’s formula.

To verify our calculation, we use the traditional formula for the area of a triangle with a base of 32 m and a height calculated from the triangle's properties. This verification reassures us that Heron’s formula produces reliable results. Finding the same area through different methods strengthens our understanding and confidence in the calculations.

Think of double-checking your homework. If you arrive at the same answer using two different methods, you can be more certain that your solution is correct, just like checking the area of a triangular park using both Heron’s formula and the base-height method.

Now using Heron’s formula, you verify this fact by finding the areas of other triangles discussed earlier: (i) equilateral triangle with side 10 cm and (ii) isosceles triangle with unequal side as 8 cm and each equal side as 5 cm.

By applying Heron's formula to different types of triangles, we can compare results and see how versatile the formula is. Each triangle type—equilateral and isosceles—has its own set of dimensions, yet Heron's formula allows us to find their areas without knowing heights.

It's like learning multiple routes to get to your friend's house. Regardless of the way you choose, you'll still reach your destination. Similarly, Heron's formula provides a consistent way to find the area of various triangle shapes.

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