Herons’ Formula

Whenever the length of all 3 sides of the triangle are called, Heron’s formula (also Well-known as Heron’s formula) yields an area of the triangle. It is named before the Hero of Alexandria. Unlike previous triangular area formulas, no angles and other dimensions inside the triangle must be calculated beforehand.

According to Heron, the area of every triangle, whether scalene, isosceles, or equilateral, can be calculated using the formula if the sides of a triangle were known.

Consider the triangle ABC, which has sides a, b, and c, respectively. As a result, the size of a triangular may be calculated as follows:

Area of triangle using three sides=s(s-a)(s-b)(s-c)

Where “s” denotes the semi-perimeter, which is equal to (a+b+c) / 2.

The triangle’s three sides are a, b, and c.

HERON  

The Hero of Alexandra became a mathematician who invented the formula for calculating the overall area of a triangle to use all three sides’ lengths. He emphasized utilizing this concept to calculate the area of quadrilaterals with greater polygons. This formulae can be used to prove the rule of cosine or even the rule of cotangents, among other things, in trigonometry.

HERON’S FORMULA FOR EQUILATERAL TRIANGLE 

The equilateral triangle, as we all know, has all of its sides equal. To get the area of an equilateral triangle, we must first determine the semi perimeter of triangle, which is:

(a+a+a)/2 = s

s=3a/2

where an is indeed the side’s length.

Now we understand, thanks to the heron’s formula: 

Area=s(s-a)(s-b)(s-c)

Because a = b = c,

Therefore,

A=s(s-a)3

which is the formula that must be used. 

SOLVED EXAMPLES 

1. Determine the area of the triangle with lengths of 5, 6, & 9 units, correspondingly.
2. We already know that a Equals 5 units, b = 6 units, & c = 9 units.

As a result, semi-perimeter s Equals (a + b + c)/2 = (5 + 6 + 9)/2 = 10 units.

Triangle area = s(s-a)(s-b)(s-c)= Square root of (10(10-5) (10-6) (10-9))

Triangle area = ((10 5 4 1)) = square root of 200 = 14.142 square unit

∴ The triangle’s area is 14.142 square unit

1. If the edges of a triangular ABC were 4 in., 3 in., and 5 in. Using heron’s formula, estimate its area. 
2. Find the area of a triangle ABC.

Assume AB Equals 4 in, BC Equal 3 in, and AC = 5 in 

By employing Heron’s Formula,

A=s(s-a)(s-b)(s-c)

As a result, s = (a+b+c)/2  = (4+3+5)/2  s = 6 units.

Put the data together: A = square root of (6(6-4)(6-3)(6-5))     A = (36) = 6 square inches 

Answer: ∴ The triangle has a surface area of 6 square inches .