## Introduction to Heron’s Formula

Heron of Alexandria was a great mathematician who was the first to derive the formula of calculation of the area of a triangle using the length of all three sides. This formula is now known as Heron’s Formula or Hero’s Formula. Heron of Alexandria also used this formula to derive the area of quadrilaterals as well as high-order polygons. This formula is widely used in trigonometry as well as proving the law of cosines, the law of cotangents, etc. However, angle measurement is not required to calculate its area.

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### What is Heron’s Formula?

Heron’s formula is used to determine the area of a triangle using the lengths of all its three sides. This formula is also used to determine the area of quadrilaterals as well as high-order polygons.

### Heron’s Formula for Finding Area of a Triangle

According to Heron and his formula, we can determine the area of any triangle be it scalene, equilateral, or isosceles with the help of the formula as long as we have the lengths of all three sides provided.

For example, if we have a triangle XYZ and the sides are x, y, and z respectively, then the area of the triangle will be as follows-

\[Area = \sqrt{s(s - x)(s - y)(s - z)}\]

Here s refers to the semi-perimeter of the triangle i.e. s = (x + y + z)/2.

### Heron’s Formula for Equilateral Triangle

To derive the area of an equilateral triangle, first, we have to find out the semi-perimeter of the triangle i.e.

s = (x + x + x)/2

s = 3x/2

Here x is the length of the sides of the triangle.

By putting it in Heron’s formula, we get

A = √[s(s-x)3]

### Heron’s Formula for Isosceles Triangle

An isosceles triangle is a triangle whose two sides are equal while those corresponding angles are congruent to each other. We can find out the area of an isosceles triangle from Heron’s Formula in the following way -

Let the length of the congruent sides be x and the length of the base be y.

So, their semi-perimeter s = (x + x + y)/2

s = (2x + y)/2

Area of a triangle according to Heron’s Formula

Area = √[s(s - x)(s - y)(s - z)]

In an isosceles triangle,

Area = √[s(s - x)(s - y)(s - x)]

= √[s(s - x)2(s - y)]

= (s - x)√[s(s - y)]

### Area of a Triangle Using Heron’s Formula

There are two steps involved in finding out the area of a triangle. The steps are as follows -

First, we have to find out the value of the semi-perimeter of the triangle given to us.

Next, we have to use Heron's formula, substitute the values obtained and given, and thus find out the area of the triangle.

### Solved Example

A triangle XYZ has three sides of length 4cm, 13cm, and 15 cm. Calculate the area of the triangle XYZ using Heron’s Formula.

Solution - Semiperimeter of the triangle XYZ s = (4 + 13 + 15)/2 = 32/2 = 16

By using Heron’s Formula, we get,

Area = √[s(s - x)(s - y)(s - z)]

= √[16(16 - 4)(16 - 13)(16 - 15)]

= √(16 * 12 * 3 * 1)

= √576

= 24 sp. cm.

1. What is Heron’s Formula?

Ans: Heron of Alexandria derived the formula of calculation of the area of a triangle using the length of all three sides. This formula is now known as Heron’s Formula or Hero’s Formula. He also used this formula to derive the area of quadrilaterals as well as high-order polygons. This formula is used in trigonometry for proving the law of cosines, the law of cotangents, etc. Heron’s formula is used to calculate the area of a triangle using the length of all its three sides. However, angle measurement is not required to calculate its area.

2. Derive Heron’s Formula.

Ans: According to Heron and his formula, we can determine the area of any triangle be it scalene, equilateral, or isosceles with the help of the formula as long as we have the lengths of all three sides provided. This is an important part in the Class 9 Notes NCERT.

For example, if we have a triangle XYZ and the sides are x, y, and z respectively, then the area of the triangle will be as follows -

Area = √{s(s - x)(s - y)(s - z)}.

Here s refers to the semi-perimeter of the triangle i.e. s = (x + y + z)/2

The formula for equilateral triangle is A = √[s(s - x)^{3}]

The formula for isosceles triangle is A= (s - x)√[s(s - y)]

It can also be used to determine the area of quadrilaterals and high-end polygons.