# Introduction to Heron’s Formula

Heron’s Formula is named after Hero of Alexandria, a Greek mathematician. Not to mention, mathematicians are heroes as well! As a matter of fact, this formula allows you to save the day in geometric terms. With this in mind, let us discuss the Heron’s formula or we can say the triangle’s area formula and study its applications.

## Triangles

A triangle is a 3 sided enclosed figure in geometry. Also, the sum of all the angles of a triangle is 180°. Furthermore, depending upon the length of the sides of triangles, they are classified into 3 types. Namely,

• Equilateral Triangle: In this type, all the sides of the triangle are equal in length. Furthermore, all angles measure 60°. In the figure below, we see an equilateral triangle.

• Isosceles Triangle: When any 2 sides of a triangle are same in length then such a triangle is called an isosceles triangle. In the figure below, we see a triangle whose 2 sides are similar in length to one another. Also, the angles opposite to these identical sides also measure the same.

• Scalene Triangle: A triangle in which no side is equivalent in length to the other is called a scalene triangle. Also, no two angles of the triangle measure the same.

## What is the Heron’s Formula about?

In geometry, Heron’s formula is used to determine the area of a triangle. If we know the length of 3 sides, then we can calculate the area of any type of triangle using this formula. Moreover, this area formula makes the calculation of the area of a triangle very easy because it eliminates the use of angles and the need to calculate the height of the triangle. However, the only drawback to this formula is that it demands the length of all 3 sides.

## Heron’s Formula

Any triangle has 3 sides. We represent the length of the 3 sides as ‘a’, ‘b’, ‘c’ units respectively. Therefore the sum of lengths of all the 3 sides (perimeter) is P = a + b+ c. Hence, the semi perimeter of the triangle is

s = P/2 = (a+b+c)/2

By Heron’s formula, the area of the triangle is

√[s(s-a)(s-b)(s-c)]

The above area of triangle formula is applicable for all triangles irrespective of the side lengths.

## Solved Examples on Heron’s Formula

Question 1: Find the area of a triangle having the length of sides as 3, 4 and 5 units respectively

Answer: Given that a=3, b=4, c=5

P= (a+b+c)= (3+4+5)= 12
S= P/2 = 12/2=6
Area= √s(s-a)(s-b)(s-c)
=√6(6-3)(6-4)(6-5)
=√6x3x2x1
=√36
=6

Area of the triangle is 6 units.

Question 2. The hypotenuse of a right-angled triangle is 41 cm and the area of the triangle is 180 sq. cm, then the difference between the lengths of the legs of the triangles must be

1. 22 cm
2. 25 cm
3. 31 cm
4. 27 cm

Answer: C. The given area is 180 cm²

Implies, (1/2)ab = 180
ab = 2 × 180 = 360

So now by applying Pythagoras theorem,

a² + b² = 41 ²
a² + b² = 1681
So, a² + b² – 2ab = 1681 – 2*360
(a-b)² = 961
a-b = √961 = 31 cm
The difference between the lengths of the legs of the triangle must be 31 cm.

Question 3: What is the Heron’s Formula about?

Answer: In geometry, we use Heron’s formula to find the area of a triangle. If the length of 3 sides is known, then we can find the area of a triangle through this formula. This area formula makes calculating the area of a triangle quite simple because it eliminates the use of angles and you need not calculate the height as well. However, the only shortcoming of this formula is that it requires the length of all 3 sides.

Question 4: What is an Isosceles Triangle?

Answer: Isosceles triangle is one in which any 2 sides are equal in length. Furthermore, the angles opposite to these uniform lengths also measure the same.

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