Chapter Notes: Heron`s Formula

# Heron’s Formula Class 9 Notes Maths Chapter 10

 Table of contents Introduction Area of a Triangle — by Heron’s Formula Verification and Examples Additional Examples

## Introduction

In geometry, calculating the area of a triangle when its sides are known is essential. While the traditional formula involves using the triangle's height, Heron's formula provides an alternative approach when the height is unknown. This becomes particularly useful for scalene triangles where finding the height is challenging.

## Area of a Triangle — by Heron’s Formula

Heron, a mathematician born around 10 AD, made significant contributions to applied mathematics. His works covered various mathematical and physical subjects. In his geometrical works, Heron derived the famous formula for the area of a triangle based on its three sides. This formula is now known as Heron's formula or Hero's formula:

Area=�(�−�)(�−�)(�−�)

Here, a, b, and c are the sides of the triangle, and s is the semi-perimeter

### �=�+�Application of Heron's Formula

Let's apply Heron's formula to find the area of a triangular park with sides 40 m, 32 m, and 24 m:

�=40+32+242=48s= 40+32+24/2=48

�−�=48−40=8sa=4840=8 �−�=48−24=24

sb=4824=24 �−�=48−32=16

sc=4832=16

Area=48×8×24×16=3842=384 m2

This matches the area calculated using the traditional method:

To ascertain the park's area, the application of the formula 12×32×241/2×32×24 square meters yields 384 m².

## Verification and Examples

Now, let's verify Heron's formula by applying it to other triangles:

### Example 1:

Given sides of 8 cm, and 11 cm, and a perimeter of 32 cm, the area is calculated using Heron's formula:

Area=16×8×5×3=30 cm2

### Example 2:

How much paper of each shade is needed to make a kite given in the figure, in which ABCD is a square with diagonal 44 cm.

Solution: According to the figure,

AC = BD = 44cm , AO = 44/2 = 22cm , BO = 44/2 = 22cm

From ΔAOB,

AB2 = AO2 + BO2

⇒ AB2 = 222 + 222

⇒ AB2 = 2 × 222

⇒ AB = 22√2 cm

Area of square = (Side)2

= (22√2)2

= 968 cm2

Area of each triangle (I, II, III, IV) = Area of square /4

= 968 /4

= 242 cm2

To find area of lower triangle,

Let a = 20, b = 20, c = 14

s = (a + b + c)/2

⇒ s = (20 + 20 + 14)/2 = 54/2 = 27.

Area of the triangle = √[s(s-a)(s-b)(s-c)]

= √[27(27-20)(27-20)(27-14)]

= √[27×7×7×13]

= 131.14 cm2

Therefore, We get,

Area of Red = Area of IV

= 242 cm2

Area of Yellow = Area of I + Area of II

= 242 + 242

= 484 cm2

Area of Green = Area of III + Area of the lower triangle

= 242 + 131.14

= 373.14 cm2Area=125×5×45×75=15×30 m2=450 m2

### Example 3:

A triangular plot has sides in the ratio 3:5:7, and its perimeter is 300 m. The area is:

Area=150×90×50×10=15003 m2

These examples illustrate Heron's formula as a powerful tool for finding triangle areas without relying on height.

The document Heron’s Formula Class 9 Notes Maths Chapter 10 is a part of the Class 9 Course Mathematics (Maths) Class 9.
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## FAQs on Heron’s Formula Class 9 Notes Maths Chapter 10

 1. What is Heron's formula for finding the area of a triangle?
Ans. Heron's formula is a mathematical formula used to find the area of a triangle when the lengths of all three sides are known. It is given by the formula: Area = √(s(s-a)(s-b)(s-c)) where s is the semi-perimeter of the triangle, and a, b, and c are the lengths of its sides.
 2. How do you calculate the semi-perimeter of a triangle?
Ans. The semi-perimeter of a triangle is calculated by adding the lengths of all three sides and dividing the sum by 2. Mathematically, it can be represented as: Semi-perimeter = (a + b + c)/2 where a, b, and c are the lengths of the sides of the triangle.
 3. Can Heron's formula be used for all types of triangles?
Ans. Yes, Heron's formula can be used to find the area of any type of triangle, whether it is equilateral, isosceles, or scalene. The formula only requires the lengths of the three sides of the triangle to calculate the area accurately.
 4. Is Heron's formula the only method to find the area of a triangle?
Ans. No, Heron's formula is not the only method to find the area of a triangle. There are other methods like using the base and height of a triangle, or using trigonometric functions like sine or cosine. However, Heron's formula is particularly useful when the lengths of all three sides of the triangle are known.
 5. Can Heron's formula be used to find the area of a triangle if only the angles are given?
Ans. No, Heron's formula cannot be directly used to find the area of a triangle if only the angles are given. It requires the lengths of all three sides of the triangle to calculate the area accurately. If only the angles are given, trigonometric functions like sine or cosine can be used in combination with the given angles to find the side lengths, and then Heron's formula can be applied to find the area.

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