Class 9 Exam  >  Class 9 Notes  >  Mathematics (Maths) Class 9  >  Chapter Notes: Heron`s Formula

Heron’s Formula Class 9 Notes Maths Chapter 10

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=�(�−�)(�−�)(�−�)Heron’s Formula Class 9 Notes Maths Chapter 10

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

Heron’s Formula Class 9 Notes Maths Chapter 10

�=�+�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 m2Heron’s Formula Class 9 Notes Maths Chapter 10

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:

Heron’s Formula Class 9 Notes Maths Chapter 10

Additional Examples

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 cm2Heron’s Formula Class 9 Notes Maths Chapter 10

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.

Heron’s Formula Class 9 Notes Maths Chapter 10

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 m2Heron’s Formula Class 9 Notes Maths Chapter 10

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.
All you need of Class 9 at this link: Class 9
48 videos|387 docs|65 tests

Up next

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.
48 videos|387 docs|65 tests
Download as PDF

Up next

Explore Courses for Class 9 exam
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Download the FREE EduRev App
Track your progress, build streaks, highlight & save important lessons and more!
Related Searches

Heron’s Formula Class 9 Notes Maths Chapter 10

,

Summary

,

video lectures

,

Semester Notes

,

Heron’s Formula Class 9 Notes Maths Chapter 10

,

practice quizzes

,

Previous Year Questions with Solutions

,

Exam

,

Free

,

pdf

,

mock tests for examination

,

Extra Questions

,

Sample Paper

,

study material

,

Heron’s Formula Class 9 Notes Maths Chapter 10

,

Objective type Questions

,

ppt

,

Important questions

,

MCQs

,

shortcuts and tricks

,

past year papers

,

Viva Questions

;