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Heron’s Formula Class 9 Notes Maths Chapter 10

Introduction

Today, we’re diving into the fascinating world of geometry with Heron's Formula. Have you ever wanted to calculate the area of a triangle without knowing its height? 

For instance, consider a triangular park with sides measuring 40 m, 32 m, and 24 m. If we were to use the conventional formula for area, ½ x base x height

We would need to know the height, which we don’t have. This is where Heron's Formula comes in, allowing us to find the area of a triangle using just the lengths of its three sides. 

In this lesson, we will explore how to use Heron’s Formula step-by-step and apply it through engaging examples. So, get ready to unlock the secrets of triangles and discover the beauty of geometry!

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, band c are the sides of the triangle, and  is the semi-perimeter i.e sum of all-side divided by 2

s= (a+b+c) /2  

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 cm, 32 cm, and 24 cm:Heron’s Formula Class 9 Notes Maths Chapter 10

�=40+32+242=48

Let us take a = 40 cm, b = 24 cm, c = 32 cm,

Semi perimeter of the triangle (s) = (a + b + c)/2 

= (40 + 32 + 24)/2 48 cm

�−�=48−40=848 − 40 8 cm�−�=48−24=24

48 − 24 24 cm �−�=48−32=16

48 − 32 16 cm

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

Heron’s Formula Class 9 Notes Maths Chapter 10Area=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 12×32×24½ × 32 × 24 square meters yields 384 cm². 

Question for Chapter Notes: Heron's Formula
Try yourself:What is the formula for calculating the area of a triangle using Heron's formula?
View Solution

Verification and Examples

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

Equilateral triangle (side =10 cm ) 

Heron’s Formula Class 9 Notes Maths Chapter 10

s= (a+b+c) /2  

=> (10+10+10) /2 

=> 30/2 =15 

=> s= 15 

replacing all values in the above area formulae we get , 

Heron’s Formula Class 9 Notes Maths Chapter 10

Additional Examples:

Example 1:

Given sides of triangle 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 ABCD = (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.

Question for Chapter Notes: Heron's Formula
Try yourself:What is the area of a triangle with sides measuring 15 cm, 18 cm, and 24 cm?
View Solution

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 calculating the area of a triangle?
Ans.Heron's Formula states that the area of a triangle can be calculated using the formula: Area = √[s(s-a)(s-b)(s-c)], where 's' is the semi-perimeter of the triangle, calculated as s = (a+b+c)/2, and 'a', 'b', and 'c' are the lengths of the sides of the triangle.
2. How do you calculate the semi-perimeter 's' of a triangle using Heron's Formula?
Ans.The semi-perimeter 's' of a triangle is calculated by adding the lengths of all three sides (a, b, and c) and dividing the sum by 2. The formula is: s = (a + b + c) / 2.
3. Can Heron's Formula be used for any type of triangle?
Ans.Yes, Heron's Formula can be used for any type of triangle, including scalene, isosceles, and equilateral triangles, as long as the lengths of the three sides are known.
4. What are some examples of using Heron's Formula to find the area of a triangle?
Ans.An example would be to find the area of a triangle with sides of lengths 5, 6, and 7. First, calculate the semi-perimeter: s = (5 + 6 + 7) / 2 = 9. Then apply Heron's Formula: Area = √[9(9-5)(9-6)(9-7)] = √[9 × 4 × 3 × 2] = √[216] = 14.7 square units.
5. Why is Heron's Formula useful in geometry?
Ans.Heron's Formula is useful because it allows for the calculation of the area of a triangle without needing to know its height, making it especially helpful when dealing with triangles that have irregular shapes or when the altitude is difficult to determine.
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