Table of contents  
Introduction  
Area of a Triangle — by Heron’s Formula  
Verification and Examples  
Additional Examples 
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.
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:
Here, a, b, and c are the sides of the triangle, and s is the semiperimeter
Let's apply Heron's formula to find the area of a triangular park with sides 40 m, 32 m, and 24 m:
s= 40+32+24/2=48
s−a=48−40=8
s−b=48−24=24
s−c=48−32=16
This matches the area calculated using the traditional method:
To ascertain the park's area, the application of the formula 1/2×32×24 square meters yields 384 m².
Now, let's verify Heron's formula by applying it to other triangles:
Given sides of 8 cm, and 11 cm, and a perimeter of 32 cm, the area is calculated using Heron's formula:
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,
AB^{2} = AO^{2} + BO^{2}
⇒ AB^{2} = 22^{2} + 22^{2}
⇒ AB^{2} = 2 × 22^{2}
⇒ AB = 22√2 cm
Area of square = (Side)^{2}
= (22√2)^{2}
= 968 cm^{2}
Area of each triangle (I, II, III, IV) = Area of square /4
= 968 /4
= 242 cm^{2}
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(sa)(sb)(sc)]
= √[27(2720)(2720)(2714)]
= √[27×7×7×13]
= 131.14 cm^{2}
Therefore, We get,
Area of Red = Area of IV
= 242 cm^{2}
Area of Yellow = Area of I + Area of II
= 242 + 242
= 484 cm^{2}
Area of Green = Area of III + Area of the lower triangle
= 242 + 131.14
= 373.14 cm^{2}
A triangular plot has sides in the ratio 3:5:7, and its perimeter is 300 m. The area is:
These examples illustrate Heron's formula as a powerful tool for finding triangle areas without relying on height.
48 videos378 docs65 tests

1. What is Heron's formula for finding the area of a triangle? 
2. How do you calculate the semiperimeter of a triangle? 
3. Can Heron's formula be used for all types of triangles? 
4. Is Heron's formula the only method to find the area of a triangle? 
5. Can Heron's formula be used to find the area of a triangle if only the angles are given? 
48 videos378 docs65 tests


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