Heron’s Formula: In geometry, Heron’s formula is named after Heron who was born in about 10 AD approximately in Alexandria in Egypt. Heron’s formula is used to find out the area of a triangle. Generally, there are formulas to find the area of a triangle. But those formulas are not suitable for all kinds of triangles. So Heron gave a formula for finding the area of a triangle for all types by a simple basic formula which is explained below:
Area of a triangle = √s (s-a) (s-b) (s-c)
Where a, b and c are the sides of a triangle and s refers to as the semi-perimeter i.e half of the perimeter of a triangle which can be calculated as :
s = a+ b + c /2
This formula is very much helpful where it is not possible to find the height of a triangle.
The heron’s formula can be used for finding the area of a triangle which is explained below.
Calculate the area of a triangle whose sides are 40m, 24m, and 32 m?
So according to the questions, sides of a triangle are given below:
a= 40 m
b= 24 m
c= 32 m
First, we find the value of s which can be find as:
s= a+ b + c /2
=40+24+32 / 2
=96/2 m
=48 m
Now we calculate further value:
S-a = (48-40) m = 8m
S-b = (48-24) m = 24 m
S-c = (48-32) m = 16 m
Therefore, the area of a triangle=√s (s-a) (s-b) (s-c)
= √48*8*24*16 m²
= 384 m²
24 cm and 10 cm are the perimeter and hypotenuse of the right triangle respectively. find=d the other two sides of a triangle. Find the area of a triangle using the formula: area of the right triangle and verify the results by using Heron’s formula.
We can assume that x and y are the two lines of a right triangle.
AB= x cm
BC= y cm
AC= 10 cm (given)
According to the question:
Perimeter = 24 cm (given)
x+y+10=24
x+y= 14 cm ------------(1)
Using Pythagoras theorem (PT)
AB² + BC² = 10²
x² + y² = 100 ----------------(2)
From equation 1:
(x+y)² = 14²
x² + y² + 2xy = 196
Therefore, 100 + 2xy = 196 ( from eq 2)
2xy = 196 - 100
xy = 96/2
xy = 48 cm
Area of triangle ABC= ½ * x * y
= ½ * 48
= 24 cm² ------------------- (3)
Again, (x-y)² = (x+y)² - 4xy
= 14² - 4 * 48
x-y = +2, -2
(1) When (x-y) =2 and (x+y) = 14 then 2x= 16
x = 16/2
x = 8
And (8 - y) = 2
y= 6
(2) When (x-y)= -2 and (x+y)= 14 then 2x = 12
x= 12/2
x= 6
And y= 8
Verification by using Heron's formula:
a= 6 cm
b= 8 cm
c= 10 cm
Now by applying Heron's formula :
Area =√s(s-a)(s-b)(s-c)
Here, s= a+b+c/2
s= 6+ 8 + 10 /2
s =12 cm
Now: (s-a) = 12 -6 = 6 cm
(s-b) = 12 -8 = 4 cm
(s-c) = 12 - 10 = 2cm
Area= √12 * 6 * 4 * 2
= 24 cm² ------------------ (4)
Therefore from both the formulas, the area of a triangle is the same which can be observed from equations 3 and 4.
Hence Verified.
The most common application of Heron’s formula is that it is used to find the area of a triangle.
We have different geometrical shapes, and we need to find the area of those figures. So Heron's Formula can be used to calculate the area of a quadrilateral. It can be calculated by dividing the quadrilateral which results in the formation of two triangles, and then we can calculate the area of both the triangles and then add the area of both triangles which results in the area of a quadrilateral.
Students can refer to the Vedantu site, for the class 9 heron’s formula MCQ, a worksheet with detailed solutions. The ncert class 9 heron's formula worksheet with answers are also available on the Vedantu site. The worksheets are available also in the form of the pdfs. The students can refer to them as well as practice the questions.
Class 9 MCQs of Herons Formula is Given Below:
1. The Edges of a Triangular Board are Given as 6cm, 8cm, and 10 cm. What will be the Cost of Painting it at the Rate of 9 Paisa/ cm ² is
Rs 3.00
Rs 2.16
Rs 5.00
Rs.2.92
Ans: Rs 2.16
2. The Sides of a Triangle are Given as 35 cm, 54 cm, and 61 cm Respectively. What will be the Length of its Longest Altitude?
16
24√5
28
40√2
Ans: 24 √5
3. The Sides of a Triangle are 56 cm, 60 cm, and 52 cm Long. Then the Area of the Triangle is
1311 cm²
1344 cm²
1392 cm²
1322 cm²
Ans: 1344 cm²
1. How Can We Find the Area of a Quadrilateral by Using Heron’s Formula?
Ans: The quadrilateral is a geometrical figure having four sides. So Heron’s formula can be used to find the area of such figures. If we are given the sides and diagonal of a quadrilateral we are able to find the area of that figure. We can divide the quadrilateral which will result in the formation of two triangles. And when we have two triangles, we can find the area by using Heron’s formula. First, we need to find the area of both the triangles and then add the area of both the triangles which will result in the area of a quadrilateral.
2. Who Invented Heron’s Formula and is Heron’s Formula Accurate?
Heron’s formula is invented by Heron who is known as Heron of Alexandria for finding the area of triangles in terms of its sides. Given the length of each side, Heron's formula calculates the area of a triangle. If you have a very thin triangle, one where two of the sides are about equal to s, and the third side is much shorter, the formula of direct implementation of Heron may not be exact only in this kind of triangle. In all the other cases, Heron’s formula provides accurate results.
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