it's urgent maths by using herons formula step by step
Answers
The semi-perimeter is
30 + 20 + 15/2
= 32.5 m
Now ,
A = √ s(s-a) * (s-b) * (s-c) ( s = 32.5 a = 30 b = 20 c = 15 )
therefore , A = √ 32.5( 32.5 - 30 ) x ( 32.5 - 20) x ( 32.5 - 15b )
= √1026.25 x 12.5 x 17.5
= √ 224,492.1875
= 473.81 m
That's it !
Answer:
Heron’s formula is used to find the area of a triangle when we know the length of all its sides. It is also termed as Hero’s Formula. We don’t have to need to know the angle measurement of a triangle to calculate its area.
Area of triangle using three sides=s(s−a)(s−b)(s−c)−−−−−−−−−−−−−−−−−√
Semiperimeter, s= Perimeter of triangle/2 = (a+b+c)/2
Hero of Alexandria was a great mathematician who derived the formula for the calculation of the area of a triangle using the length of all three sides. It is also termed as Hero’s Formula. He also extended this idea to find the area of quadrilateral and also higher-order polygons. This formula has its huge applications in trigonometry such as proving the law of cosines or law of cotangents, etc.
Table of contents:
Hero’s Formula
For Quadrilateral
For Equilateral Triangle
For Isosceles Triangle
Proof
Using Cosine Rule
Using Pythagoras Theorem
Problems and Solutions
Questions
Heron’s Formula For Area of Triangle
According to Heron, we can find the area of any given triangle, whether it is a scalene, isosceles or equilateral, by using the formula, provided the sides of the triangle.
Suppose, a triangle ABC, whose sides are a, b and c, respectively. Thus, the area of a triangle can be given by;
Area=s(s−a)(s−b)(s−c)−−−−−−−−−−−−−−−−−√
Where “s” is semi-perimeter = (a+b+c) / 2
And a, b, c are the three sides of the triangle.
How to Find the Area Using Heron’s Formula?
To find the area of a triangle using Heron’s formula, we have to follow two steps:
The first step is to find the value of the semi-perimeter of the given triangle.
S = (a+b+c)/2
The second step is to use Heron’s formula to find the area of a triangle.
Let us understand that with the help of an example.
Example: A triangle PQR has sides 4 cm, 13 cm and 15 cm. Find the area of the triangle.
Semiperimeter of triangle PQR, s = (4+13+15)/2 = 32/2 = 16
By heron’s formula, we know;
A = √[s(s-a)(s-b)(s-c)]
Hence, A = √[16(16-4)(16-13)(16-15)] = √(16 x 12 x 3 x 1) = √576 = 24 sq.cm
This formula is applicable to all types of triangles. Now let us derive the area formula given by Heron.
Heron’s Formula For Quadrilateral
Let us learn how to find the area of quadrilateral using Heron’s formula here.
If ABCD is a quadrilateral, where AB||CD and AC & BD are the diagonals.
AC divides the quad.ABCD into two triangles ADC and ABC.
Now we have two triangles here.
Area of quad.ABCD = Area of ∆ADC + Area of ∆ABC
So, if we know the lengths of all sides of a quadrilateral and length of diagonal AC, then we can use Heron’s formula to find the total area.
Hence, we will first find the area of ∆ADC and area of ∆ABC using Heron’s formula and at last, will add them to get the final value.