By heron formula the area of triangle abc is given by triangle is _ sq. Unit
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Answer:
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.
Heron’s Formula for Equilateral Triangle
As we know the equilateral triangle has all its sides equal. To find the area of equilateral triangle let us first find the semi perimeter of the equilateral triangle will be:
s = (a+a+a)/2
s=3a/2
where a is the length of the side.
Now, as per the heron’s formula, we know;
Area=
s(s−a)(s−b)(s−c)
− − − − − − − − − − − − − − − − −
√
Since, a = b = c
Therefore,
A = √[s(s-a)3]
which is the required formula.
Heron’s Formula for Isosceles Triangle
An isosceles triangle has two of its sides equal and the angles corresponding to these sides are congruent. To find the area of isosceles triangle, we can derive the heron’s formula as given below:
Let a be the length of the congruent sides and b be the length of the base.
Semi-perimeter (s) = (a + a + b)/2
s = (2a + b)/2
Using the heron’s formula of a triangle,
Area = √[s(s – a)(s – b)(s – c)]
By substituting the sides of an isosceles triangle,
Area = √[s(s – a)(s – a)(s – b)]
= √[s(s – a)2(s – b)]
Or
= (s – a)√[s(s – b)]
which is the required formula to find the area for the given isosceles triangle.
Proof
There are two methods by which we can derive the Hero’s formula. First, by using trigonometric identities and cosine rule. Secondly, solving algebraic expression using Pythagoras theorem.