Introduction
In the previous class, we studied many different types of shapes such as triangles, squares, rectangles, quadrilaterals, etc., and formulae to find their areas. We have already studied how to find the area of a triangle if its base and height are given. We use the formula written below to find the area.
Area of a triangle = ½ × Base × Height
This formula is directly useful for right-angled triangles. How can we find the area of any other triangle if its height is not given? In this section, we shall study the formula to find the area of a triangle without using the above formula.
What is Heron’s Formula
Heron’s formula is derived by the mathematician Heron. According to heron, if the three sides of a triangle are given then we can find its area with the help of a formula. This formula is known as Heron’s Formula.
According to the formula,
Area of a triangle = √s(s – a)(s – b)(s – c)
Where, a, b, and c = sides of the triangle
s = semi-perimeter of the triangle = (a + b + c)/2
Note – 1) Semi-perimeter of the triangle is half the perimeter of the triangle.
2) The three sides of the triangle are a, b, and c. Where side a represents the side opposite to vertex A means BC. Similarly, sides b and c represent the sides opposite to vertex B and C means AC and AB respectively.
3) Heron’s formula is useful when the height of the triangle is not given or cannot be found easily.
Area of Triangle by Heron’s Formula
With the help of Heron’s formula, we can easily find the area of a triangle. In any triangle, if all the three sides are given then first, we find the semi-perimeter and then we use the formula. Let us understand with the help of an example.
Example – Find the area of the triangle whose three sides are 3 cm, 5 cm, and 6 cm.
Solution – Let three sides be a = 3 cm, b = 5 cm, and c = 6 cm.
Semi-perimeter of the triangle, s = (a + b + c)/2 = (3 + 5 + 6)/2 = 14/2 = 7 cm
Now, area of the triangle = √s(s – a)(s – b)(s – c)
= √7(7 – 3)(7 – 5)(7 – 6)
= √7(4)(2)(1)
= √7×2×2×2×1
= 2√7×2×1
= 2√14 sq. cm. Ans.
Note – We know that triangle with all three sides of different measurements is Scalene Triangle. In the above example, the given triangle is a scalene triangle.
Area of an Isosceles Triangle by Heron’s Formula
If two sides of the given triangle are equal then that is an Isosceles Triangle. We can compress the formula to find the area of an isosceles triangle. Let’s see how can we do it.
Let △ABC be an isosceles triangle and its sides are a, b, and b. In this isosceles △ABC, sides AB and AC are equal sides.
First of all, we shall find the semi-perimeter of the isosceles △ABC.
Semi-perimeter of isosceles △ABC, s = (a + b + b)/2 = (a + 2b)/2
Now, area of isosceles △ABC = √s(s – a)(s – b)(s – b)
= √s(s – a)(s – b)2
= (s – b)√s(s – a)
Since s = (a + 2b)/2 therefore,
= {(a + 2b/2) – b}√{(a + 2b)/2}[{(a + 2b)/2} – a]
= {(a + 2b – 2b)/2}√{(a + 2b)/2}[{(a + 2b – 2a)/2}]
= (a/2)√{(a + 2b)/2}[{(2b – a)/2}]
= (a/2)√{(2b + a)/2}{(2b – a)/2}
= (a/2)√[{(2b)2 – a2}/4]
= (a/2)(1/2)√{4b2 – a2}
= (a/4)√(4b2 – a2)
Area of an isosceles triangle = (a/4)√(4b2 – a2)
In the above formula, b is the equal side of the isosceles triangle. This formula is very helpful to find the area of an isosceles triangle without finding the semi-perimeter. We can directly use this formula to solve the questions related to isosceles triangles.
Area of an Equilateral Triangle by Heron’s Formula
The triangle with all three sides of equal measure is an Equilateral Triangle. We can minimize the formula to find the area of an equilateral triangle. Let’s see how can we do it.
Let △ABC be an equilateral triangle and its three sides are a, a, and a.
First of all, we shall find the semi-perimeter of the equilateral △ABC.
Semi-perimeter of equilateral △ABC, s = (a + a + a)/2 = 3a/2
Now, area of equilateral △ABC = √s(s – a)(s – a)(s – a)
= √s(s – a)(s – a)2
= (s – a)√s(s – a)
Since s = 3a/2 therefore,
= (3a/2 – a)√(3a/2)(3a/2 – a)
= {(3a – 2a)/2}√(3a/2){(3a – 2a)/2}
= (a/2)√(3a/2)(a/2)
= (a/2)√(3a2/4)
= (a/2)(a/2)√3
= (a2/4)√3
Area of an equilateral triangle = (a2/4)√3
In the above formula, a is the equal side of the equilateral triangle. We can use the above formula to find the area of an equilateral triangle without finding the semi-perimeter. The formula can be directly used if the side of the equilateral triangle is given.
