Scalene Triangle

Scalene Triangle

Scalene Triangle

A triangle having non of the sides equal is called an scalene triangle. In the given figure of triangle ABC, all three sides AB, BC and AC have different measurements, therefore ΔABC is a scalene triangle.
Scalene ΔABC

Area of Scalene Triangle (Heron’s Formula)

If all sides of a scalene triangle are given, then we can find out the area of the triangle by Heron’s formula. Heron of Alexandria gave this formula.

Let a, b and c be three sides BC, AC and AB respectively of a scalene triangle ABC given in the figure. AD is the height which is denoted by h.
scalene triangle ABC
Let, BD=x, then DC= a – x
Perimeter of ΔABC = a + b + c
Let, 2s = a + b + c where s = semi-perimeter

Area of ΔABC = ½ a × h ………...... (i)  [Because, Area of Δ = ½ × base × height]

In right angled ΔADC,                              
h2 = b2 – (a – x)2 ………………. (ii)
In the right angled ΔADB,                              
h2 = c2 – x2 ……………………. (iii)
From (i) and (ii),
From (i) and (ii), c2 – x2 = b2 – (a – x)2

Substituting the value of x from above in equation (iii), we get        
Area of Scalene Triangle (Heron’s Formula)

Putting the value of h in equation (i), we get  
Area of ΔABC

Thus, when three sides of a triangle are given, then its area is given by the formula

Heron’s Formula
where a, b and c are the three sides of the triangle and s is the semi perimeter i.e.                                           
semi perimeter

This formula is known as the Heron’s formula.

Workout Examples

Example 1: Find the area of triangle ABC.
Example 1: triangle ABC
Solution: In ΔABC, a = 10cm
           b = 8cm
           c = 6cm
area of triangle ABC

You can comment your questions or problems regarding the scalene triangle or area of triangles here.

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