# Scalene Triangle

A triangle whose all sides are unequal is a Scalene Triangle.

In Latin, the word “Scalene” mean unequal.
Hence the term “Scalene Triangle” signifies triangle with unequal sides

Examples of Scalene Triangle

## Properties of Scalene Triangle

(01) In Scalene Triangle, all the angles are also of different measure

The above image is of scalene triangle where all sides are different.
Note the value of all angles are different

(02) The angle opposite to the longest side will be the largest and Vice-Versa

Example 01
Below is the image of Scalene Triangle

Note that side BC = 5cm is the longest in the triangle
So, the angle opposite to side BC will be the largest.
Thus, Angle A is the largest

Example 02
Below image is of Scalene Triangle

The largest angle is \angle B=125\ degree
Then the largest side will the the one opposite of angle B (i.e. AC)
Hence AC is the longest side 7.5 cms

(03) Scalene Triangles have no line of symmetry

It means that there is no line which can divide the Scalene Triangle into equal halves.
This happens because the triangle has no equal sides.

Let us try to understand with examples

Observe the above figures of two scalene triangles.
Note the line MN cannot divide the triangle into two equal halves because of its unequal sides.
Hence in Scalene triangle, there is no line of symmetry

## Area of Scalene Triangle

There are two methods to calculate the area of scalene triangle:
(a) Formula when length of one side and perpendicular distance from opposite sides are given
(b) Length of all sides are given

Area of Triangle when base length and perpendicular distance is given

Observe the above image of scalene triangle provided with following details:
⟹ length of one side (i.e. b)
⟹ Perpendicular distance from the side to opposite vertex (i.e. h)

When the above details are provided, you following formula for Area calculation:
Area of Triangle = \frac{1}{2} \times b\times h

Let us solve some examples related to the concept:

Example 01
Find the area of following scalene triangle

Length of side (b) = 6 cm
Perpendicular distance (h) = 3 cm

Area =\frac{1}{2} \times b\times h\\\ \\ Area =\ \frac{1}{2} \times 6\times 3\\\ \\ Area= \ 9

Hence, 9 sq cm is the area of the above triangle

Example 02
Find the area of following scalene triangle

In the above image, following elements are given:

Side Length (b) = 4 cm
Height from side to opposite vertex (h) = 2.5 cm

Area\ =\frac{1}{2} \times b\times h\\\ \\ Area\ =\ \frac{1}{2} \times 4\times 2.5\\\ \\ Area\ =5\ cm^{2}

Hence, 5 sq cm is the area of above triangle

Area formula when lengths of all sides are given (Heron’s Formula)

Above image is of Scalene triangle with sides a, b and c

You can find the area of above triangle using Heron’s Formula.

(a) Calculate Semi-perimeter of triangle
S\ =\frac{a+b+c}{2}

(b) Now use following formula
Area\ =\sqrt{S\ ( S-a)( S-b)( S-c)}
Where a, b and c are side length of triangle

Let us solve some questions related to this concept:

Example 01
Find the area of following scalene triangle with length 3, 4 and 5 cms

S\ =\ \frac{a+b+c}{2}\\\ \\ S\ =\ \frac{3+4+5}{2}\\\ \\ S=\ 6\\\ \\ \\\ \\ Area\ =\sqrt{S\ ( S-a)( S-b)( S-c)}\\\ \\ Area\ =\ \sqrt{6\ ( 6-3)( 6-4)( 6-5)}\\\ \\ Area\ =\ \sqrt{6\ ( 3)( 2)( 1)}\\\ \\ Area\ =\sqrt{36}\\\ \\ Area\ =\ 6\ cm^{2}

Hence, the area of given triangle is 6 sq cm