Scalene Triangle: Definition, Properties, Formula, Examples

Scalene Triangle is a type of triangle where all three sides are different lengths, and all three angles have different measures, a scalene triangle is unique in its irregularity and it does not have any symmetry.

Classification of Triangles

We can classify the triangles into various categories by comparing their sides. On the basis of the measure of the side of the triangles, they are categorized into three types, which include,

• Scalene Triangle
• Isosceles Triangle
• Equilateral Triangle

Scalene Triangle Types

Scalene triangles are based on the measure of their interior angles. They can be further classified into three categories that are,

• Acute-Angled Scalene Triangle
• Obtuse-Angled Scalene Triangle
• Right-Angled Scalene Triangle

Acute-Angled Scalene Triangle: An acute-angled scalene triangle is a scalene triangle in which all the interior angles of the triangle are acute angles. I

Obtuse-Angled Scalene Triangle: An obtuse-angled scalene triangle is a scalene triangle in which any one of the interior angles of the triangle is an obtuse angle(i.e. its measure is greater than 90°). The other two angles are acute angles.

Right-Angled Scalene Triangle: A right-angled scalene triangle is a scalene triangle in which any one of the interior angles of the triangle is a right angle (i.e. its measure is 90°). The other two angles are acute angles.

Properties of Scalene Triangle

Key properties of a scalene triangle are,

• All three sides of a scalene triangle are not equal. (for a scalene triangle △ABC AB ≠ BC ≠ CA)
• No angle of the Scalene triangle is equal to one another. (for a scalene triangle △ABC ∠A ≠ ∠B ≠ ∠C)
• Interior angles of a scalene triangle can be either acute, obtuse, or right angle, but some of all its angle is 180 degrees. (for a scalene triangle △ABC ∠A+∠B+∠C = 180°)
• No line of Symmetry exists in the Scalene triangle

Difference between Scalene, Equilateral and Isosceles Triangles

The main differences between Scalene, Equilateral and Isosceles Triangles are tabulated below:

Equilateral Triangle	Isosceles Triangle	Scalene Triangle
In an Equilateral triangle, all three sides of a triangle are equal.	In an Isosceles triangle, any two sides of the triangle are equal.	In a Scalene triangle, no sides of a triangle are equal to each other.
All angles in an equilateral triangle are equal they measure 60 degrees each.	Angles opposite to equal sides of an Isosceles triangle are equal.	No two angles are equal in Scalene triangles.
The equilateral triangle is shown in the image added below,	The isosceles triangle is shown in the image added below,	The scalene triangle is shown in the image added below,

Read More:

• Right Angle Formula
• Area of Triangle
• Area of Equilateral Triangle

Perimeter of Scalene Triangle

Perimeter of any figure is the length of its total boundary. So, the perimeter of a scalene triangle is defined as the sum of all of its three sides.

From the above figure,

Perimeter = (a + b + c) units

Where a, b and c are the sides of the triangle.

Area of Scalene Triangle

Area of any figure is the space enclosed inside its boundaries for the scalene triangle area is defined as the total square unit of space occupied by the Scalene triangle.

Area of the scalene triangle depends upon its base and height of it. The image added below shows a scalene triangle with sides a, b and c and height h units.

When Base and Height are Given

When the base and the height of the scalene triangle is given then its area is calculated using the formula added below,

A = (1/2) × b × h sq. units

Where,

• b is the base and 
• h is the height (altitude) of the triangle.

When Sides of a Triangle are Given

If the lengths of all three sides of the scalene triangle are given instead of base and height, we calculate the area using Heron's formula, which is given by,

A = √(s(s - a)(s - b)(s - c)) sq. units

Where,

• s denotes the semi-perimeter of the triangle, i.e, s = (a + b + c)/2, and
• a, b, and c denotes the sides of the triangle.

Read More:

• Types of Triangles
• Area of an Equilateral Triangle
• Perimeter of a Triangle

Solved Examples Scalene Triangle

Example 1: Find the perimeter of a scalene triangle with side lengths of 10 cm, 15 cm, and 6 cm.
Solution:

We have, 

a = 10
b = 15
c = 6

Using the Perimeter Formula 

Perimeter (P) = (a + b + c)

⇒ P = (10 + 15 + 6)
⇒ P  = 31 cm

Thus, the required perimeter of the triangle is 31 cm.

Example 2: Find the length of the third side of a scalene triangle with two side lengths of 3 cm and 7 cm and a perimeter of 20 cm.
Solution:

We have, 

a = 3
b = 7
P = 20

Using the Perimeter Formula 

Perimeter (P) = (a + b + c)

⇒ P = (a + b + c)
⇒ 20 = (3 + 7 + c)
⇒ 20 = 10 + c
⇒ c = 10 cm

Thus, the required length of third side of the triangle is 10 cm

Example 3: Find the area of a scalene triangle with side lengths of 8 cm, 6 cm, and 10 cm.
Solution:

We have, 

a = 8
b = 6
c = 10

Semi-Perimeter (s) = (a + b + c)/2

⇒ s = (8 + 6 + 10)/2
⇒ s = 24/2
⇒ s = 12 cm

Using the Heron's formula 

Area = √(s(s - a)(s - b)(s - c))

⇒ A = √(12(12 - 8)(12 - 6)(12 - 10))
⇒ A  = √(12(4)(6)(2))
⇒ A  = √576
⇒ A  = 24 sq. cm

Thus, the required area of the scalene triangle is 24 cm²

Example 4: Find the area of a scalene triangle whose base is 20 cm and altitude is 10 cm.
Solution:

We have, 

b = 20 
h = 10

Area of Scalene Triangle (A) = 1/2 × b × h

⇒ A  = 1/2 × 20 × 10
⇒ A = 100 sq. cm

Thus, the area of the given scalene triangle is 100 sq. cm.