Scalene Triangle is one of the types of triangles which is mentioned in geometry.  We are going to discuss here its definition, formulas for perimeter and area and its properties. The triangles are basically defined on the basis of its sides and angles. In geometry, a triangle is a closed two-dimensional plane figure with three sides and three angles and is shown as a three-sided polygon. It has three vertices and three edges.

Based on the sides and the interior angles of a triangle, a triangle has different types. According to the interior angles of the triangle, it can be classified as three types, namely:

• Acute Angle Triangle
• Right Angle Triangle
• Obtuse Angle Triangle

According to the sides of the triangle, the triangle can be classified into three types, namely;

• Scalene Triangle
• Isosceles Triangle
• Equilateral Triangle

Scalene Triangle Definition

It is defined as, a triangle in which all the sides of a triangle are of different length and all the angles of a triangle are of different measures where the sum of the interior three angles of a triangle is always equal to 180 degrees.

Formulas

Let us learn the formulas to find the area and perimeter of a triangle which has unequal sides and angles or we can which is scalene in nature.

Area of a Scalene Triangle

Area  = (1/2) x b x h square units

Where,

“b” refers to the base of the triangle

“h” refers to the height of a triangle

If the sides of the triangle are given, then apply the Heron’s formula

Area of the triangle =  \(A = \sqrt{S (S-a)(S-b)(S-c)}\) square units

Where S is the semiperimeter of a triangle, which can be found using the formula

S = (a+b+c)/2

Here,

a, b, and c denotes the sides of the triangle

Perimeter of a Scalene Triangle

The perimeter of a triangle is equal to the sum of the length of sides of a triangle and it is given as

Perimeter  = a + b + c units

Example: Consider a given triangle

To find the perimeter for the given triangle, add the sides of a triangle

Therefore, perimeter = 15 + 34 + 32 = 81 cm

Properties

Some of the important properties of the scalene triangle are as follows:

• It has no equal sides
• It has no equal angles
• It has no line symmetry
• It has no point symmetry.
• The angles inside this triangle can be an acute, obtuse or right angle.
• If all the angles of the triangle are less than 90 degrees(acute), then the center of the circumscribing circle will lie inside a triangle.
• In a scalene obtuse triangle, the circumcenter will lie outside the triangle.

Sample Problem

Question: Find the area of the scalene triangle ABC with the sides 8cm, 6cm and 4cm

Solution:

Let a= 8 cm

b = 6 cm

c = 4 cm

If all the sides of a triangle are given, then use Heron’s formula

Area of triangle =  \( \sqrt{S (S-a)(S-b)(S-c)}\)

Now, find the semiperimeter value

S = (a+b+c)/2

S = (8+6+4)/2

S = 18/2

S = 9

Now substitute the value of S in the area formula,

Area = \(\sqrt{9(9-8)(9-6)(9-4)}\)

\(=\sqrt{9(1)(3)(5)}\)

=√135

=11.6

Therefore, the area of the scalene triangle = 11.6 square units.