# Scalene Triangle Formula

What is the Scalene Triangle?

In geometry, a triangle is a closed two-dimensional geometric figure with three sides and angles. A triangle has three vertices and three edges. There are three different types of triangles based on their sides and interior angles.

Classification of the triangle is as simple as comparing the sides of the triangle. A triangle with all three sides equal in length is known as an equilateral triangle.  A triangle with two sides equal in length and the third side different is known as an isosceles triangle. If the measurement of all the three sides of a triangle is unequal, then the triangle is known as the scalene triangle.

Along with the unequal side, the interior angles of a scalene triangle are also different. There are also some notable properties of the scalene triangle such that it has no line and point symmetry. The angle of a scalene triangle can be acute, right, or obtuse, but the sum of all the interior angles of scalene triangles should always be 180 degrees.

What is the Definition of the Scalene Triangle?

A scalene triangle is a type of triangle with all the three sides unequal in length, and also the measurement of all three angles of a scalene triangle is different. However, the sum of all the interior angles of a scalene triangle is always equal to 180 degrees.

Area of Scalene Triangle Formula

The following information is required to calculate the area of a scalene triangle

1. The length of one side and the perpendicular distance of the side to the opposite angle.

2. The length of all three sides.

Area of Scalene Triangle Formula When Base and Height are Given.

Area of scalene triangle formula when any side considered as base ‘b’ and height ‘h’ (a perpendicular drawn from the base) is given as:

Area = ½ × base ×height

If all the three sides of a triangle are given, then the area of a triangle can be calculated using Heroes’ formula.

How to Calculate the Area of a Triangle Using Heroes’ Formula?

The following are the steps to calculate the area of the triangle using Heroes’ formula.

1. Consider the length of all the three sides of a triangle say (a,b,c).

2. Calculate the semi perimeters of a triangle, S

The formula to calculate the semi perimeter of the triangle is

a + b + c /2

1. After calculating the semi-perimeter (S)of a triangle, apply the heroes' formula to find the area of a scalene triangle.

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

Area of a Scalene Triangle Formula without Height

When anyone angles of a triangle ( suppose ∠C)  along with the length of the two sides (a,b) is given, then the area of a triangle is measured s

Area = ab/2

The Perimeter of Scalene Triangle

The perimeter of a scalene triangle equals the sum of all the length of the sides of a triangle, and it is given as :

Perimeter of scalene triangle  = a + b + c units

For example, Consider the below scalene triangle.

Solution: To find the perimeter of the above scalene triangle, we will add all the sides of a given triangle.

Hence perimeter = 15 + 34 + 32 = 81 cm

Properties of Scalene Triangle

The different properties of a scalene triangle are given below:

•  The measurement of all three sides of a scalene triangle is different.

• The measurement of all three interior angles of a triangle is different.

• There is no line of symmetry in scalene triangles.

• The scalene triangle cannot be divided into two halves as there is no line of symmetry.

• The scalene triangle has no point symmetry.

• If all the angles of scalene triangles are less than 90 degrees (acute), then the center of the circumscribing circle will fall inside a triangle.

• The circumcentre of an obtuse scalene triangle lies outside the triangle.

Solved Examples for the Scalene Triangle Formulas

Some solved examples for the scalene triangle formulas are given below:

1. Calculate the Area of a Triangle with Two Sides as 20 cm, and 30 cm Angle Between the Two Sides of a Triangle is 30°.

Solution:

Area of triangle = ab/2 × Sin C

Area = 20  30/2 ×sin 30 cm

= 300 × 1/2

= 150 cm²

2. Calculate the Area of the Triangle Whose Sides are 10 cm , 12 cm and 18 cm

Solution:

As we know, the formula for the semi-perimeter of a triangle  = (a + b + c) / 2. Accordingly, the semi-perimeter for the above triangle is calculated as

= 10 + 12 + 18/2 cm

= 40 / 2

S = 20 cm

Area = $\sqrt{20(20-10)(20-12)(20-18)}$

= $\sqrt{20(10)(8)(2)}$

= $\sqrt{20(160)}$

= $\sqrt{320}$

= 17.8 cm²

3. Find the Area of a Triangle with Base 18 cm and Height 20 cm.

Solution:

As we know, the formula for the area of triangle = ½ * base * height. Accordingly, the area for the above triangle is calculated as

= ½ ×18 ×20

=  ½ × 360

=  180 cm²

Quiz Time

1. A Triangle with No Equal Sides and No Equal Angles is Known as

1. Isosceles triangle

2. Scalene triangle

3. Right angle triangle

4. Acute angle triangle

2. A Triangle with Three Equal Sides and Three Equal Angles is Known as

1. Isosceles triangle

2. Scalene triangle

3. Right angle triangle

4. Acute angle triangle

1. What are the Three Types of Triangles Based on the Length of their Sides?

The three types of a triangle based on the length of their slides are:

1. Scalene Triangle -  If the measurement of all the three sides and interior angles of a triangle are different, then the triangle is said to be a scalene triangle. It also implies that three vertices of a triangle are different.

2. Isosceles Triangle - If the measurement of any two sides of a triangle are equal and the third side has a different length, then the triangle is said to be an isosceles triangle. The measurement of the angle opposite to the equal sides of an isosceles triangle is the same.

3. Equilateral Triangle - If the measurement of all the three sides and interior angles of a triangle is the same then the triangle is said to be an equilateral triangle. As, we know the sum of all the three angles of a triangle is 180 degrees. Hence the measurement of each angle of an equilateral triangle is 60 degrees.

2. What are the Three Types of Triangle Bases on their Interior Angles?

The three types of triangle bases on their interior angles are:

1. Acute Angle Triangle - A triangle consisting of three acute angles is known as an acute angle triangle. An acute angle is an angle that measures below 90 degrees.

2. Obtuse Angle Triangle - A triangle consisting of two acute angles and one obtuse angle is known as an obtuse angle triangle. An obtuse angle is an angle that measures above 90 degrees.

3. Right Angle Triangle -  A triangle consisting of two acute angles and one right angle are known as the right-angle triangle. An acute angle is an angle that measures below 90 degrees.