Exams Academic Scalene Triangle: Definition, Properties, Area & More

Written By Prince
Last Modified 10-03-2023

Scalene Triangle: Definition, Properties, Area & More

In geometry, a triangle is a closed two-dimensional plane figure, which is in the form of a three-sided polygon with three sides, three angles, three vertices and three edges. A scalene triangle is a triangle which has three sides of three different lengths, and three different angles at the vertices. However, the sum of all the interior angles of the triangle is always 180°, satisfying the angle sum property of a triangle. In this article, we are going to discuss the definition, formulas for perimeter, area and properties of a scalene triangle. Scroll down to learn more about this interesting and important geometric concept.

Also, Check:

Triangles	Properties of Triangles
Area of Triangle	Area of Equilateral Triangle
Area of Right Angled Triangle	Geometry Formulas

What is a Scalene Triangle?

Scalene Triangle is a triangle that has no equal sides and no two equal (similar) angles. Some of the real-life examples of this kind of triangle are roof truss as used in the building roofs, frame of a bicycle, nachos, set squares, etc. Look at △ABC in the diagram given below as an example:

In the diagram above, we have: AB ≠ AC ≠ BC and ∠A ≠ ∠B ≠ ∠C

Properties of Scalene Triangle

The important properties of a scalene triangle are given below:

All sides of the triangle are unequal.
All angles of the triangle are unequal.
The triangle has no line of symmetry.
The angle opposite to the longest side would be the greatest angle and vice versa.
The triangle can be acute-angled or obtuse-angled or right-angled.
With the knowledge of two angles, the remaining one angle can be calculated as we know that the sum of angles in a triangle is 180 degrees (as per the properties of triangle).

Perimeter of Scalene Triangle

The perimeter of a triangle is the sum of the length of all three sides. For a scalene triangle, the perimeter can be found using the length of all three sides. The formula to calculate the perimeter of the isosceles triangle is given by:

$$Perimeter\,of\,Scalene\,Triangle\,{\rm{ = }}\,Sum\,of\,lengths\,of\,All\,Three\,Sides$$

For example,

For the above triangle,
Perimeter = 7 cm + 12 cm + 15 cm
= 34 cm

Area of Scalene Triangle (Heron’s Formula)

The area of a scalene triangle is given by Heron’s formula which is a 2-step process that is explained below:

Step 1: Calculate “s” which is the semi-perimeter of a triangle i.e, perimeter divided by 2

$$s = {{a + b + c} \over 2}$$

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

Step 2: Then calculate the area using the formula provided below:

$$Area = \sqrt {s(s – a)(s – b)(s – c)} $$

Thus, we can obtain the area of a triangle if we know the length of all its three sides.

Also check..

Students can access the following study materials on Embibe for their preparation:

NCERT Solutions	NCERT Books
Class 8 Mock Test Series	Class 8 Practice Questions
Class 9 Mock Test Series	Class 9 Practice Questions
Class 10 Mock Test Series	Class 10 Practice Questions

Solved Examples on Scalene Triangle

Here are some of the solved examples to understand this type of triangles better:

Question 1: Find the area of the scalene triangle ABC with the sides 8 cm, 6 cm and 4 cm.

Solution: Let a= 8 cm
b = 6 cm
c = 4 cm
If all the sides of a triangle are given, then we use Heron’s formula to calculate the area.
So,
$$Area\,of\,triangle\, = \,\sqrt {s(s – a)(s – b)(s – c)} $$
Here,
$$s = {{a + b + c} \over 2} = {{8 + 6 + 4} \over 2} = 9$$
Putting the value of s, a, b, and c in the formula, we get:
$$Area = \sqrt {9(9 – 8)(9 – 6)(9 – 4)} = \sqrt {135} = 11.6$$
Therefore, the area of the triangle = 11.6 cm²

Question 2: If the sides of a triangle are 8cm, 15cm and 9cm. Find its perimeter.

Solution: Perimeter of a triangle = Sum of all its sides
Hence, Perimeter = (8 + 15 + 9)cm
= 32 cm

Question 3: Find the area of this triangle:

Solution: In this example, we have the three sides of the triangle as:
a = 7 cm
b = 13 cm
c = 14 cm

First, we calculate s which using the formula:

$$s = {{a + b + c} \over 2} = {{7 + 13 + 14} \over 2} = {{34} \over 2} = 17$$

Now using Heron’s formula, we can calculate the area of this triangle.

$$Area = \sqrt {s(s – a)(s – b)(s – c)} $$

Putting the values of s, a, b, and c in the above equation, we get:

$$ = \sqrt {17(17 – 7)(17 – 13)(17 – 4)} = \sqrt {8840} = 94.021$$

Hence area of the triangle is 94.02 cm².

Frequently Asked Questions (FAQs)

Some of the frequently asked questions about this topic are answered below:

Q1: What is the definition of a scalene triangle?
A: A scalene triangle is a triangle that has all its sides unequal in length and all its angles unequal in measure.

Q2: What are the properties of a scalene triangle?
A: Some of the important properties are:
(i) It has all sides unequal
(ii) It has no line of symmetry
(iii) Interior angles can be acute, obtuse or right-angle.

Q3: What is the angle sum property of a scalene triangle?
A: As per the angle sum property, the sum of the three interior angles equals 180 degrees.

Q4: What is the formula for the area and perimeter of a scalene triangle?
A: Area of scalene triangle is equal to half of the product of its base-length and height. Perimeter is equal to the sum of its three unequal sides.

Q5: What is a right-angled scalene triangle?
A: When one of the three angles measure 90 degrees and the angles or lengths of the other two sides are not congruent, then the scalene triangle is called a right scalene triangle.

Related Concepts:

Area of Triangle Concepts

Area of Triangle in Coordinate Geometry

Area of Triangle & Properties

Properties of Triangle Concepts

Area of Triangle Vectors Cross Product Concepts

Important Points of Triangle

Examples o f Properties of Triangle

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