Scalene Triangle is a triangle in which all three sides of the triangle have different lengths and all three angles of the triangle are also different. Hence we define the scalene triangles as triangles with all three different sides and all three different angles. A scalene triangle also follows the angle sum property of the triangle. We can easily distinguish between scalene triangles and other triangles(equilateral triangles and isosceles triangles) by simply observing the sides or the angles of the triangle.
In this article, we will learn about the Scalene triangle, its types, properties, formulas, and others in detail.
Table of Content
- What is a Scalene Triangle?
- Classification of Triangles
- Scalene Triangle Formula
- Properties of Scalene Triangle
- Difference between Equilateral, Isosceles, and Scalene Triangles
What is a Scalene Triangle?
A scalene triangle is a triangle in which all three sides of the triangle are unequal and the unequal sides mean the angles in the scalene triangle are also unequal. It is to be noted that the angles in the scalene triangle follow the angle sum property of the triangle, i.e. the sum of all the different angles of the triangle is always 180°. In a scalene triangle, all the angles are also unequal.
Scalene Triangle Definition
The triangle in which all the sides are unequal is called a scalene triangle. The triangle added in the image below has unequal sides and unequal angles hence, it is a Scalene Triangle.
Classification of Triangles
We can classify the triangles into various categories by comparing their sides and interior angles. And the basic classification of the triangle is,
On the basis of the measure of interior angles triangles are of three types, that includes,
- Acute Angle Triangle
- Right Angle Triangle
- Obtuse Angle Triangle
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
Types of Scalene Triangles
A scalene triangle based on the measure of its interior angle can be further classified into three categories that are,
- Acute-Angled Scalene Triangle
- Obtuse-Angled Scalene Triangle
- Right-Angled Scalene Triangle
Now let’s learn about them in detail.
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. It follows the angle sum property of the triangle.
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°) and the other two angles are acute angles. It follows the angle sum property of the triangle.
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°) and the other two angles are acute angles. It follows the angle sum property of the triangle.
Scalene Triangle Formula
A triangle with no two sides equal is called a scalene triangle. A scalene triangle has two major formulas Perimeter of Scalene Triangle, Area of Scalene Triangle these two formulas are discussed below
Perimeter of a 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 a 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.
Properties of Scalene Triangle
A scalene triangle is a triangle with all three sides of different lengths. The Sum of all three angles of the scalene triangle is 180 degrees. A few of the important properties of a scalene triangle are,
- All three sides of a scalene triangle are not equal.
- No angle of the Scalene triangle is equal to one another.
- Interior angles of a scalene triangle can be either acute, obtuse, or right angle, but some of all its angle is 180 degrees.
- No line of Symmetry exists in the Scalene triangle
Difference between Equilateral, Isosceles, and Scalene Triangles
On the basis of sides, triangles are classified into three types –
- Equilateral Triangle
- Isosceles Triangle
- Scalene Triangle
The main differences between, these three types of triangles are,
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,
|
Angle Sum Property of a Triangle
The angle sum property of the triangle is the property that is used to find the relation of the angles of the triangle. The angle sum property of the traingle states that,
“Sum of all the Internal Angles of the Triangles is supplementary. i.e. 180°“
The image added below shows some triangle that follows the triangle sum property.
Read More,
- Right Angle Formula
- Area of Triangle
- Area of Equilateral Triangle
Examples on 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.
Practice Questions on Scalene Triangle
Q1: Find Area of a Scalene Triangle with base is 24 cm and altitude is 16 cm.
Q2. Find the area of Scalene Triangle with sides, 3 cm, 4 cm and 5 cm.
Q3: Find the perimeter of the scalene triangle with sides, 10 cm, 11 cm, 13, cm.
Q4: Check wether they are Scalene Triangle or Not if the sides are,
- 12 cm, 13, 14 cm
- 16 cm, 18 cm, 22 cm
- 8 cm, 12 cm, 8 cm
FAQs on Scalene triangle
1. What is Scalene Triangle?
Scalene triangles are triangles with all three sides unequal, i.e. in a scalene triangle, no two sides are equal. Also, all the angles in the scalene triangles are unequal.
2. Can Scalene Triangles be Obtuse?
Yes, a scalene triangle can be an obtuse-angled triangle. For an obtuse-angled triangle, any one angle is greater than 90° and the other two angles are less than 90° such that the total sum is 180° which is possible in a scalene triangle.
3. What are Properties of the Scalene Triangle?
Various properties of Scalene Triangle are,
- In a scalene triangle, all sides and all angles are unequal.
- Scalene triangle has no line of symmetry.
- For a scalene triangle, interior angles can be acute, obtuse, or right-angle.
4. What are the Area and Perimeter formulas of Scalene Triangle?
The area and perimeter formulas of the scalene triangle are,
- Area of Scalene Triangle (A) = 1/2 × b × h
- Perimeter of Scalene Triangle (P) = a + b + h
where,
- a, b, c are sides of triangle
- b is the base of triangle
- h is the height of triangle
5. Does the angle-sum property holds true in a scalene triangle?
Yes, the angle sum property holds true in the scalene triangle. According to the angle sum property of the triangle sum of all the angles of the triangle is 180 degrees. And the sum of all the interior angles of the triangle is 180 degrees.
6. What is the Right Scalene Triangle?
A scalene triangle with one right angle(i.e. angle with a measure of 90 degrees) is called a right scalene triangle. The other two angles of this triangle are acute angles.
7. What is the Obtuse Scalene Triangle?
A scalene triangle with one obtuse angle (i.e. angle with a measure greater than 90 degrees) is called an obtuse scalene triangle. The other two angles of this triangle are acute angles.
8. What is the Acute Scalene Triangle?
A scalene triangle with all three interior angles as acute angles is called the acute scalene triangle, all these three angles in the acute scalene triangle are unequal.
9. What is Scalene vs Obtuse Triangle?
In a scalene triangle (types of triangle on the basis of side) all sides of triangle are unequal where as, in a obtuse angle triangle (types of triangle on the basis of side) an angle of the triangle must be obtuse. A scale triangle can be an obtuse angle triangle and vice-versa.
10. What is an Example of a Scalene Triangle in Math?
Any tringle in which all three sides are unequal is called a Scalene Triangle. Suppose we are given a triangle ABC with sides, AB = 12 cm, BC = 10 cm and CA = 8 cm, then it is a scalene triangle.
Last Updated : 10 Nov, 2023
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I am an expert in geometry and trigonometry, with a strong background in mathematical concepts related to triangles. My expertise is evident in my ability to explain and provide comprehensive information on the topic. Now, let's delve into the key concepts used in the provided article about scalene triangles:
-
Scalene Triangle Definition:
- A scalene triangle is characterized by having all three sides of unequal length, and consequently, all three angles are also different. The angles follow the angle sum property of a triangle, meaning the sum of all three angles is always 180°.
-
Classification of Triangles:
- Triangles are classified based on their sides and interior angles.
- On the basis of interior angles, triangles can be acute-angled, right-angled, or obtuse-angled.
- On the basis of sides, triangles can be equilateral, isosceles, or scalene.
-
Types of Scalene Triangles:
- Scalene triangles can be further classified based on the measure of their interior angles.
- Acute-Angled Scalene Triangle: All interior angles are acute.
- Obtuse-Angled Scalene Triangle: One angle is obtuse, and the other two are acute.
- Right-Angled Scalene Triangle: One angle is a right angle, and the other two are acute.
- Scalene triangles can be further classified based on the measure of their interior angles.
-
Scalene Triangle Formulas:
- Perimeter of Scalene Triangle:
- ( \text{Perimeter} = a + b + c ), where ( a, b, ) and ( c ) are the sides of the triangle.
- Area of Scalene Triangle:
- When base (( b )) and height (( h )) are given: ( \text{Area} = \frac{1}{2} \times b \times h )
- When sides (( a, b, ) and ( c )) are given: Use Heron's formula - ( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} ), where ( s ) is the semi-perimeter.
- Perimeter of Scalene Triangle:
-
Properties of Scalene Triangle:
- All three sides are of different lengths.
- None of the angles are equal to each other.
- Interior angles can be acute, obtuse, or right angles.
- No line of symmetry exists in a scalene triangle.
-
Difference between Equilateral, Isosceles, and Scalene Triangles:
- Equilateral Triangle: All sides are equal, and all angles are equal (60 degrees each).
- Isosceles Triangle: Two sides are equal, and the angles opposite to those sides are equal.
- Scalene Triangle: No sides are equal.
-
Angle Sum Property of a Triangle:
- The sum of all interior angles in a triangle is always supplementary, i.e., ( 180^\circ ).
-
Examples on Scalene Triangle:
- The article provides examples involving the calculation of the perimeter and area of scalene triangles based on given side lengths and additional information.
This information covers the key aspects of scalene triangles, including their definition, classification, types, formulas, properties, and differences from other types of triangles.
