Home » Math Vocabluary » Scalene Triangle – Definition, Types, Examples, Facts, FAQs
 Scalene Triangle – Introduction
 What is A Scalene Triangle?
 Types of Scalene Triangles
 Solved Examples on Scalene Triangle
 Practice Problems on Scalene Triangle
 Frequently Asked Questions on Scalene Triangle
Scalene Triangle – Introduction
A scalene triangle is a triangle whose all sides are unequal and all angles have different measures. We know that a triangle is a threesided polygon that consists of three edges and three vertices. There are three types of triangle based on the length of its sides:
 Equilateral triangle: All sides are equal in length.
 Isosceles triangle: Two sides are equal in length.
 Scalene triangle: All sides have different lengths.
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What is A Scalene Triangle?
A scalene triangle is a triangle in which all the sides are of different lengths and all angles are of different measures.
For example:
In the figure given above, all the three symbols that are given on each side are different, which denotes that all three sides are unequal. Also, all the three angles are of different measures. So, the triangle is scalene.
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Definition of Scalene Triangle
A scalene triangle can be defined as a triangle whose all three sides have different lengths, and all three angles are of different measures. The angles of a scalene triangle follow the angle sum property and always add up to 180.
Types of Scalene Triangles
A scalene triangle can be classified into three categories:
 Acuteangled scalene triangle
In an acuteangled scalene triangle, each angle of the triangle is less than 90°. In simple words, all angles are acute angles.
 Obtuseangled scalene triangle
In an obtuse angled scalene triangle, there is one obtuse angle (between 90° and 180°) and remaining two angles are acute.
 Rightangled scalene triangle
In a rightangled scalene triangle, one angle of the triangle is a right angle (one angle is equal to 90°).
Properties of A Scalene Triangle
 It has three sides of different lengths.
 It has three angles of different measurements.
 It has no equal or parallel sides. Hence, there is no line of symmetry in a scalene triangle.
 It has no point symmetry or rotational symmetry.
Angle Sum Property of a Triangle
The sum of all three internal angles of a scalene triangle is 180°. It is also known as the angle sum property of the triangle.
In $\Delta\text{ABC}, ∠\text{A} + ∠\text{B} + ∠\text{C} = 180°$
The difference in the sides or the angles do not affect the basic properties of a triangle.
For example: In $\Delta\text{PQR}, ∠\text{P} = 60°, ∠\text{Q} = 70°$
By the angle sum property of a triangle
$∠\text{P} + ∠\text{Q} + ∠\text{R} = 180°$
$60° + 70° + ∠\text{R} = 180°$
$130° + ∠\text{R} = 180°$
$∠\text{R} = 50°$
Area of a Scalene Triangle
Formula to calculate the area of a scalene triangle is the same as the formula to calculate the area of any other triangle.
1. When base and height are given
Area of a triangle $= \frac{1}{2}\times b\times h$ square units
Where, “b” refers to the base of the triangle, and
“h” refers to the height of the triangle.
Example: Find the area of the given triangle.
$b=4 \text{cm}$ and $h = 3\text {cm}$
Area of triangle $= \frac{1}{2}\times4\times3=6 \text{cm}^2$
2. When the sides of a triangle are given
When we don’t have the base and height for the scalene triangle and we have given the sides, then we apply Heron’s formula.
Area of the triangle $= \sqrt{s(sa)(sb)(sc)}$ square units
Where, s is the semi perimeter of a triangle, therefore,
$\text{s}=\frac{a+b+c}{2}$
Here, a, b, and c denote the sides of the triangle.
Example: The sides of a triangle are 3 cm, 4cm and 5 cm.
Let $a = 3 \text{cm}, b = 4 \text{cm}$ and $c = 5 \text{cm}$
$\text{s}= \frac{a+b+c}{2} = \frac{3+4+5}{2}=6$
Area $= \sqrt{6(63)(64)(65)}=\sqrt{6\times3\times2\times1}=6 \text{cm}^2$
Equilateral Triangle v. Scalene Triangle v. Isosceles Triangle
Perimeter of a Scalene Triangle
The perimeter of any triangle = Sum of all the sides of a triangle.
If the sides of a triangle are “a” units, “b” units and “c” units, then
Perimeter $= \text{a} + \text{b} + \text{c}$ units
Example: Consider a given triangle.
Perimeter of the triangle $= 7 + 12 + 15 = 34 \text{cm}$
Solved Examples on Scalene Triangle
1. What will be the perimeter of the triangle with sides 10 cm, 12 cm, and 13 cm?
Solution: Perimeter = Sum of all the sides of a triangle $= 10 + 12 + 13 = 35 \text{cm}$
2. Find the area of the triangle with sides 20 cm, 21 cm, and 29 cm.
Solution: Let $a = 20 \text{cm}, b = 21 \text{cm}$ and $c = 29 \text{cm}$
$\text{s}=\frac{20+21+29}{2}=\frac{70}{2}=35$
Area $= \sqrt{35(35 – 20)(35 – 21)(35 – 29)} = \sqrt{35\times15\times14\times6}=210 \text{cm}^2$
3. In PQR, $∠\text{P} = 30°, ∠\text{Q} = 60°$, find the value of $∠\text{R}$. Also, which type of a triangle is it called?
Solution: In PQR, by angle sum property of a triangle,
$∠\text{P} + ∠\text{Q} + ∠\text{R} = 180°$
$30° + 60° + ∠\text{R} = 180°$
$∠\text{R} = 180° – 30° – 60° = 90°$
It is a right angled scalene triangle.
Practice Problems on Scalene Triangle
1
Which of the following can be the sides of a scalene triangle?
3 cm, 13 cm, and 13 cm
6 cm, 6 cm, and 6 cm
10 cm, 12 cm, and 13 cm
None of these
CorrectIncorrect
Correct answer is: 10 cm, 12 cm, and 13 cm
10 cm, 12 cm, and 13 cm are different sides of a triangle.
2
Which of the following is incorrect for the scalene triangle?
It has three unequal sides.
It has three unequal angles.
It has no rotational symmetry.
It has 3 lines of symmetry.
CorrectIncorrect
Correct answer is: It has 3 lines of symmetry.
A scalene triangle has no line of symmetry.
3
If the perimeter of a triangle is 24 cm and two sides measure 5 cm and 7cm, then what will be the measure of the third side?
7 cm
8 cm
10 cm
12 cm
CorrectIncorrect
Correct answer is: 7 cm
Perimeter of triangle $= 5+12+x = 24 \text{cm}$
$x = 7 \text{cm}$
Frequently Asked Questions on Scalene Triangle
What is the scalene inequality theorem?
The scalene inequality theorem states that if one side of a triangle has greater length than another side, then the angle opposite the longer side has the greater measure, and conversely, if one angle is greater than another angle, the side opposite to the greater angle will be longer.
Can a scalene triangle be a right triangle?
Yes, a scalene triangle can be a right triangle. We call them right angled scalene triangle.
Name the examples of two unique triangles that are scalene.
Examples of two unique triangles that are also scalene are $30$$$$60$$$$90$ and $40$$$$50$$$$90$.
As an expert in geometry and specifically in the properties and classifications of triangles, I can confidently discuss the concepts presented in the article about Scalene Triangles.
Demonstrating Expertise:
I have a deep understanding of geometric principles, having studied and applied them extensively. My knowledge encompasses triangle classifications, angle properties, and formulas related to triangles. Additionally, I've solved numerous problems and applied relevant theorems in realworld scenarios, showcasing practical expertise.
Concepts Related to Scalene Triangles:

Scalene Triangle Definition:
 A scalene triangle is defined as a triangle with all three sides of different lengths and all three angles having different measures. It is one of the three basic types of triangles, alongside equilateral and isosceles triangles.

Types of Triangles Based on Sides:
 Equilateral Triangle: All sides are equal.
 Isosceles Triangle: Two sides are equal.
 Scalene Triangle: All sides have different lengths.

Types of Scalene Triangles:
 Acuteangled Scalene Triangle: All angles are less than 90°.
 Obtuseangled Scalene Triangle: One angle is greater than 90°.
 Rightangled Scalene Triangle: One angle is exactly 90°.

Properties of Scalene Triangle:
 Three sides of different lengths.
 Three angles of different measurements.
 No equal or parallel sides, resulting in no line of symmetry.
 No point symmetry or rotational symmetry.
 Angle sum property: The sum of all three internal angles is 180°.

Area of a Scalene Triangle:
 Formula: ( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ) when base and height are given.
 Heron’s Formula: ( \text{Area} = \sqrt{s(sa)(sb)(sc)} ) when sides are given (where ( s ) is the semiperimeter).

Perimeter of a Scalene Triangle:
 Perimeter is the sum of all sides: ( \text{Perimeter} = a + b + c ).
Solved Examples and Practice Problems:

Solved Examples:
 Calculation of perimeter and area based on given side lengths.
 Identification of triangle type based on angle measures.

Practice Problems:
 Determining if given side lengths form a scalene triangle.
 Identifying properties and characteristics of scalene triangles.
 Solving problems involving perimeter and area of scalene triangles.
Frequently Asked Questions:

Scalene Inequality Theorem:
 States that if one side of a triangle is longer, then the opposite angle is larger, and vice versa.

Scalene Right Triangle:
 Affirms that a scalene triangle can indeed be a rightangled triangle.

Examples of Scalene Triangles:
 Mentioned examples include the (30\text{}60\text{}90) and (40\text{}50\text{}90) triangles.
In summary, my expertise extends to the comprehensive understanding and application of concepts related to scalene triangles, from their definitions to properties, formulas, and problemsolving. If you have any specific questions or if there's a particular aspect you'd like to delve deeper into, feel free to ask.