Scalene Triangle

Last updated on 25 October 2025

Table of contents

Definition of scalene triangle
Characteristics of a scalene triangle
Solved Examples on Scalene Triangle
FAQs on Scalene Triangle

A scalene triangle is a type of triangle in which all three sides have different lengths and all three angles have different measures. This distinguishes it from other types of triangles, such as equilateral triangles where all three sides are equal and isosceles triangles where at least two sides are equal.

Definition of scalene triangle: A scalene triangle is a type of triangle that has three unequal sides and three unequal angles. In other words, all the sides and angles of a scalene triangle have different measures. The term "scalene" comes from the Greek word "skalenos", meaning "uneven" or "unequal".

In the figure above, all the three sides and angles of △PQR are unequal, hence it is scalene triangle.

Characteristics of a Scalene Triangle:

Below are some characteristics and properties of scalene triangles:

Side Lengths: In a scalene triangle, each side has a different length. Let's label the sides as $a$ , $b$ and $c$ . So, $a \neq b \neq c$
Angle Measures: The angles of a scalene triangle are also different from one another. Let's label the angles as $∠ A$ , $∠ B$ and $∠ C$ . So, $∠ A \neq ∠ B \neq ∠ C$
Triangle Inequality: The triangle inequality theorem applies to scalene triangles, just as it does for all triangles. It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In mathematical terms, if $a$ , $b$ and $c$ are the lengths of the sides of a scalene triangle, then:
$a + b > c$ $a + c > b$ $b + c > a$
Angle Relationships: Since the angles of a scalene triangle are all different, there are no special relationships between the angles. Each angle have different measure.
Perimeter: The perimeter of a scalene triangle is simply the sum of the lengths of its three sides, i.e. $a + b + c$ .
Area: The area of a scalene triangle can be calculated using various methods, such as Heron's formula or by calculating the height and base. Heron's formula states that the area (A) of a scalene triangle with side lengths $a$ , $b$ and $c$ is given by:
$A = \sqrt{s (s - a) (s - b) (s - c)}$
where s is the semi-perimeter of the triangle, calculated as $s = \frac{(a + b + c)}{2}$
Congruence: Scalene triangles are not congruent to each other unless they have the same side lengths and angle measures. Congruent triangles have exactly the same shape and size.
Symmetry: Scalene triangles do not possess any line or rotational symmetry. They can be rotated and reflected in various ways.

Scalene triangles are commonly encountered in real-world objects and situations. Understanding their properties and characteristics helps in various mathematical and geometric applications, such as solving triangles, calculating areas and analyzing shapes and structures.

Solved Examples on Scalene Triangle

Example 1: Find perimeter and area of a scalene triangle if it's sides are $3 cm$ , $4 cm$ and $5 cm$ .

Given: Sides of scalene triangle: $3 cm$ , $4 cm$ , $5 cm$

To find: Perimeter( $P$ ) and Area( $A$ ) of triangle

Solution: First lets find the perimeter of the triangle.

\begin{array}{rcl} ∵ & P & = & a + b + c \\ ∴ & P & = & 3 + 4 + 5 \\ ∴ & P & = & 12 cm \end{array}

Therefore, the perimeter of the given triangle is

12 cm

Now lets find the semi perimeter of the triangle.
As we know that, semi perimeter is half of the perimeter,

\begin{array}{rcl} ∵ & s & = & \frac{P}{2} \\ ∴ & s & = & \frac{12}{2} \\ ∴ & s & = & 6 cm \end{array}

Now, as per the Heron’s formula, we know;

\begin{array}{rcl} A & = & \sqrt{s (s - a) (s - b) (s - c)} \\ = & \sqrt{6 (6 - 3) (6 - 4) (6 - 5)} \\ = & \sqrt{6 \times 3 \times 2 \times 1} \\ = & \sqrt{36} \\ = & \sqrt{6^{2}} \\ = & 6 {cm}^{2} \end{array}

Therefore, the area of the given scalene triangle is

6 {cm}^{2}

Example 2: Find perimeter and area of a scalene triangle if it's sides are $9 cm$ , $10 cm$ and $11 cm$ .

Given: Sides of scalene triangle: $9 cm$ , $10 cm$ , $11 cm$

To find: Perimeter( $P$ ) and Area( $A$ ) of triangle

Solution: First lets find the perimeter of the triangle.

\begin{array}{rcl} ∵ & P & = & a + b + c \\ ∴ & P & = & 9 + 10 + 11 \\ ∴ & P & = & 30 cm \end{array}

Therefore, the perimeter of the given triangle is

30 cm

Now lets find the semi perimeter of the triangle.
As we know that, semi perimeter is half of the perimeter,

\begin{array}{rcl} ∵ & s & = & \frac{P}{2} \\ ∴ & s & = & \frac{30}{2} \\ ∴ & s & = & 15 cm \end{array}

Now, as per the Heron’s formula, we know;

\begin{array}{rcl} A & = & \sqrt{s (s - a) (s - b) (s - c)} \\ = & \sqrt{15 (15 - 9) (15 - 10) (15 - 11)} \\ = & \sqrt{15 \times 6 \times 5 \times 4} \\ = & \sqrt{2^{2} \times 450} \\ = & 2 \times \sqrt{450} \\ = & 2 \times \sqrt{225 \times 2} \\ = & 2 \times \sqrt{15^{2} \times 2} \\ = & 2 \times 15 \times \sqrt{2} \\ = & 30 \sqrt{2} {cm}^{2} \end{array}

Therefore, the area of the given scalene triangle is

30 \sqrt{2} {cm}^{2}

FAQs on Scalene Triangle

How is a scalene triangle different from other triangles?
Unlike equilateral (all sides equal) or isosceles (two sides equal) triangles, a scalene triangle has no equal sides or angles.
What are the properties of a scalene triangle?
- The sum of the interior angles is always 180°.
- There is no line of symmetry since no sides or angles are equal.
- The longest side is opposite the largest angle and the shortest side is opposite the smallest angle.
How do I calculate the area of a scalene triangle?
- The area of a scalene triangle can be calculated using Heron's formula:
  $A = \sqrt{s (s - a) (s - b) (s - c)}$
  where
  - $s$ is the semi-perimeter of the triangle, $(s = \frac{a + b + c}{2})$
  - $a$ , $b$ and $c$ are the three sides of the triangle.
- If the base and height of the scalene triangle is known, then it's area can be calculated using below formula:
  $Area = \frac{1}{2} \times base \times height$
Can a scalene triangle have an equal angle and different sides?
No. If any two angles are equal, the sides opposite those angles must also be equal — making it isosceles, not scalene.
How do scalene triangles relate to the Triangle Inequality Theorem?
Like all triangles, scalene triangles must satisfy the Triangle Inequality Theorem. This means that the sum of the lengths of any two sides must be greater than the length of the third side.

Scalene Triangle

Characteristics of a Scalene Triangle:

Solved Examples on Scalene Triangle

Example 1: Find perimeter and area of a scalene triangle if it's sides are 3cm, 4cm and 5cm.

Example 2: Find perimeter and area of a scalene triangle if it's sides are 9cm, 10cm and 11cm.

FAQs on Scalene Triangle

How is a scalene triangle different from other triangles?

What are the properties of a scalene triangle?

How do I calculate the area of a scalene triangle?

Can a scalene triangle have an equal angle and different sides?

How do scalene triangles relate to the Triangle Inequality Theorem?

Triangle

Isosceles Triangle

Example 1: Find perimeter and area of a scalene triangle if it's sides are $3 cm$ , $4 cm$ and $5 cm$ .

Example 2: Find perimeter and area of a scalene triangle if it's sides are $9 cm$ , $10 cm$ and $11 cm$ .