**Introduction**

A triangle is a two-dimensional (2D) shape with three sides, three angles and three vertices. It can be classified based on its angle measurements or lengths of sides. Triangles may be equilateral, isosceles, or scalene based on the lengths of sides.

We shall discuss the description of the scalene triangle, its properties, and a few examples with solutions about its area and perimeter in this article.

**Scalene Triangle**

**Definition**

A triangle with three different side lengths and three different angle measurements is called a scalene triangle. The total of all interior angles of a scalene triangle is always equal to 180 degrees.

An illustration of a scalene triangle is shown in the picture below. The hatch “|” symbol on each side denotes that each side has different measurements. The angles A, B, and C also have different measurements.

**Measuring Scalene Triangles**

It will be helpful to use tools to identify whether a triangle is a scalene since a scalene triangle is one with three different side lengths and three different angle measurements.

We can easily measure each side of the triangle using a ** ruler.** If all sides have different lengths, then the triangle is scalene. To measure angles, we may use a

*protractor.***Properties of Scalene Triangles**

The following are the properties of scalene triangles:

*A scalene triangle has no equal sides.*

All sides of scalene triangles have different lengths. The figure below shows some examples of scalene triangles.

The triangle on the left uses the hatch mark “|” to indicate the triangle’s side lengths, whether they are equal or if there are none if the exact length is not specified. It is a scalene triangle because no two sides of the triangle have an equal number of hatch marks. The right triangle uses numbers to clearly represent the triangle’s side lengths.

*The Angles of a scalene triangle are not equal.*

In a scalene triangle, each angle is not equal. As shown in the illustration below, angles A, B, and C do not have the same measurement.

*Scalene Triangles have rotational symmetry of order 1.*

If a shape is rotated and still retains its original appearance, this is known as rotational symmetry. When a shape is rotated at a full turn, the number of times its original shape fits within the shape is the order of rotational symmetry.

The diagram below shows an example of a scalene triangle rotated at different angles 90 degrees, 180 degrees, 270 degrees and 360 degrees. Since a scalene triangle fits onto its original shape when rotated 360 degrees or fully, it has rotational symmetry of order 1.

*Scalene triangles have no line of symmetry.*

A triangle’s symmetry line must pass through just one of its vertices. Only if the lengths of the two sides that meet at that vertex are equal can there be a line of symmetry. The line of symmetry at a vertex that connects two sides when they do have the same length also travels through the middle of the opposite side. As shown below, an equilateral triangle has 3 lines of symmetry, an isosceles triangle has 1 line of symmetry, while a scalene triangle has no line of symmetry.

A line of symmetry cannot exist in a scalene triangle because its sides are not equal. Furthermore, it cannot be split into two identical pieces.

*The greatest angle would be the angle that is opposite the longest side and vice versa.*

The greatest measurement in a triangle is the angle across from the longest side and vice versa. As shown below, the longest side AC has the measure of 9 cm, and the angle opposite to it measures 87 degrees which is the greatest among the three angles of the scalene triangle.

*The smallest angle would be the angle that is opposite the shortest side and vice versa.*

The smallest measurement in a triangle is the angle across from the smallest side and vice versa. As shown below, the shortest side BC has the measure of 5 cm, and the angle opposite to it measures 33 degrees which is the smallest among the three angles of the scalene triangle.

**Types of Scalene Triangles **

The three types of scalene triangles are as follows:

**Acute Scalene Triangle**

A triangle with angles that are less than 90 degrees and different measurements for each of its three sides and angles is known as an acute scalene triangle. The image below is an example of an acute scalene triangle whose angles are 40 degrees, 60 degrees and 80 degrees.

**Obtuse scalene triangle**

If one of a triangle’s angles is greater than 90 degrees but less than 180 degrees, and the other two angles are both less than 90 degrees, the triangle is said to be an obtuse scalene triangle. Simply put, the other two angles are acute, whereas the one is obtuse. All three sides and angles have different measurements.

In the figure below, angle A is an obtuse angle since its measure is greater than 90 degrees but less than 180 degrees. Angles B and C are both acute angles whose measures are both less than 90 degrees.

**Right scalene triangle**

A right scalene triangle combines the characteristics of both a right angle and a scalene triangle, where one angle is 90 degrees, the other two are less than 90 degrees, and all of the sides are of varying lengths.

An illustration of a right scalene triangle is shown below. Angle B and C are both acute angles that are less than 90 degrees, but Angle A is a right angle that is exactly 90 degrees. The sides have a different measure, as shown by the hatch markings “|”.

The comparison of the three varieties of scalene triangles according to their sides and angles is shown in the table below:

Types of Scalene Triangles | Sides | Angles |

Acute Scalene Triangle | The sides of an acute scalene triangle are different in measure. | An acute scalene triangle has acute angles of various measurement, adding up to 180 degrees in total. |

Obtuse Scalene Triangle | The sides of obtuse scalene triangles are different in measure. | An obtuse scalene triangle has two other angles that are both less than 90 degrees and a third angle that is more than 90 degrees but less than 180 degrees. The sum of the three angles is equal to 180 degrees. |

Right Scalene Triangle | The sides of a right scalene triangle are different in measure. | One of the angles in a right scalene triangle is 90 degrees, while the other two are less than 90 degrees. These angles add up to a total of 180 degrees. |

**How do you determine the perimeter of a scalene triangle?**

The total distance surrounding a triangle or its boundary is known as its perimeter. The formula for calculating a scalene triangle’s perimeter uses the formula for calculating any triangle’s perimeter, which is the total of all the triangles’ side lengths.

P=a+b+c

where P is the perimeter, and a, b, and c are the side lengths

**Example 1**

Find the perimeter of a scalene triangle whose sides are 6, 8, and 9 cm in length.

*Solution:*

Let a=6 cm , b=8 cm and c=9 cm.

Using the formula in getting the perimeter of a triangle, we have,

Perimeter=a+b+c

Perimeter=6 cm+8 cm+9 cm

Perimeter=23 cm

Hence, the perimeter of the triangle is 23 cm.

**Example 2**

In reference to the figure below, calculate its perimeter.

*Solution:*

Let us calculate the perimeter of ∆ABC by getting the sum of its side lengths. We have,

Perimeter=12 in+15 in+20 in

Perimeter=47 in

Therefore, the perimeter of ∆ABC is 47 inches.

**How do you calculate a scalene triangle’s area?**

The space occupied by a flat surface inside a scalene triangle is known as its area. We may use specific formulas in calculating a scalene triangle’s area depending on the given measurements of the triangle.

**Base and Height of a Scalene Triangle are Known **

When the base and height of a scalene triangle are known, the formula for getting its area is given by

Area of a Triangle=$\frac{1}{2}$×base×height

In the figure above, the height is the perpendicular distance from the base to the opposite vertex and notice that it forms a right angle with the base.

**Example 1**

Calculate the area of a scalene triangle with a base of 10 units and a height of 5 units.

*Solution:*

Let us use the formula, Area=½×base×h, to calculate the area of the given triangle.

Area=½×10 units×5 units

Area=½×50 units^{2}

Area=25 units^{2}

Therefore, the area of the scalene triangle is 25 square units.

**Example 2**

With a base of 12 cm and a height of 7 cm, calculate the area of the scalene triangle.

*Solution:*

Let us substitute the given values to the formula for finding the area of a triangle. We have,

Area=½×base×height

Area=½ ( 12 cm )( 7 cm )

Area=½ ( 84 cm^{2})

Area=$\frac{84}{2}$ cm^{2}

Area=42 cm^{2}

Therefore, the area of the scalene triangle is 42 square centimetres.

**All Three sides of a scalene triangle are Known (using Heron’s Formula)**

Given by is Heron’s formula for calculating the area of a scalene triangle.

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

where the variable S is the semi-perimeter, and a,b, and c are the side lengths

The semi-perimeter S is given by the formula S=$\frac{a+b+c}{2}$.

**Example 1**

Calculate the area of a scalene triangle with sides that are 80, 92, and 120 cm long.

*Solution:*

Let us find the semi perimeter of the scalene triangle by using the formula S=$\frac{a+b+c}{2}$.. Hence, we have,

S=$\frac{80+92+120}{2}$

S=$\frac{292}{2}$

S=146 cm.

Since we already know that the semi-perimeter of the given scalene triangle is 146 cm., let us now use Heron’s formula to find its area.

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

Area=$\sqrt{146 (146-80)(146-92)(146-120)}$

Area=$\sqrt{146 (66)(54)(26)}$

Area=$\sqrt{13528944}$

Area≈3678.17 cm^{2}

Therefore, the area of the given scalene triangle is 3678.17 square centimetres.

**Example 2**

Calculate the area of a scalene triangle with side lengths 3 cm, 6 cm, and 7 cm.

*Solution:*

Let us find the semi perimeter of the scalene triangle by using the formula S=$\frac{a+b+c}{2}$. Hence, we have,

S=$\frac{3+6+7}{2}$

S=$\frac{16}{2}$

S=8 cm.

Let us now have Heron’s formula to calculate the area of the given scalene triangle. Substituting the computed value of the triangle’s semi-perimeter, we have,

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

Area=$\sqrt{8 (8-3)(8-6)(8-7)}$

Area=$\sqrt{8 (5)(2)(1)}$

Area=$\sqrt{80}$

Area=4$\sqrt{5}$ cm2

Therefore, the area of the given scalene triangle is 4$\sqrt{5}$ square centimetres.

**How to Identify Scalene Triangles Using Distance Formula?**

Let us say that we are asked to determine whether a scalene triangle is formed by the three provided coordinate points. Since a scalene triangle has unequal sides, we must obtain different measures when comparing the distances of each side of the triangle using the distance formula. We will use the distance formula d=$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ to determine the distance between two points (x_{1},y_{1}) and (x_{2},y_{2}).

**Example 1**

Show that the triangle with the vertices, L (0,1), M (2, 3 ), and N (4, -5) is scalene.

*Solution:*

The figure below shows the three points when graphing and forming a triangle.

We must apply the distance formula to determine how far apart the three sides LM, MN, and LN are, as the objective is to determine whether the three points form a scalene triangle.

*Distance of side LM*

Let us have the coordinates of point L (0,1) be (x_{1}, y_{1}) while the coordinates of point M 2, 3 be (x_{2},y_{2}). Using the distance formula, we have,

LM=$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

LM=$\sqrt{(2-0)^2+(3-1)^2}$

LM=$\sqrt{2^2+2^2}$

LM=$\sqrt{4+4}$

LM=$\sqrt{8}$

LM=2$\sqrt{2}$

*Distance of side MN*

Let us use the coordinates of point M (2,3) as (x_{1}, y_{1}), and the coordinate of point N 4, -5 be (x_{2},y_{2}). Using the distance formula, we have,

MN=$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

MN=$\sqrt{(4-2)^2+(-5-3)^2}$

MN=$\sqrt{2^2+(-8)^2}$

MN=$\sqrt{4+64}$

MN=$\sqrt{68}$

MN=2$\sqrt{17}$

*Distance of side LN*

Let us use the coordinates of point L (0,1) as (x_{1}, y_{1}), and the coordinate of point N 4, -5 be (x_{2},y_{2}). Using the distance formula, we have,

LN=$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

LN=$\sqrt{(4-0)^2+(-5-1)^2}$

LN=$\sqrt{4^2+(-6)^2}$

LN=$\sqrt{16+36}$

LN=$\sqrt{52}$

LN=2$\sqrt{13}$

∆LMN is a scalene triangle since the lengths of the sides of the triangle are 2$\sqrt{2}$, 2$\sqrt{17}$, and 2$\sqrt{13}$, which are all different measures.

**Example 2**

Using the distance formula, show that the given triangle below is scalene.

*Solution:*

The coordinates of the vertices are P 6, 7, Q 7, -1 and R (2, 5 ). To show that the triangle is scalene, we must obtain different lengths of the three sides, PR, PQ, and QR, using the distance formula.

*Distance of side PR*

Let us have the coordinates of point P (6,7) be (x_{1}, y_{1}) while the coordinates of point R 2, 5 be (x_{2},y_{2}). Using the distance formula, we have,

PR=$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

PR=$\sqrt{(2-6)^2+(5-7)^2}$

PR=$\sqrt{(-4)^2+(-2)^2}$

PR=$\sqrt{16+4}$

PR=$\sqrt{20}$

PR=$2\sqrt{5}$

*Distance of side PQ*

Let us use the coordinates of point P (6,7) as (x_{1}, y_{1}), and the coordinate of point Q 7, -1 be (x_{2},y_{2}). Using the distance formula, we have,

PQ=$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

PQ=$\sqrt{(7-6)^2+(-1-7)^2}$

PQ=$\sqrt{(-1)^2+(-8)^2}$

PQ=$\sqrt{1+64}$

PQ=$\sqrt{65}$

*Distance of side QR*

Let us use the coordinates of point Q (7, -1) as (x_{1}, y_{1}), and the coordinate of point R 2, 5 be (x_{2},y_{2}). Using the distance formula, we have,

QR=$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

QR=$\sqrt{(2-7)^2+(5-(-1))^2}$

QR=$\sqrt{(2-7)^2+(5+1)^2}$

QR=$\sqrt{(-5)^2+(6)^2}$

QR=$\sqrt{25+36}$

QR=$\sqrt{61}$

∆QPR is a scalene triangle since the lengths of the sides of the triangle are $2\sqrt{5}$, $\sqrt{65}$, and $\sqrt{61}$, which are all different measures.

**Summary**

**Definition**

A triangle with three different side lengths and three different angle measurements is called a scalene triangle. The total of all interior angles of a scalene triangle is always equal to 180 degrees.

We may use a ruler or a protractor to measure a scalene triangle’s side lengths and angle.

**Properties of Scalene Triangles**

The following are the basic properties of scalene triangles:

A scalene triangle has no equal sides.

The Angles of a scalene triangle are not equal.

Scalene Triangles have rotational symmetry of order 1.

Scalene triangles have no line of symmetry.

The greatest angle would be the angle that is opposite the longest side and vice versa.

The smallest angle would be the angle that is opposite the shortest side and vice versa.

**Types of Scalene Triangles**

These are the types of scalene triangles:

Acute scalene triangle

Obtuse Scalene triangle

Right Scalene Triangle

**Formulas **

P = a + b + c

where a,b, and c are the side lengths of a triangle.

Semi-Perimeter of a Triangle

S=$\frac{a+b+c}{2}$

Area of a Triangle

Area=$\frac{1}{2}$×base×height

Heron’s Formula

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

**Frequently Asked Questions on Scalene Triangles**

**How do equilateral, isosceles, and scalene triangles differ from one another?**

The basic difference between the three triangles, equilateral, isosceles, and scalene, is presented below. The information shows a comparison of their angles and sides.

Type of Tringle | Angles | Sides |

Equilateral Triangle | An equilateral triangle has angles that are each 60 degrees | in measure.An equilateral triangle has three equal sides. |

Isosceles Triangle | In an isosceles triangle, the angles that face equal sides are also equal. | Any two sides of an isosceles triangle are equal. |

Scalene Triangle | All three angles of a scalene triangle are unequal. | All three sides of a scalene triangle are unequal. |

**How to identify a scalene triangle?**

We need to be familiar with a scalene triangle’s properties in order to recognize it. The following is a list of scalene triangles’ properties.

( a ) A scalene triangle has no equal sides.

( b ) Angles of a scalene triangle are not equal.

( c ) Scalene Triangles have rotational symmetry of order 1.

( d ) Scalene triangles have no line of symmetry.

( e ) The greatest angle would be the angle that is opposite the longest side and vice versa.

( f ) The smallest angle would be the angle that is opposite the shortest side and vice versa.

**What are the types of scalene triangles?**

The table below shows the three types of scalene triangles and their characteristics.

Acute Scalene Triangle | Obtuse scalene triangle | Right scalene triangle |

A triangle with angles that are less than 90 degrees and different measurements for each of its three sides and angles is known as an acute scalene triangle. | If one of a triangle’s angles is greater than 90 degrees but less than 180 degrees, and the other two angles are both less than 90 degrees, the triangle is said to be an obtuse scalene triangle. | A right scalene triangle combines the characteristics of both a right angle and a scalene triangle, where one angle is 90 degrees, the other two are less than 90 degrees, and all of the sides are of varying lengths. |

**What are the different classifications of triangles?**

Based on the angle and side length measurements, there are various types of triangles. If we classify triangles according to the interior sides, the types are acute triangle, right triangle, and obtuse triangle. If triangles are classified according to the side lengths, we have equilateral, isosceles, and scalene.

The table below shows the types of triangles according to the interior sides.

Types of Triangles | Angles |

Acute Triangle | When all three of the triangle’s internal angles are acute, a triangle is said to be an acute triangle. An acute angle is one with a measurement between 0° and 90°. |

Right Triangle | If one of a triangle’s angles is 90 degrees, the triangle is said to be a right triangle. |

Obtuse Triangle | When one of the triangle’s internal angles is obtuse, a triangle is said to be an obtuse triangle. Obtuse angles are those that are greater than 90 degrees. |

The table below shows the type of triangle according to the side lengths.

Types of Triangles | Angles | Sides |

Equilateral Triangle | An equilateral triangle has angles that are each 60 degrees in measure. | An equilateral triangle has three equal sides. |

Isosceles Triangle | In an isosceles triangle, the angles that face equal sides are also equal. | Any two sides of an isosceles triangle are equal. |

Scalene Triangle | All three angles of a scalene triangle are unequal. | All three sides of a scalene triangle are unequal. |

**What are examples of side lengths that form a scalene triangle?**

The following are examples of side lengths that will form a scalene triangle.

(a) 10 , 12 , 15

(b) 8 , 4, 7

(c) 11, 17, 19

(d) 21, 18, 12

(e) 13, 17, 10

While the side lengths of a scalene triangle differ, we must also be aware of the Triangle Inequality Theorem, which stipulates that the total of any two sides of a triangle is higher than the length of the third side, in order to determine whether the provided side lengths create a triangle.

Let us say we have side lengths a, b, and c; the following inequalities hold:

a+b>c

a+c>b

b+c>a

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