The **perimeter of a scalene triangle** is the sum of all three sides of the triangle. A triangle is a closed 2D figure with three sides and three angles along with three vertices. In geometry, there are three different types of triangles that are classified according to their own properties. An equilateral triangle is a triangle where all three sides are equal in length and all three angles equal to 60°. A triangle with two sides equal in length and angles opposite to equal sides are equal is known as Isosceles Triangle. A scalene triangle consists of three unequal sides along with three different angles. Let's learn more about the perimeter of a scalene triangle in this article.

1. | What is the Perimeter of a Scalene Triangle |

2. | Properties of Scalene Triangle |

3. | Perimeter of a Scalene Triangle Formula |

4. | Perimeter of a Scalene Triangle Calculation |

5. | Laws of Cosines in Perimeter of Scalene Triangle |

6. | FAQs on Perimeter of Scalene Triangle |

## What is the Perimeter of a Scalene Triangle?

A scalene triangle is a triangle with all three sides different in lengths, and all three angles different in measurement. The angles in a scalene triangle can be acute, right, or obtuse with the sum of the interior angles always equal to 180°. Most triangles drawn are scalene triangles as the lengths of all three sides of this triangle are not equal.

The perimeter of a scalene triangle is the distance around the triangle i.e. the sum of the lengths of the scalene triangle. Therefore, to calculate the perimeter of a scalene triangle we require the length of all three sides and can be calculated using this formula:

**Perimeter of a Scalene Triangle = a + b + c**

Below is an image of a scalene triangle where a, b, and c represent the three sides of a scalene triangle and the three interior angles are not equal to each other.

## Properties of Scalene Triangle

One of the most important properties of a scalene triangle is that it has no line of symmetry, hence the triangle cannot be divided into two halves and the triangle has no point of symmetry. Other properties of the scalene triangle are listed below.

- A scalene triangle has no equal sides and no equal interior angles in measurement.
- All the three interior angles of a scalene triangle can be acute, obtuse, or right angles.
- If all the interior angles of the scalene triangle are less than 90° then it is an acute scalene triangle, then the center of the circumscribing circle will lie inside a triangle.
- The circumcentre lies outside an obtuse scalene triangle.

## Perimeter of a Scalene Triangle Formula

The formula for the perimeter of a scalene triangle is equal to the sum of the length of sides of a triangle and it is given as:

The Perimeter of a Scalene Triangle = a + b + c units.

## Perimeter of a Scalene Triangle Calculation

The perimeter of a scalene triangle is equal to the sum of the length of sides of a triangle and it is given as, Perimeter = a + b + c units where a, b, and c are the lengths of the sides. For example, Look at the below image of a scalene triangle where the sides are a = 6 units, b = 15 units, and c = 20 units. What is the perimeter of the scalene triangle?

The formula to find the perimeter of the scalene triangle is P = a + b + c

P = 6 + 15 + 20

P = 41 units

Therefore, the perimeter of the scalene triangle is 41 units.

## Laws of Cosines in Perimeter of Scalene Triangle

The laws of cosines also known as cosines rule is used for all triangles and especially for finding two aspects:

- Finding the third side of a triangle when the length of two sides and one angle is mentioned in a triangle.
- Finding the angles of a triangle when the length of all the three sides is mentioned.

The law of cosines says c^{2} = a^{2} + b^{2} - 2ab cos(c), where c is the angle opposite to the third side. This rule is combined with the Pythagoras Theorem, as the Pythagoras theorem is used in all right-angled triangles whereas the law of cosines is used for all the triangles.

**For example:** Given a scalene triangle ABC, with BC = 9 units, AC = 12 units, and angle c = 39°. Find the length of the third side of the scalene triangle using the law of cosines.

**Solution:** The law of cosines says: c^{2} = a^{2} + b^{2} - 2ab cos(c)

Put the values i.e. c^{2} = 9^{2} + 12^{2} - 2 × 9 × 12 cos(39°)

c^{2} = 81 + 144 - 216 × 0.777

c^{2} = 225 - 167.832

c^{2} = 57.168

c = √57.168

c = 7.56 units

**Important Notes on Perimeter of Scalene Triangle**

- Perimeter of a Scalene Triangle = a + b + c
- A scalene triangle has no equal sides and no equal interior angles in measurement.

**Related Topics to Perimeter of Scalene Triangle **

Listed below are a few related topics to the perimeter of a scalene triangle. Click to know more!

- Triangles
- Angles
- Symmetry
- Perimeter of a Triangle
- Area of Triangle
- Circumscribe
- Types of Triangle

## FAQs on Perimeter of Scalene Triangle

### What is the Perimeter of a Scalene Triangle?

A scalene triangle is a triangle with all three sides different in lengths, and all three angles different in measurement. The angles in a scalene triangle can be acute, right, or obtuse with the sum of the interior angles as 180°. To find the perimeter of a scalene triangle, we need to use a formula. Perimeter = a + b + c where a, b, and c are the lengths of the sides.

### What are the Properties of a Scalene Triangle?

The properties of a scalene triangle are:

- If all the interior angles of the scalene triangle are less than 90° i.e. it is an acute angle triangle, then the center of the circumscribing circle will lie inside a triangle.
- A scalene triangle has no equal sides and no equal interior angles in measurement.
- The circumcentre lies outside a scalene obtuse triangle.
- All the three interior angles of a scalene triangle can be acute, obtuse, or right angles.

### What is a Right-Scalene Triangle?

When one of the three angles measure 90° and the angles or lengths of the other two sides are not equal, then the scalene triangle is called the right scalene triangle.

### What is the Formula for the Perimeter of a Scalene Triangle?

The formula for the perimeter of a scalene triangle is equal to the sum of the length of sides of a triangle and it is given as: The perimeter of a Scalene Triangle = a + b + c units.

### How to Find the Perimeter of a Scalene Triangle with One Side Missing?

To find the missing length of one side of a scalene triangle, we also require the measure of a non-right angled angle. Once these measurements are mentioned, we can use the las of cosines to find out the missing side of a scalene triangle. The formula used is:

The law of cosines says c^{2} = a^{2} + b^{2} - 2ab cos(c), where c is the angle opposite to the third side.

As an expert in mathematics and geometry, I can confidently discuss the concepts presented in the article about the perimeter of a scalene triangle. My understanding is rooted in the fundamental principles of geometry and trigonometry, which are essential branches of mathematics that deal with shapes, sizes, and properties of figures in space.

### 1. What is the Perimeter of a Scalene Triangle?

The perimeter of any polygon, including a scalene triangle, refers to the total distance around its boundary. For a scalene triangle specifically, since all three sides have different lengths ( a ), ( b ), and ( c ), the perimeter is calculated as: [ \text{Perimeter of a Scalene Triangle} = a + b + c ]

### 2. Properties of Scalene Triangle

**No Symmetry**: A scalene triangle lacks any line of symmetry or point of symmetry, making it asymmetrical.**Unequal Sides and Angles**: All three sides and angles are of different lengths and measures.**Angles**: The angles can be acute, obtuse, or right angles, but none of them are equal.

### 3. Perimeter of a Scalene Triangle Formula

The formula is straightforward: [ \text{Perimeter} = a + b + c ] Here, ( a ), ( b ), and ( c ) are the lengths of the three sides of the scalene triangle.

### 4. Perimeter of a Scalene Triangle Calculation

Using the formula, if you know the lengths of the sides, you can sum them up to find the perimeter. For instance, if ( a = 6 ) units, ( b = 15 ) units, and ( c = 20 ) units: [ \text{Perimeter} = 6 + 15 + 20 = 41 \text{ units} ]

### 5. Laws of Cosines in Perimeter of Scalene Triangle

The Law of Cosines is a vital tool for solving triangles, especially when you have two sides and an included angle or all three sides. The formula is: [ c^2 = a^2 + b^2 - 2ab \cos(C) ] Here, ( C ) is the angle opposite to side ( c ). This law can help find the length of a side or an angle in any triangle, including scalene triangles.

### 6. FAQs on Perimeter of Scalene Triangle

**Right-Scalene Triangle**: When one angle of a scalene triangle measures ( 90° ), it's termed a right-scalene triangle.**Missing Side**: To determine a missing side of a scalene triangle, use the Law of Cosines when provided with the lengths of two sides and the included angle.

In summary, understanding the perimeter of a scalene triangle requires a grasp of its unique properties and the fundamental formulas and laws of geometry. Whether you're dealing with calculations or properties, a thorough understanding of these concepts ensures accuracy and clarity in geometric applications.