Contents

# Included Angle|Definition & Meaning

## Definition

The **angle** created at the vertices of two **neighboring** triangle legs is an included angle. If two legs meet at a **vertex,** they are **considered** adjacent. This angle’s **measurement** can be **utilized** to determine the **triangle’s** other **characteristics** and values.

An **included** angle is a type of angle that can be produced in **geometry** by using 2 lines that **meet** at the **same** location.Â

Figure 1 – The included angle of a triangle

The **endpoint** that is shared is **referred** to as the **vertex,** as well as the two lines that make up the **angle** are referred to as the sides of that same **angle.** Calculated in degrees as well as radians, with 360 **degrees** as well as 2 **radians** corresponding to a full circle, an angle can be expressed using either system.

The **included angle** is frequently used to define the angle formed by **two planes** within **three-dimensional** space or by two lines in a plane. In addition to that, you can use it to talk about the angle that **exists** only **between** sides of just a **polygon** or even the **angle** that exists among the faces of such a **polyhedron.**

In geometry, one of the most significant concepts is called the included angle, and its purpose is to **define** the **shape** and **size** of **different** figures and objects. In addition to this, it is utilized in a variety of **scientific** and **engineering** applications, as well as in the field of **trigonometry** and other sub fields of **mathematics.**

## What Is Included Angle and Included Side?

A **common** leg **between** 2 angles is known as an **included side.** In **contrast** to the **included angle,** this will be **expressed** as a length in **units** of **measure** like **inches** or **centimeters.**

The **included side** joins two **angles,** whereas the **included angle includes** two **neighboring** legs. The **opposite** component is on either side of the included angles and **legs, which** are in the midst of the **three measurements.**

## Examples of Included Angles

An **angle** that **isn’t included** in the **definition** of the leg is called a **non-included** angle. This **means** that any one of the **legs** that are next to each other does not **possess** a value. Both of the **legs** that are **next** to one **another** need to have a value **specified** for the angle to be **regarded** as an **included angle.**

In **order** for the angle to be **considered** part of the **triangle,** it must be located between the two **sides** of the **triangle.** If the angle is just **adjacent** to one leg, then it is **omitted** from **consideration.**

Figure 2 – Right angle triangle showing the included angle

The **following** two **lines** showing the **included** angle.

Figure 3 – Two lines showing the included angle

## Uses of the Included Angle

**Included angles** are used in **several** applications.

Before **going** through them, know that just **like** the angle **formed** by two **lines** or planes that **meet constitutes** the included angle, the **angle** formed by the intersection of 2 faces of a **solid object** is also the same thing. It can also be known as the inner angle or even the **“angle between.” Both** of these names refer to the same thing.

- When
**defining**the**angle**that exists between two lines in geometry that meet at a point, an included angle is**frequently**the method of choice. - When
**determining**the length of either side of such a triangle in trigonometry, an included angle is one of the three angles that must be considered together with the length of the**remaining**2**sides**& the**included**angle. - When it comes to
**engineering**and**building design,**the included angle is one of the most important factors**considered**when evaluating the robustness and steadiness of structures like**bridges, buildings,**and**machinery.**For instance, the**load-carrying**capability of a truss construction is affected by the angle that is**included between**two of the**beams**that make up the structure.

## A Numerical Example of the Included Angle

### Example 1

**Determine** the range of **values** that can be assigned to an included angle of a triangle whose side lengths are 5 **centimeters** and 8 **centimeters** and whose area is **15** **centimeters** square.

### Solution

**Given that:**

**The side length of the triangle is = 5 cm**

**The side length of the triangle is = 8 cm**

**The area of a triangle is = 15 cm ^{2}**

We have to **find** the **included** angle.

So, if the required included angle is **Î¸**, then:

**0.5 x 5 x 8 x sin(Î¸) = 15**

By simplifying, we get:

**sin(Î¸) = 15 / 20**

By calculating the sin inverse, we get:

**= 48.6 degrees**

The included angle for the given values is **48.6 degrees**.

### Example 2

Determine the range of **values** that can be assigned to an included angle of a triangle whose side lengths are 5 **centimeters** and 10 **centimeters** and whose area is 15 centimeters square.

### Solution

**Given** that:

**The side length of the triangle is = 5 cm**

**The side length of the triangle is = 10 cm**

**The area of a triangle is = 15 cm ^{2}**

We **have** to find the included angle.

So, if the **required included** angle is Î¸, then:

**0.5 x 5 x 10 x sin(Î¸) = 15**

By **simplifying,** we get:

**sin(Î¸) = 15 / 25**

By **calculating** the sin inverse, we get:

**= 36.86 degree**

The included angle for the given values is **36.86 degrees**.

### Example 3

**Determine** the range of values that can be assigned to an included angle of a triangle whose side lengths are 5 **centimeters** and 20 **centimeters** and whose area is 15 **centimeters** square.

### Solution

**Given** that :

**The side length of the triangle is = 5 cm**

**The side length of the triangle is = 20 cm**

**The area of a triangle is = 15 cm ^{2}**

We **have** to find the included angle.

So, if the **required** included angle is Î¸, then:

**0.5 x 5 x 20 x sin(Î¸)Â = 15**

By **simplifying,** we get:

**sin(Î¸) = 15 / 50**

By calculating the sin inverse, we get:

**= 17.45 degree**

The **included** angle for the given values is** 17.45 degrees**.

*All mathematical drawings and images were created withÂ GeoGebra.*