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How to find the measure of an **angle** is an essential skill in various fields, ranging from **mathematics** and **engineering** to **architecture** and **navigation**.

Whether you’re a **student**, a **professional**, or simply an **enthusiast** eager to delve into the world of angles, this comprehensive guide will equip you with the knowledge and techniques to confidently determine the measure of any angle.

From **fundamental concepts** to practical methods, we will unravel the mysteries of **angle measurement**, empowering you to unlock new dimensions in **problem-solving**, **precision**, and **geometric analysis**. Join us on this enlightening journey as we demystify angles, explore **measurement tools**, and unravel the secrets to accurately determining the measure of an angle.

**How to Find Measure of an Angle**

To **find the measure of an angle**, use a protractor placing its midpoint on the angle’s vertex. Read the scale where the angle’s sides intersect the protractor’s edge.

The **measure** of an **angle** refers to the degree of** rotation** between two **intersecting lines**, **rays**, or** line segments**, typically measured in **degrees (°)**.

Determining the **measure of an angle** is a process of quantifying the amount of** rotation** or** deviation** between these g**eometric elements.** It allows us to express the **angular relationship** between objects, understand their **spatial positioning**, and solve various mathematical and practical problems.

By employing **mathematical principles**, **measurement tools**, and **geometric concepts**, we can accurately calculate the measure of an angle, enabling us to analyze shapes, construct diagrams, and make informed decisions based on **angular information**.

**Types of Angles**

### Acute angles

#### Definition

**Acute angles** are angles that measure less than **90 degrees**.

#### Explanation

**Acute angles** are characterized by their **sharpness** and **compactness**. They are smaller than a **right angle** and do not extend beyond it.

#### Examples

An angle of **30 degrees,** **45 degrees**, or **60 degrees** is considered** acute**. Examples of real-life acute angles include the corners of a **triangle**, the hands on a clock at certain times, or the **slope** of a **steep hill**.

### Obtuse angles

#### Definition

**Obtuse angles** are angles that measure between **90** and **180 degrees**.

#### Explanation

**Obtuse angles** are wider than a **right angles**. They have a greater degree of **openness** and appear more **spacious** than **acute angles**.

#### Examples

An angle of **100 degrees**, **120 degrees**, or **150 degrees** is considered obtuse. Examples of real-life obtuse angles include the opening of a wide **V-shape**, the angle between the hands on a clock at certain times, or the angle formed by a** ladder** leaning against a **wall**.

### Right angles

#### Definition

**Right angles** are angles that measure exactly **90 degrees**.

#### Explanation

**Right angles** are characterized by their **L-shape**, forming a perfect quarter turn. They are often associated with **perpendicular** lines and provide a basis for many **geometric** **constructions** and **calculations**.

#### Examples

The **angle** formed by the corner of a square, the intersection of the edges of a **cube**, or the meeting of two **perpendicular** lines are all examples of right angles. **Right angles** are commonly used in **architectural drawings**, **engineering designs,** and **navigation**.

### Straight angles

#### Definition

**Straight angles** are angles that measure exactly **180 degrees**.

#### Explanation

**Straight angles** are the largest possible angles, forming a straight line. They are completely flat and lack any **curvature** or **bending**.

#### Examples

A **straight angle** can be visualized as a **line segment** or a ray that extends in a straight path without any **deviation**. It represents a full turn or revolution around a point. Straight angles are commonly **encountered** in **geometry**, **trigonometry**, and **circular motion concepts**.

### Reflex angles

#### Definition

**Reflex angles** are angles that measure between **180** and **360** **degrees**.

#### Explanation

**Reflex angles** exceed the measurement of a** straight angle** and continue beyond it. They extend in a** clockwise direction** from the initial position of a straight angle.

#### Examples

An angle of **200 degrees**,** 270 degrees**, or **320 degrees** is considered reflex. Examples of real-life reflex angles include the angle formed by the hands on a **clock pointing** beyond the 6 o’clock mark or the angle between the blades of a **rotating windmill**.

### Full circle

#### Definition

A **full circle** is an angle that measures **360 degrees**.

#### Explanation

A **full circle** represents a complete revolution or a complete** trip around** a central point. It is comprised of infinite points or **angles**, all of which sum up to **360 degrees**.

#### Examples

A **clock’s dial**, a compass, or a circular track are examples of situations where a full circle is encountered. **Full circles** are fundamental in **trigonometry**, **navigation**, and **circular geometry**, and they are used to calculate **bearings**, **directions**, and **rotations**.

Below we present all the aforementioned angles in figure-1.

Figure-1.

## Common Angle Measurement Techniques

The measure of an **angle** is typically measured in **degrees** or **radians**, and it tells you how much rotation is required to get from one of the angle’s **rays** (or **sides**) to the other. Here’s a simple way to measure an angle:

**Use a Protractor**

A **protractor** is a **semi-circular** tool often used in **geometry** to measure angles. It has degree marks from **0** to **180**.

Steps to using a protractor:

**Position**the**center**point of the**protractor**at the**vertex**(the point of intersection) of the two sides of the angle.**Align**the zero line of the**protractor**with one side of the angle.- Read the value of the
**protractor**where the second side of the angle**intersects**the protractor. This is the measure of the angle.

**Use Trigonometry**

If you have a **right triangle**, you can use the **sides** of the triangle to compute the measures of the **non-right angles**.

For example, the **tangent** of an angle is equal to the length of the **side opposite** the angle divided by the length of the **side adjacent** to the angle. So if you know the lengths of these sides, you can use the **arctangent function** to find the **measure of the angle**.

**Use Geometric Properties**

Some **angles** have measures that can be determined from their **geometric** properties.

- For example, the angles in a
**triangle**always add up to**180 degrees**, so if you know the measures of two of the**angles**, you can subtract their sum from**180**to find the measure of the third**angle**. - As another example,
**vertical angles**(angles across from each other when two lines intersect) are always**equal**.

- For example, the angles in a

**Applications **

**Measurements** of **angles** are used in many fields and contexts. Here are some of them:

**Geometry**

**Geometry** is almost entirely about the properties of **shapes**, many of which involve **angles**. Angles are used to **classify shapes** (like triangles and polygons), to find missing sides or angles in triangles, and to **prove relationships** between shapes.

**Trigonometry**

The branch of mathematics that deals with the relationships between the sides and angles of triangles is called **trigonometry**. Trigonometric functions like **sine**, **cosine**, and **tangent** are ratios that relate the angles of a right triangle to the lengths of its sides. These functions have many applications in other fields as well.

**Physics**

Many aspects of **physics**, such as the study of forces and motion, use **angles**. For example, angles are used to calculate the components of **vectors**, to study **pendulum motion**, and to analyze the behavior of **light** and **sound waves**.

**Astronomy**

The position of **celestial bodies** in the sky is defined using **angles** – typically **declination** (angle from the celestial equator) and **right ascension** (angle along the celestial equator).

Additionally, the principle of **parallax**, which is used to measure the distance of nearby stars, is based on observing the **apparent shift** of a star from two different points of **Earth’s orbit** around the sun. The angle of this **apparent shift** gives a triangle with a known base and allows **astronomers** to calculate the distance to the **star**.

**Navigation**

Traditionally, **sailors** would use a **sextant** to measure the angle between the horizon and a** celestial body** (like the Sun at noon or a star) to find their **latitude**. Nowadays, in modern GPS technology, the receiver calculates the distance to satellites based on **signal travel time**.

Knowing these **distances** and the position of the** satellites** in space, the receiver can calculate its own position, essentially by **triangulation**. This is a form of **angle measurement** in three dimensions.

**Architecture and Engineering**

In these fields, the principles of **geometry** and **trigonometry** are applied to design and construct **buildings**, **bridges**, and other **structures**. For example, the **angles** between beams or supports can affect the **load-bearing capacity** of a structure. Also, angles are crucial in designing **roof slopes**, **stair inclines**, or any structure or part that is not perpendicular or parallel to others.

**Computer Graphics and Game Design**

Angles are used in **3D modeling** and **rendering**, where the geometry of a scene is defined in terms of points (or vertices) in three-dimensional space and their connections (forming polygons like triangles). **Rotation**, **scaling**, and **translation** of objects are transformations that involve angle measurements.

In **video games**, physical simulations like object collisions, reflections, lighting, and shadows all involve computations with angles.

**Geography and Cartography**

Angles are used in the field of **geography** to study the Earth’s physical features and climate. In **cartography**, angles help in making accurate maps and determining distances between locations.

**Sports**

Many **sports** involve some understanding of angles, whether implicitly or explicitly. For example, in **billiards** or **pool**, players must think about the angles at which to hit the balls to make them go where they want.

**Art**

**Artists** use angles to create perspective and depth in their work, especially in **drawing** and **painting.**

**Exercise **

**Example 1**

In a right triangle, one of the angles is **45 degrees**. What is the measure of the other **non-right angle**?

**Solution**

In a triangle, the angles add up to **180 degrees**. In a right triangle, one of the angles is **90 degrees**. So, the other two angles must add up to **90 degrees** (because 180 – 90 = 90). If one of these angles is **45 degrees**, then the other must also be **45 degrees** (because 90 – 45 = 45).

Figure-2.

**Example 2**

You have an angle that measures** 35 degrees**. What would be the measure of its** supplementary angle**?

**Solution**

Supplementary angles are two angles that add up to **180 degrees**. So, to find the measure of an angle that is supplementary to a **35-degree angle**, subtract **35** from** 180**. The result is **145 degrees**.

**Example 3**

You have an angle that measures **80 degrees**. What would be the measure of its **complementary angle**?

**Solution**

**Complementary angles** are two angles that add up to **90 degrees**. So, to find the measure of an angle that is complementary to an **80-degree angle**, subtract **80** from **90**. The result is **10 degrees**.

**Example 4**

Two angles are **vertical angles,** and one measures **120 degrees**. What is the measure of the other angle?

**Solution**

**Vertical angles** are always equal. Therefore, if one angle measures **120 degrees**, the vertical angle to it also measures **120 degrees**.

**Example 5**

In an **isosceles triangle**, the **vertex angle** measures **40 degrees**. What are the measures of the **base angles**?

**Solution**

In an **isosceles triangle**, the base angles are equal. Furthermore, the sum of the angles in any triangle is **180 degrees**. Therefore, the sum of the base angles is **180 – 40 = 140 degrees.** Since the base angles are equal, each one measures** 140 / 2 = 70 degrees**.

Figure-3.

**Example 6**

A **straight angle** is split into two angles. One angle measures **110 degrees**. What is the measure of the other angle?

**Solution**

A straight angle measures** 180 degrees**. If one angle is **110 degrees**, then the other angle is **180 – 110 = 70 degrees**.

**Example 7**

What is the measure of each angle in an **equilateral triangle?**

**Solution**

An equilateral triangle has all three sides of equal length, and all three angles are also equal. Since the sum of the angles in any triangle is **180 degrees**, each angle in an equilateral triangle measures **180 / 3 = 60 degrees**.

**Example 8**

The measures of the two angles of a triangle are **35 degrees** and **65 degrees**. What is the measure of the third angle?

**Solution**: The sum of the angles in a triangle is **180 degrees**. If two angles measure **35** and **65 degrees**, their sum is **35 + 65 = 100 degrees**. Therefore, the third angle measures **180 – 100 = 80 degrees**.

*All images were created with GeoGebra.*