**Angles are measured in:**

Angles are measured in a variety of units and systems, depending on the context and the precision required. They are fundamental to mathematics, geometry, and physics, and are used in many practical applications, such as engineering, architecture, navigation, astronomy, and sports. In this article, we will explore the different ways angles can be measured, and some of the terminology, symbols, and conventions involved.

## Angles and Unit Chart:

Unit | Abbreviation | Definition | Main Use |
---|---|---|---|

Degree | ° | 1/360 of a full rotation | Common unit for measuring angles in geometry and trigonometry. |

Radian | rad | The angle subtended at the center of a circle by an arc equal in length to the radius of the circle | Preferred unit in calculus and physics. |

Grad | grad | 1/400 of a full rotation | Used in some engineering and military applications. |

Mil | mil | 1/6400 of a full rotation | Used in military applications. |

Revolution | rev | One full rotation | Used in machinery and mechanical engineering. |

## Degrees: The Most Common Unit:

The most common unit of measuring angles is the degree, denoted by the symbol °. A degree is defined as 1/360th of a full circle, which is a complete revolution around a point. A circle has 360 degrees, and each degree is divided into 60 minutes (‘) and each minute is divided into 60 seconds (“). Thus, a degree is equal to 60 minutes or 3600 seconds.

Degrees are used in many everyday situations, such as measuring temperature, the direction of a compass, or the position of the sun or the moon. They are also used in mathematics and geometry to express the size of angles and to calculate trigonometric functions such as sine, cosine, and tangent. For example, the angle between two lines, planes, or vectors can be expressed in degrees and can be compared, added, subtracted, or converted to other units.

## Radians: The Natural Unit:

Radians are another unit of measuring angles, which are widely used in mathematics and physics, especially in calculus, trigonometry, and mechanics. A radian is defined as the ratio between the length of an arc of a circle and the radius of the circle. In other words, a radian is an angle subtended by an arc whose length is equal to the radius of the circle. The symbol for radians is rad, or sometimes c (for circular measure).

The advantage of radians over degrees is that they are more natural and convenient for calculus and trigonometry, as they simplify many formulas and identities. For example, the derivative of sin(x) with respect to x is cos(x), and the derivative of cos(x) is -sin(x), when x is measured in radians. Moreover, many physical laws and equations are expressed in terms of radians, such as the period of a pendulum, the frequency of a wave, or the rotational speed of a motor.

## Grads: The Metric Unit

Grads, also known as grades or gon, are a metric unit of measuring angles, which are based on the circumference of a circle, rather than its radius or diameter. A grad is defined as 1/400th of a full circle, which means that a circle has 400 grads. Each grad is divided into 100 centigrade, or simply cents, which are equivalent to 0.9 degrees or π/200 radians. The symbol for grads is g or grad.

Grads are used in some countries, especially in Europe, for practical and educational purposes, such as land surveying, construction, and military. They are also used in some specialized fields, such as watchmaking or astronomy, where small angles need to be measured with high precision. Grads have the advantage of being more compatible with the metric system, and of being easier to convert to degrees or radians, as they have simple factors of 0.9 or π/200.

## Other Units and Systems:

Besides degrees, radians, and grads, there are many other units and systems of measuring angles, which are less common or specialized. Some of these are:

### Turns:

A turn, also known as a revolution or a cycle, is a complete rotation around a point, which is equal to 2π radians, or 360 degrees. Turns are used in some contexts, such as navigation or aviation, where headings or bearings are expressed as fractions or multiples of turns, such as quarter-turn, half-turn, or full-turn.

### Mil:

A mil, short for mil-radian or milliradian, is a unit of measuring angles that is commonly used in military or sniper scopes, as well as in long-range shooting or mapping. A mil is defined as 1/6400th of a full circle, or approximately 0.0573 degrees, or 0.001 radians. The advantage of mils over degrees or radians is that they provide finer and more consistent graduations for aiming or measuring distances, especially at long ranges or in adverse conditions.

### Gradients:

A gradient, also known as a grade or a slope, is a measure of the steepness or incline of a surface or a path, which is expressed as the ratio of the vertical rise or drop to the horizontal run or distance. Gradients are often used in civil engineering, road construction, or railway planning, to ensure safety, stability, and efficiency of transportation. The unit of gradients is usually a percentage, which is equivalent to 100 times the tangent of the angle in degrees.

### Sextant:

A sextant is a navigation instrument that is used to measure the angle between two celestial objects, such as the sun, the moon, or the stars, and the horizon or the meridian. Sextants are essential for marine navigation, especially in the absence of GPS or other electronic devices, and rely on the principles of trigonometry and optics to determine the position of a ship or an aircraft. The unit of sextants is usually degrees or minutes, and the accuracy can be as high as a few seconds.

## Terminology and Symbols

Angles are described and represented in various ways, depending on the context and the convention. Some of the terms and symbols used for angles are:

### Vertex:

The vertex of an angle is the common endpoint of the two rays or line segments that form the angle. The vertex is usually marked with a dot or a letter, such as A, B, or O.

### Arms:

The arms of an angle are the two rays or line segments that extend from the vertex to the endpoints, and define the magnitude and the direction of the angle. The arms are usually labeled with letters, such as AB and AC, or with symbols, such as →AB and →AC.

### Interior and Exterior:

The interior of an angle is the region between the two arms, and is often shaded or colored to distinguish it from the exterior, which is the complement of the interior, and includes all the points outside the angle.

### Adjacent, Opposite, and Hypotenuse:

These terms are used in trigonometry and geometry to describe the relationship between angles and sides of triangles. The adjacent side is the one that is adjacent to the angle and forms one of the arms. The opposite side is the one that is opposite to the angle and does not form any of the arms. The hypotenuse is the longest side of a right triangle, and is opposite to the right angle.

### Complementary and Supplementary:

These terms are used to describe pairs of angles that add up to 90 degrees (complementary) or 180 degrees (supplementary). For example, if angle A is complementary to angle B, then A+B=90°, and if angle C is supplementary to angle D, then C+D=180°.

## FAQs

## Can angles be negative or imaginary?

No, angles are usually considered to be positive and real and can range from 0 to 360 degrees or from 0 to 2π radians. However, in some contexts, such as complex analysis or signal processing, angles can be extended to the complex plane and can have negative or imaginary values.

## How can I convert angles from one unit to another?

To convert angles from one unit to another, you need to use the appropriate conversion factor or formula, depending on the units involved. Here are some examples:

- To convert degrees to radians, multiply the degree measure by π/180 or divide by 57.2958.
- To convert radians to degrees, multiply the radian measure by 180/π or multiply by 57.2958.
- To convert degrees to radians, multiply the degree measure by 10/9 or divide by 0.9.
- To convert radians to degrees, multiply the gradian measure by 9/10 or multiply by 0.9.
- To convert mils to degrees, divide the mil measure by 17.7778 or multiply by 0.05625.
- To convert degrees to mils, multiply the degree measure by 17.7778 or divide by 0.05625.

## Conclusion:

In conclusion, angles are measured in various units, such as degrees, radians, gradians, mils, and gradients, depending on the context and the precision required. Angles are essential for many areas of mathematics, science, engineering, and navigation, and can be described and represented in different ways, such as vertices, arms, interiors, and exteriors. By understanding the concepts and properties of angles, you can enhance your problem-solving skills, your spatial reasoning abilities, and your appreciation of the beauty and complexity of the world around us.

## FAQs

### What can angles be measured with? ›

Angles are measured in **degrees**. We can use a protractor to measure how many degrees an angle is.

**Why do we measure angles in radians? ›**

Radians have the following benefits: They are dimensionless, which means that they can be treated just as numbers (although you still do not want to confuse Hertz with radians per second). **Radians give a very natural description of an angle** (whereas the idea of 360 degrees making a full rotation is very arbitrary).

**What units is angle usually measured in? ›**

"**Degrees**" is the measuring unit of an angle.

**What are the 3 ways to measure angles? ›**

There are three units of measure for angles: **revolutions, degrees, and radians**. In trigonometry, radians are used most often, but it is important to be able to convert between any of the three units.

**What is not used to measure angles? ›**

**A ruler** is used to measure length, and it can not be used to measure angles. A protractor is used to measure angles.

**Where are angles always measured from? ›**

The angle measure is the amount of rotation between the two rays forming the angle. Rotation is measured from **the initial side to the terminal side of the angle**. Positive angles (Figure a) result from counterclockwise rotation, and negative angles (Figure b) result from clockwise rotation.

**How were angles first measured? ›**

The history of the mathematical measurement of angles, possibly dates back to 1500BC in Egypt, where **measurements were taken of the Sun's shadow against graduations marked on stone tables**, examples of which can be seen in the Egyptian Museum in Berlin.

**How are the angles measured and why? ›**

An angle can be measured using a protractor, precisely. An angle is measured in degrees, hence its called 'degree measure'. One complete revolution is equal to 360 degrees, hence it is divided into 360 parts. Each part of the revolution is a degree.

**Do radians measure angles? ›**

**One way to measure angles is in radians**. To define a radian , use a central angle of a circle (an angle whose vertex is the center of the circle). One radian is the measure of a central angle that intercepts an arc s equal in length to the radius r of the circle.

**What are two common units used to measure angles? ›**

The two most common units for measuring angles are **degrees and radians**.

### What is called angle? ›

What is an angle? In Plane Geometry, **a figure which is formed by two rays or lines that shares a common endpoint** is called an angle. The word “angle” is derived from the Latin word “angulus”, which means “corner”. The two rays are called the sides of an angle, and the common endpoint is called the vertex.

**What are the two units of angle measure? ›**

Since **degree and radian** measures are the two most commonly used units in angle measurement, there is a convention in place for writing them.

**How many types of angle measurement are there? ›**

The names of basic angles are **Acute angle, Obtuse angle, Right angle, Straight angle, reflex angle and full rotation**.

**How many types of measurement are of the angle? ›**

There are **three** systems for measuring angles: Sexagesimal system. Centesimal system. Circular system.

**How do you measure angles in real life? ›**

Answer: An angle is formed by two lines or line segments or rays with a common point. We make use of the symbol ∠ to denote an angle. We measure it in an anti-clockwise manner by using a protractor in a degree.

**What is the most common measurement for angles? ›**

For exactly a half rotation, the angle is said to be a straight angle. The most familiar unit for measuring angles is **the degree**. An angle containing exactly one complete rotation is defined to be equal to 360 degrees. A quarter rotation, producing a right angle, is equal to 360/4 or 90 degrees (90°).

**What are some facts about angle measure? ›**

**Angles are measured in degrees, which is a measure of circularity or rotation**. A full rotation, which would bring you back to face in the same direction, is 360°. A semi-circle or half-circle is therefore 180°, and a quarter-circle, or right-angle, is 90°. Two or more angles on a straight line add up to 180°.

**How many degrees is a square? ›**

A square can also be referred to as a rectangle with two opposite sides having an equal length. The interior angles of a square are all equal and sum up to **360°**.

**How many angles is a radian? ›**

On the other hand, to scientists, engineers, and mathematicians it is usual to measure angles in radians. The size of a radian is determined by the requirement that there are 2 radians in a circle. Thus 2 radians equals 360 degrees. This means that **1 radian = 180/ degrees**, and 1 degree = /180 radians.

**How do radians describe angles? ›**

Radians **measure angles by distance traveled**. or angle in radians (theta) is arc length (s) divided by radius (r). A circle has 360 degrees or 2pi radians — going all the way around is 2 * pi * r / r. So a radian is about 360 /(2 * pi) or 57.3 degrees.

### Are angles and radians the same? ›

**A radian is just another way to measure the same angle**. For example, you can measure a circle as being 360° or as being about 6.28 radians.

**What are radians used for? ›**

You should use radians **when you are looking at objects moving in circular paths or parts of circular path**. In particular, rotational motion equations are almost always expressed using radians. The initial parameters of a problem might be in degrees, but you should convert these angles to radians before using them.

**How many radians is a circle? ›**

There are **2π radians** in a full circle. (So 2π radians should equal 360°.

**How is a radian different from a degree? ›**

**A degree is a unit of measurement which is used to measure circles, spheres, and angles while a radian is also a unit of measurement which is used to measure angles**. A circle has 360 degrees which are its full area while its radian is only half of it which is 180 degrees or one pi radian.

**How do you convert from degrees to radians? ›**

To convert from degrees to radians, **multiply the number of degrees by π/180**. This will give you the measurement in radians. If you have an angle that's 90 degrees, and you want to know what it is in radians, you multiply 90 by π/180. This gives you π/2.

**What are the 4 types of angles? ›**

Angles: **Acute, Obtuse, Straight and Right**

There are four types of angles depending on their size in degrees.

**Where do angles come from? ›**

Bede gave a precise date, 449AD, for the first arrival of the Anglo-Saxons and he said they came from three tribes: the Angles, Saxons and Jutes, who themselves came from different parts of Germany and Denmark – **the Angles were from Angeln, which is a small district in northern Germany**; the Saxons were from what is now ...

**What is a property of angle? ›**

Properties of Angles

**The sum of all the angles on one side of a straight line is always 180 degrees**. For example, The sum of ∠1, ∠2, and ∠3 is 180 degrees.

**What is the symbol for angle? ›**

Symbol of Angle

The symbol **∠** represents an angle. Angles are measured in degrees (°) using a protractor.

**Which term refers to the side of an angle that rotates? ›**

The endpoint point about which the ray rotates is the vertex. The amount of rotation determines the measure of the angle. The ray in the initial position, before the rotation, is called the initial side of the angle. The ray in the terminal position, after the rotation, is called the **terminal side** of the angle.

### Is an angle measures exactly 180? ›

**Angles that are 180 degrees (θ = 180°) are known as straight angles**. Angles between 180 and 360 degrees (180°< θ < 360°) are called reflex angles.

**How do you find a 45 degree angle? ›**

Solution: One-fourth of 180 degree angle = 180/4 = 45. Answer: One-fourth of a 180-degree angle is a 45-degree angle.

**What angle is 120 degrees? ›**

120º angle

This is an **obtuse angle** because 120 is a number greater than 90 and less than 180. So this angle measurement is between 90º and 180º: 90º < 120º < 180º.

**What angle is 75 degrees? ›**

75° is less than 90°. So it is **acute angle**.

**What angle is 100 degrees? ›**

Answer and Explanation: Since 100° > 90°, this would be an **obtuse angle**. Its supplement (the angle added to the original angle to create a straight angle) would be an acute angle (80°).

**What is a 93 degree angle called? ›**

Obtuse Angle. An obtuse angle is the opposite of an acute angle. It is the angle which lies between 90 degrees and 180 degrees or in other words; an obtuse angle is greater than 90 degrees and less than 180 degrees.

**Is an angle measure exactly 90? ›**

Acute angles measure less than 90 degrees. **Right angles measure 90 degrees**. Obtuse angles measure more than 90 degrees.

**What are the 7 types of angles? ›**

Angle Type | Angle measure |
---|---|

Right angle | 90° |

Obtuse angle | Greater than 90°, less than 180° |

Straight angle | 180° |

Reflex angle | Greater than 180°, less than 360° |

**What angle is 60 degrees? ›**

60 degree angle is an **acute angle**, as angles smaller than a right angle (less than 90°) are called acute angles. In the case of a geometric angle, the arc is centered at the vertex and constrained by the sides.