Measurements of Angles

According to Euclidean Geometry, an angle is formed if two rays intersect and share a common endpoint. The rays are denoted as sides of the angle, whereas the common endpoint is the vertex. Angles are formed by the rays that lie in the same plane. In other words, angles are formed at the exact point of intersection of the two planes. The measurement of the space between the rays is an angle measurement. It is the direction of the rays in relation to one another that determines the measurement of an angle. Protractor is the most commonly used angle measuring tool. Angles are useful in defining and studying polygons such as triangles and quadrilaterals and are used in multiple disciplines such as physics, carpentry, shipping and animation.

An angle is universally denoted by the symbol’∠’, followed by three capital letters. 

The middle stands for the vertex or origin of the angle, and two others are points on the sides. Alternatively, one can also be denoted by one capital letter for the vertex.

Measurement of Angles in Trigonometry

Trigonometry derived from the Greek word ‘Trigon’ and ‘Metron’ is a branch of mathematics that deals with specific functions and measurement of angles and their application to calculations. In trigonometry, the measurement of angles is often defined in terms of rotation. An angle in trigonometric terms is the rotation of a ray from an initial point to a point where it terminates, which is also known as the terminal point. The endpoint about which the ray rotates is called the vertex. The amount or measure of rotation is the measurement of the angle. Some important terminologies regarding the angle are as follows:

• Initial side: The ray in the initial position before the rotation
• Terminal side: It is the final position of the ray upon rotation
• Vertex: Common endpoint of rotation
• Positive angle: When the direction of the rotation is anti-clockwise
• Negative angle: When the direction of the rotation is clockwise

Different Systems used for the Measurement of Angles

If a straight line stands on another straight line and the two adjacent angles formed on either side of the line are equal in measurement, then the angles are denoted as right angles. In geometrical terms, when two rays intersect and form a 90˚ angle at the intersection, they are said to form a right angle. The acute angle degree is less than 90°. Obtuse angles are angles that are greater than 90°. The straight angle is similar to a straight line and hence the name. The measure of a straight angle is exactly 180 °. 

The right angle forms the basis for defining the different systems used for measuring angles.

Following are the three main systems used in the measurement of angles:

1. Sexagesimal System (English system)
2. Centesimal System (French system)
3. Circular system

Sexagesimal System

In the Sexagesimal system, the basic premise or measurement of angles in degrees, minutes and seconds. The angle measurement is divided into 90 equal parts called degrees (°). Each degree is divided into 60 equal parts called minutes (‘), and each minute is again divided into 60 equal parts called seconds (“). So, degrees, minutes and seconds can be represented by:

One right angle = 90 degrees (or 90°)

1 degree (or 1°)= 60 minutes (or 60′)

1 minute (or 1′)= 60 seconds (or 60″).

 Generally, this system is the most commonly used but is not convenient for the multipliers 60 and 90.

Centesimal System

In this system, angles are measured in grades, minutes and seconds. The right angle is divided or measured in 100 equal parts called grades (g). Each grade is then further divided into 100 equal parts called minutes, and each minute is then further divided into 100 equal parts known as seconds. So, in short,

1 right angle=100 grades (or, 100g)

1 grade (or 1g)= 100 minutes (or, 100‵)

 1 minute (or 1‵)= 100 seconds (or, 100‶).

It is to be noted that the Sexagesimal and Centesimal systems are stated differently for the measurement of angles in minutes and seconds. 

For example,

1 right angle = 90 × 60 = 5400 sexagesimal minutes = (5400)’

and 1 right angle = 100 × 100 = 10000 centesimal minutes = (10000) ‶

(ii) Since, 1 right angle = 90° = 100g

Therefore, 90° = 100g

or, 1° = (10/9) g and 1g = (9/10) °

Based on the information above, the first relation is applied to reduce the angle of the sexagesimal system to the centesimal system. The second is applied to reduce an angle of the centesimal system to the sexagesimal system. To change degrees into grades, add on one-ninth and to change grades into degrees, subtract one-tenth.

For example,: 36° = (36 + 1/9 * 36) g = 40 g,

And 64 g = (64 – 1/10 * 64) ° = (64-6.4) ° = 57.6 °

When the angle does not contain an integral number of degrees, any angle can be reduced to a fraction of the right angle. In theory, this system is much more convenient than the Sexagesimal system; however, for practical applications, many tables would have to be recalculated. Hence, the system is never used in practice to measure angles.

Circular System

In this system, measurement of angles is done in radians. It is the SI unit for measuring angles. It is considered a standard unit for angle measurements in advanced mathematics. The angle subtended at the centre by an arc of a circle whose length is equal to the circle’s radius is called a radian. 

In theory, the circular measure of an angle is the number of radians it contains. Thus, the circular (radian) right angle measure is π/2.

If an angle is given without mentioning units, it is assumed to be radians.

The Formula for Measurement of Angles

Degree measure = 

1 ° = (1/360) th of a complete rotation

Radian measure = 

ϴ = l/r where l is the arc length and r is the circle’s radius.

It is a known fact that a complete rotation or the angle subtended by a circle at the centre is considered as 360 ° as per the degree measure and is 2π radians in radian measure.

2π (radians) = 360° (degrees) or π (radians) = 180° (degrees)

Assuming, π = 3.14159

1 radian = 180°/ π

1 degree = π/180°

The following convention is followed for writing degree and radian measures which are the two most commonly used units in angle measurement,

• If you write angle θ°, it means an angle whose degree measure is θ
• If you write angle β, it implies an angle whose radian measure is β

Also, note that the term ‘radian’ is usually omitted while writing the radian measure. Hence, π radian = 180° is simply written as π = 180°. Further, the relationship between radian and degree measure can be established through the following formula:

 The relation between degree measures and circular (radian) measures of some standard angles are given below:

Degrees and their Radians

0°=0

30°=π /6

45°=π /4

60°=π /3

90°=π /2

120°=2 π /3

135°=3 π /4

150°=5 π /6

180°=Π

270°=3 π /2

360°=2 π

Also, we know that,

Radian measure = π/180° x Degree measure

Therefore, the radian measure of 40° 20′ = π/180 x 121/3 = 121π/540 radian.

Conclusion

Trigonometry and geometry deal with the study of angles and their practical applications for calculations. It is used for many real-world and useful applications like sea navigation by engineers to develop a physical design or prototype. It is used in seismology, predicting the height of tides in the ocean, analysing a musical tone, and many other areas. Hence, studying the measurement of angles is used in advanced mathematics and physics and real-world applications. The relation and spatial understanding between numbers and measurements can be learned effectively by measuring angles. Measurement of angles not only help us academically but also to understand and use things around us.