Systems of Measuring Angles | Relation between Sexagesimal and Circular – Examples

An angle is a shape formed by joining two rays through a common vertex. In trigonometry, there are various ways of measuring angles. An angle is usually measured in degrees. Here we will learn about the systems of measuring angle and the relationship between sexagesimal and circular systems. Also, get the solved questions related to the angles.

Systems of Measuring Angles

There are 3 different systems of measuring trigonometric angles.

Sexagesimal System: In this type, the angle is measured in degrees, minutes and seconds. One complete revolution is divided into 360 units. So, one complete angle is divided by 360 equal parts and each part is measured with the unit degree.

The relation of degree, minute and seconds are here.

  • 1 right angle = 90 degrees = 90°
  • 1° = 60 minutes = 60′
  • 1′ = 60 seconds (60″)

Centesimal System: In this system, the angle is measured in grades, minutes and seconds. In the centesimal system, one right angle is split into 100 parts, each part is grade. 1 grade (1g) is divided into 100 equal parts, each part is minute. Minute is again divided into 100 equal parts, each is second.

  • 1 right angle = 100 grades = 100
  • 1 grade (1g) = 100 minutes (100′)
  • 1 minute (1′) = 100 seconds (100″)

Circular System: In a circular system, the angle is measured in radians. Radian is complicated to measure compared to the degree measure.

Also, Find

Relation between Sexagesimal and Circular

The relation between the units of two systems i.e sexagesimal and circular are here:

Sexagesimal Circular
360° 2 radian = 2πc
180° π radian = πc
90° π/2 radian = πc/2

The important points regarding the relationship between sexagesimal and circular are as follows:

  • In the sexagesimal system, to indicate measures of angles in degrees (°), for example, 30 degrees = 30°. In the circular system, to indicate measures of angles in radians (c), for example, 2 radians = 2c
  • In trigonometry, we do not write the magnitude of angles like 1c, but write them in terms of π.

Questions on Sexagesimal, Circular Systems

Question 1:
The circular measure of an angle is π/8, find its value in sexagesimal systems.

Solution:
Given angle is πc/8
= 180°/8   [πc = 180°]
= 22°30′

Question 2:
Find the sexagesimal, circular units and internal angle of a regular hexagon.

Solution:
The sum of the internal angles of a polygon of n sides = (2n – 4)
Sum of six internal angles of a regular hexagon = (2 x 6 – 4) = 8 right angles
Each internal angle of Hexagon = 8/6 = 4/3 right angles
Therefore, each internal angle of regular hexagon in sexagesimal system = 4/3 x 90 = 120°
Circular symstem = (4/3 x π/2) = (2π/3)c

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