Convert Angle Units

Angle Dimensional Analysis#

An angle quantifies the extent of rotation between intersecting lines. It's a dimensionless measure often expressed in degrees (deg, º) or radians (rad).

From the perspective of dimensional analysis, an angle translates to a ratio of two lengths, typically the arc length of a circle divided by its radius.

Angle = length / length
      = L¹ / L¹
      = L⁰

where L represents the fundamental base unit of length.

Commonly Used Angle Units#

Both metric and US systems use degrees (deg or º) as the main unit of measuring angles. A degree is a fraction of a full circle, which is divided in a total of 360 fractions. Degrees (deg or º) take the spotlight as the most prevalent angle unit in daily affairs as well.

Radians (rad) are widely used in engineering and science, specially as a function of pi (π) for its convenience.

On the other hand, the gradian (gon, grade, or grade) was an attempt to introduce a more decimal-like measurement system. However, the unit never took off, and today is barely used.

How to Convert Angles from Degrees to Radians#

By definition, a full circle is 2π rad or 360 deg:

2π rad = 360 deg
π rad = 180 deg

Using this known relationship, we can work out a conversion factor for converting deg into rad. Let's divide both sides of the equation by 360 deg:

π rad / (180 deg) = 1

This simple trick allows us to take the left side of the equation as a conversion factor. Applying this conversion factor to an angle in deg, the degree units cancel out, and we're left with the rad unit we're aiming for.

Here's an example of how to convert an angle of 90 deg into radians:

90 deg · conversion_factor
= 90 deg · π rad / (180 deg)
= 90 / 180 · π deg
= 0.5 π deg
= π/2 deg

How to Convert Angles from Radians to Degrees#

Using the same procedure, we can convert rad to deg. We just have to invert the fraction to have in the denominator the units we need to cancel out:

180 deg / (π rad) = 1

Here's how to convert an angle of π/2 rad into degrees:

π/2 rad · conversion_factor
= π/2 · 180 deg / (π rad)
= π/2 · 180/π deg
= 90 deg

Angle Equivalent Values#

Here you'll find a handy comparison of the most prevalent angle units, aiding your understanding of their proportional relationships.