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How to Calculate Angular Dimensions

Angles are dimensioned in units of degrees and radians. Each unit uses its own constant of proportionality which is different from but geometrically related to the other. A calculation of either of these two dimensions exploits the geometric property of a circle that the ratio of its circumference to its diameter is equal to the constant pi (3.14). You can calculate an angle in units of radians and degrees for a given arc-length of the unit circle, or the circle whose radius is equal to one.

Things You'll Need

Pencil
Paper
Calculator

Instructions

Arc-length = 1.57
- 1
  Write the circumference of the circle C = 2*pi*r where "r" is the radius. For the case of the unit circle, r = 1. Therefore C = 2pi(1) = 2(3.14)(1) = 6.28.
- 2
  Write down the constant of proportionality that defines the unit. One full rotation of a circle is subtended by 360 degrees and 2*pi radians.
- 3
  Divide the circumference of the circle by the constant of proportionality to get the amount of circumference each angular unit subtends. In radians, the amount of circumference subtended by one radian is C/(2*pi) = 2*pi*r/(2*pi) = r = 1. In degrees, the arc-lengthwise proportion of the circumference per degree is C/360 = 2*pi*r/360 = 2*pi(1)/360 = pi/180.
- 4
  Scale (multiply) the given arc-length by the inverse of the proportional units that were calculated in Step 3. This operation gives dimensions of length times dimensions of angular units divided by dimensions of length to render angular dimensions. In radians, the angle is 1.57(1/1) = 1.57 or pi/2 radians. In degrees, the angle is 1.57(180/pi) = 90 degrees.
- 5
  Check the result. A 90 degree (or pi/2) angle is exactly one quarter of the angular units subtended by one full rotation of the circle. The proportionality must hold for the ratio of the arc-length to the circumference of the circle: 1.57/C = 1.57/6.28 = 0.25 = 1/4.

Nature