When measuring the angular diameter of an object, what expression is used to find the actual diameter?

To determine the actual diameter of an object based on its angular diameter and the distance from the observer to the object, it's essential to understand the relationship between these quantities in terms of angular measurements.

The angular diameter is measured in radians, and the formula to find the actual diameter from the angular diameter involves the use of the small angle approximation for angles measured in radians. This approximation states that for small angles, the angular diameter in radians can be approximately equal to the actual size divided by the distance to the object.

When you convert degrees to radians, you use the factor of ( \frac{\pi \text{ radians}}{180 \text{ degrees}} ). Since 1 radian is approximately 57.3 degrees, the conversion from degrees to radians involves dividing the angular diameter in degrees by 57.3. To find the diameter, one rearranges the small angle formula:

[ \text{Diameter} = \text{Distance} \times \text{Angular Diameter in Radians} ]

To adapt this for angular diameter in degrees, you would take into account the conversion factor. Thus, the expression becomes:

[ \text{Diameter} = \text{Distance} \times \left(\frac{\text{Angular Diameter}}