Mathematically, angular diameter, linear size, and distance can be combined in an extremely useful and simple equation called the small-angle approximation. As seen in Figure 6.3, the angular diameter, θ, depends on the distance to the object, d, and its actual linear size, S, according to:
\begin{equation} tan(\frac{\theta}{2}) = (\frac{S}{2d}) \end{equation}
For very small values of θ measured in radians, tan(θ) = θ. Using this approximation, we can simplify the equation relating distance and linear size to:
\begin{equation} \theta/2 = S / 2d \end{equation}
or more simply
\begin{equation}\theta = S/d \end{equation}
In the small-angle approximation, if any two of the quantities are known, the third can be calculated. In astronomy, the angular diameter is usually measured directly, and the equation is used to calculate the distance to or physical size of the object. Since distances to astronomical objects are usually much larger than their linear sizes, this approximation is of great use in all branches and at all levels of astronomy!
The table below shows the relationship between the angle (in degrees and radians) and the tangent of the angle. Also shown is the difference between the angle in radians and the tangent of the angle. The close correspondence between these two quantities is the basis of the small-angle approximation.
| Angle (degrees) | Angle (radians) | Tangent (angle) | Difference | % Difference |
| 0.5 | 0.0087 | 0.0087 | 0.0000 | 0.0025 |
| 1.0 | 0.0175 | 0.0175 | 0.0000 | 0.0102 |
| 2.0 | 0.0349 | 0.0349 | 0.0000 | 0.0406 |
| 4.0 | 0.0698 | 0.0698 | 0.0000 | 0.1628 |
| 8.0 | 0.1396 | 0.1405 | 0.0009 | 0.6550 |
| 15.0 | 0.2618 | 0.2679 | 0.0066 | 2.349 |
| 20.0 | 0.3491 | 0.3640 | 0.0149 | 4.270 |
| 25.0 | 0.4363 | 0.4663 | 0.0300 | 6.870 |
| 30.0 | 0.5236 | 0.5774 | 0.0538 | 10.27 |
Based on looking at the table, when do you think the approximation breaks down?