The Small Angle Formula

In astronomy, the sizes of objects in the sky are often given in terms of their angular size as seen from Earth, rather than their actual sizes. For a given observer, the distance to the object D, the size of the object (or separation) d, and angle θ in radians (as portrayed in the picture above) form a right triangle with the trigonometric relationship:

Since these angular diameters are often small, we can use the small angle approximation which will give us:

So we can rewrite our small angle approximation as:

When dealing with astronomically distant objects, where angle sizes are extremely small, it is often more practical to present our angles in terms of arcseconds, which is 1/3600th of one degree. Since one radian equals 3600⋄(180/π) ≈ 206265 arcseconds, we can then rewrite this as the Small Angle Formula:

where θ is now measured in arcseconds, d is the physical size or separation, and D is the distance to the object.

Since it is easy to measure the angular size of astronomical objects, we often use this to solve for other unknowns, such as the distance or the diameter of a celestial body. If two objects are roughly the same distance from the observer, you can also use the formula to find the distance between the two objects. As you are aware from the background section of this lab (and hopefully your own experience), an object's angular size depends on its physical size (in feet, meters, etc.) and its distance.

Lab Exercise

Using trigonometry, if you know the length of two sides of a right triangle, you can find the angles. Suppose the building in the picture to the left is 50 feet tall and 300 feet away from you (A=50ft, B=300 ft), and that you can use the small angle approximation.

1. What is the angular size of the building in degrees?

2. Now find the angular size of something in the lab. Be sure to record how large it is and how far away it is from you.

3. Does the angular size depend on where you stand in relation to the object?