Angular Size Formula
Calculate the angular size of any object using its physical size and distance.
Includes the small angle approximation and examples with planets and the Moon.
The Formula
Angular size (also called angular diameter) is the angle subtended by an object as seen from the observer. It depends on both the object's physical size and its distance — a small nearby object can appear larger than a huge distant one. This formula applies to any object: planets, galaxies, telescope fields of view, or everyday objects.
Small Angle Approximation
For small angles (less than about 10°), the approximation θ ≈ d/D is excellent and much easier to compute. Converting to arcseconds: θ″ = 206,265 × (d / D).
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| θ | Angular size (angular diameter) | radians, degrees, or arcseconds |
| d | Physical diameter of the object | any unit (km, m, AU, ly) |
| D | Distance from observer to object | same unit as d |
Example 1 — The Moon
Moon diameter = 3,474 km, distance = 384,400 km
θ = 3,474 / 384,400 = 0.009036 radians
Convert: 0.009036 × (180/π) = 0.518°
θ ≈ 0.52° or about 31 arcminutes — roughly the same as the Sun
Example 2 — Jupiter through a Telescope
Jupiter diameter = 139,820 km, distance at opposition = 628,000,000 km
θ = 139,820 / 628,000,000 = 2.226 × 10⁻⁴ radians
In arcseconds: 2.226 × 10⁻⁴ × 206,265 = 45.9″
θ ≈ 46 arcseconds at closest approach — visible disc in small telescope
Example 3 — Rearranged for Distance
A galaxy has angular size 0.5° and known diameter of 100,000 light-years. How far is it?
θ in radians = 0.5 × π/180 = 0.008727 rad
D = d / θ = 100,000 / 0.008727
D ≈ 11,460,000 light-years ≈ 11.5 million light-years
When to Use It
- Predicting how large planets and nebulae will appear through a telescope
- Calculating the field of view needed for astrophotography
- Estimating distances to objects of known size (standard rulers)
- Describing apparent sizes of craters, star clusters, and galaxies in star atlases
- Camera and lens selection — calculating image scale in photography