Angular size
An object’s angular size (or angular diameter) is the angle it covers in your field of view. It depends on both its real size and its distance:
θ = 2 × arctan(size ÷ (2 × distance))
For small angles, θ in radians is simply size ÷ distance. Use the same unit for size and distance; the result is the same whether both are in meters, miles or kilometers.
Units
- 1 degree (°) = 60 arcminutes (′)
- 1 arcminute = 60 arcseconds (″)
- A full circle is 360°, or 1,296,000 arcseconds.
Example: the Moon
The Moon is 3,475 km across and on average 384,400 km away:
- θ ≈ 0.518°, or about 31 arcminutes, half a degree.
The Sun, 400 times larger but also about 400 times farther away, has almost the same angular size. That coincidence is why total solar eclipses are possible.
Everyday references
- The width of your little finger at arm’s length: about 1°.
- Your fist at arm’s length: about 10°.
- The Moon and the Sun: about 0.5°.
- Jupiter at its closest: about 50 arcseconds.
- The naked eye’s resolution: about 1 arcminute.
Uses
- Astronomy: deciding whether a planet or galaxy will fit in a telescope’s field of view, or how much magnification is needed to see detail.
- Photography: matching a subject to a lens’s angle of view.
- Vision and displays: a 20/20 letter subtends 5 arcminutes; the “retina” display threshold is about 1 arcminute per pixel at viewing distance.
- Surveying and shooting: estimating distance from the known size of an object (MOA, or minutes of angle, are used in rifle scopes; 1 MOA is about 1.05 in at 100 yards).
Reverse calculation
If you know an object’s angular size and distance, its real size is 2 × distance × tan(θ ÷ 2). If you know its size and angular size, its distance is size ÷ (2 × tan(θ ÷ 2)). This is how astronomers estimate the sizes of planets and the distances of standard objects.