Angular size calculator
How big does the Moon look, how tall is a building that fills 2° of your view, how far is a car that looks 10 arcminutes wide? Enter any two of size, distance and angle to get the third, using the exact formula rather than the small-angle shortcut.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Choose what to find: the angle, the size or the distance.
- Enter the other two values. Size and distance use the same unit; angles can be in degrees, arcminutes or arcseconds.
- Read the result; the small-angle value next to it shows how much the shortcut would differ.
For an object of size s seen face-on from a distance d (measured to its middle):
- angle θ = 2 × atan(s ÷ 2d)
- size s = 2d × tan(θ ÷ 2)
- distance d = s ÷ (2 × tan(θ ÷ 2))
Small-angle approximation: θ (radians) ≈ s ÷ d. 1 rad = 57.2958°.
Worked example
The Moon: s = 3,474.8 km, d = 384,400 km.
θ = 2 × atan(3,474.8 ÷ 768,800) = 2 × atan(0.0045198) = 0.009039 rad = 0.51792°, or 31.075′ and 1,864.53″. The small-angle shortcut gives 31.076′ – practically the same.
The other way: a person 1.75 m tall who looks 1° tall is 1.75 ÷ (2 × tan 0.5°) = 100.2651 m away.
Tips
- Your fist at arm’s length covers roughly 10°, a fingertip about 1° – handy for rough estimates in the sky.
- For astronomy the distance changes along the orbit, so the Moon’s apparent size varies by over 10 % between perigee and apogee.
- Telescope magnification multiplies the angular size: 50× makes the Moon look about 26° wide.
Frequently asked questions
What is angular size?
It is the angle an object takes up in your field of view. It depends on both its real size and how far away it is, so a coin at arm's length and the Moon can look about the same size.
How big do the Moon and the Sun look?
The Moon, 3,474.8 km across at an average 384,400 km, spans about 0.518° or 31 arcminutes. The Sun, about 1,392,700 km at 149.6 million km, spans about 32 arcminutes – which is why total solar eclipses are possible.
What is the small-angle approximation?
For small angles, angle in radians ≈ size ÷ distance. It is shown for comparison; under a few degrees the difference from the exact formula is tiny, but it grows for wide angles.
Which unit should I use for size and distance?
Any unit, as long as both are the same – metres and metres, kilometres and kilometres. The size or distance you get back is in that same unit.
How many arcseconds are in a degree?
One degree is 60 arcminutes (′) and one arcminute is 60 arcseconds (″), so a degree is 3,600 arcseconds. A full circle is 360° or 2π radians.