How to use the angular size calculator
- Choose what to find: the angular size, the object size or the distance.
- Enter the two values you know, each with its unit. Lengths run from millimeters to light-years.
- For a known angle, pick degrees, arcminutes, arcseconds or milliradians.
- Read the answer in several angle units, with the small-angle estimate beside it.
Formula
Angle = 2 atan(size / (2 x distance))
Size = 2 x distance x tan(angle / 2)
Distance = size / (2 tan(angle / 2))
Small angle: angle in radians is about size / distance; 1 radian = 206,265 arcseconds
The object is treated as a flat disc or bar facing you. Splitting the isosceles triangle in half gives a right triangle with the half-angle at the observer.
The Moon, the Sun and a coin
The Moon is about 3474.8 km across at an average distance of 384,400 km, so it spans 2 atan(3474.8 / 768,800) = 0.517924 degrees, which is 31.075 arcminutes or 1864.53 arcseconds. The Sun, 1,391,400 km across at 1 astronomical unit, spans 0.5329 degrees. They look nearly the same size in the sky.
A coin 24 mm across held 65 cm from your eye covers 2.1153 degrees. Turning it around, an object that looks 0.5 degrees wide at 384,400 km must be about 3354.54 km across.
Degrees, arcminutes and arcseconds
A degree is split into 60 arcminutes and an arcminute into 60 arcseconds, so one degree is 3600 arcseconds. A milliradian is a thousandth of a radian, about 0.0573 degrees, and is used in optics and shooting. At a distance of 1000 units, one milliradian spans about 1 unit, which is a convenient way to size things.
The same triangle logic sits behind the triangle calculator. To see how a telescope magnifies an angle use the telescope magnification calculator, and for how far you can see over the Earth, the distance to horizon calculator.
When the small-angle estimate is good
For angles under a few degrees the tangent is almost equal to the angle in radians, so size divided by distance times 206,265 gives the angle in arcseconds within a tiny fraction of a percent. For the Moon the difference is about one part in ten thousand. The page shows both, so you can see when the shortcut starts to drift.
Assumptions and limits
- Size and distance are measured to the same point on the object, and the object is seen face on.
- Angles are limited to under 180 degrees. Lengths must be above zero, up to 1e30.
- Distances for the Moon and Sun are mean values; the real distance changes during the year, so the real angle does too.
- No atmosphere or lens effects are included, and the angle is the true geometric angle.
Frequently asked questions
How do I calculate the angular size of an object?
Divide the size by twice the distance, take the inverse tangent and double it. For small angles, multiply size over distance by 206,265 to get arcseconds.
How big is the Moon in the sky?
About half a degree, or 31 arcminutes, on average. The exact value changes by a few percent as the Moon moves closer to and farther from Earth.
What is an arcsecond?
It is 1/3600 of a degree. A coin 24 mm across seen from 5 km away spans about one arcsecond, and a full degree is 3600 arcseconds.
Can I find the distance from the apparent size?
Yes, if you know the true size. Distance equals the size divided by twice the tangent of half the angle, which the Distance option calculates.