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Angular Size Calculator

Find how large an object appears from a distance.

How it is calculated

The apparent angle is θ = 2 × arctan(size ÷ (2 × distance)). size is the true diameter or width of the object and distance is how far away it is. Both can be in any unit as long as they are the same one: the unit cancels and only the ratio matters. The result is shown in degrees (four decimals), arcminutes (θ × 60) and arcseconds (θ × 3600). The exact arctangent form is used, not the small-angle shortcut, so it stays correct for large angles too. Both numbers must be greater than zero, otherwise there is no result. The distance is measured from the observer to the centre of the object, and the object is treated as a flat disc or a line facing you.

Worked example

The Moon has a diameter of about 3474.8 km and an average distance of about 384400 km. Divide size by twice the distance: 3474.8 ÷ 768800 = 0.0045198. The arctangent of that is almost the same number, 0.0045198 radians, and doubling it gives 0.0090396 radians. Converting to degrees gives 0.5179°, which is 31.08' in arcminutes and 1,864.53" in arcseconds. The Moon therefore spans roughly half a degree of sky.

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Common mistakes

  • Entering size and distance in different units, for example a diameter in kilometres and a distance in metres or miles. The ratio is then off by the conversion factor and the angle is wrong by the same amount.
  • Using the radius instead of the diameter. The tool expects the full width of the object; a radius gives an angle half as large.
  • Entering the distance from the Earth's surface or from an orbit's altitude instead of the distance to the object itself. For nearby objects such as a satellite, the difference matters.
  • Mixing up arcminutes and arcseconds. One degree is 60 arcminutes and 3600 arcseconds, so read the line that matches the unit your eyepiece or chart uses.
  • Using an average distance for objects whose distance changes a lot, like the Moon or a planet. The size on the sky changes with the distance you enter, so use the distance for the date you care about.

About this calculator

This calculator finds the angular size (apparent angular diameter) of an object as seen from a given distance: θ = 2·arctan(size / (2·distance)), or for small angles simply θ ≈ size/distance in radians. The result can be shown in degrees, arcminutes, or arcseconds.

Amateur and professional astronomers use it to predict how large the Moon, a planet, or a deep-sky object will appear in an eyepiece or camera sensor, while photographers and surveyors use it to work out framing and field of view at a given distance.

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Comments (1)

  • Editorial team

    A neat use is comparing the Sun and the Moon. With the Sun at 1,391,400 km across and 149,597,870.7 km away, the tool gives 0.5329 degrees. The Moon at 356,500 km comes out at 0.5585 degrees, larger than the Sun, while at 406,700 km it is 0.4895 degrees, smaller. That gap is why some solar eclipses are total and others leave a ring. Which object and distance are you checking?

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