Angular Size Calculator

Background.

Angular size is how big something looks rather than how big it is: the angle it spans as seen from where you stand. It is the quantity that makes the Moon and the Sun appear the same size despite one being four hundred times larger, and it is the only size an astronomer can measure directly. This calculator solves the relation in all three directions — angle from size and distance, size from angle and distance, and distance from angle and size — and it does two things most angular size calculators do not.

The first is that it lets you choose the geometry, because there are genuinely two relations and they are not the same. For a flat disc, or any chord drawn across the line of sight, the angle is twice the arctangent of half the size over the distance. For a sphere it is twice the arcsine, because what you actually see is the tangent limb rather than a diameter drawn through the centre, and the distance is measured to the centre. Almost every calculator online silently uses the arctangent for everything. For the Moon the difference is 0.019 arcseconds out of 1864 — one part in a hundred thousand, which is why nobody notices. Stand on the surface of a sphere and the difference is the entire answer: the sphere relation gives 180 degrees, which is correct, and the flat one gives 90 degrees, which is not.

The second is the unit policy. Size and distance are entered in whatever unit you like, as long as it is the same unit for both, because the angle depends only on their ratio. Kilometres for the Moon, millimetres for a coin, astronomical units for a planet, light-years for a galaxy — all work, and solving for a size returns it in whatever unit the distance was given in. This is not evasion. It removes an entire category of unit-mismatch error, and it makes a strong test possible: scale both inputs by a factor of a billion and every angle output must come back identical.

The small-angle shortcut is reported next to the exact answer along with its own error, rather than being described in words. The familiar approximation says the angle in arcseconds is roughly 206265 times the size divided by the distance. Its error is about the square of half the size-to-distance ratio, divided by three: it stays under a tenth of a percent out to a size-to-distance ratio of 0.1096, which is a subtended angle of about 6.27 degrees, and grows quadratically after that. For the Moon it is 0.00068 percent high. The shortcut always over-estimates, never under, because the tangent function grows faster than its argument.

Three situations are genuine singularities rather than merely awkward, and the calculator refuses them with a message naming the field. A size or distance of zero or less has no meaning. An angle of 648000 arcseconds is 180 degrees, and nothing subtends half a turn or more when you are outside it, so any angle at or above that is rejected. And in sphere geometry a size more than twice the distance would put you inside the body: the arcsine is undefined there and the question has no answer. A size exactly twice the distance — you standing on the surface — is accepted and correctly returns 180 degrees.

The defaults describe the Moon, and they are ordinary editable inputs rather than baked-in constants. The diameter of 3474.8 kilometres is twice JPL's mean radius of 1737.4 kilometres from ephemeris DE440; the distance of 384400 kilometres is the conventional mean, and the real Earth–Moon distance swings between roughly 356500 and 406700 kilometres over a month. That seven-percent monthly variation dwarfs every other uncertainty on this page, and it is the reason the Moon's apparent size visibly changes between perigee and apogee.

What is angular size calculator?

The angular size, or angular diameter, of an object is the angle it subtends at the observer's eye. It is measured in degrees, arcminutes (1/60 of a degree) or arcseconds (1/3600 of a degree), and it depends on both the object's true size and its distance — never on either alone.

A useful set of reference points: the full Moon and the Sun are both close to half a degree, or about 31 to 32 arcminutes. The planet Jupiter at its best is about 50 arcseconds. A typical star is well under a thousandth of an arcsecond, which is why stars look like points in any telescope. At the other end, an outstretched fist at arm's length covers roughly 10 degrees and a thumb about 2.

For a sphere the correct relation involves the arcsine rather than the arctangent, because the visible edge of a sphere is where your line of sight is tangent to it, slightly closer than a diameter through the centre would suggest. The two forms converge for distant objects and diverge sharply for nearby ones.

How to use this calculator.

1. 01Choose what you are solving for: the angle, the physical size, or the distance.
2. 02Choose the geometry. Use 'sphere' for a planet, moon or star; use 'flat' for a disc, a crater, a building or anything measured across the line of sight.
3. 03Enter the two quantities you know. Size and distance must be in the same unit — any unit at all, as long as it is the same one.
4. 04If you are entering an angle, give it in arcseconds: multiply degrees by 3600 or arcminutes by 60.
5. 05Read the answer in arcseconds, arcminutes and degrees together.
6. 06Check the shortcut error before quoting the small-angle approximation. Under 0.1 % it is safe; above that, use the exact value.

The formula.

Drop a perpendicular from the observer to the object's centre line. For a flat object of diameter d at distance D, half the object subtends an angle whose tangent is (d/2)/D, so the full angle is twice the arctangent of d/2D. For a sphere the geometry is different: your line of sight touches the surface tangentially, forming a right angle with the radius at that point, so half the angle has sine (d/2)/D and the full angle is twice the arcsine. That is the entire derivation, and it is why the sphere always subtends the larger of the two angles.

Work through the default configuration. The Moon's diameter is 3474.8 km and its mean distance 384400 km, so d/2D = 0.0045197711. The arctangent series gives 0.0045197403, and doubling that gives 0.0090394806 radians. Multiplying by 206264.806247 arcseconds per radian gives 1864.5267124920 arcseconds — 31.0754452082 arcminutes, or 0.5179240868 degrees, which is the familiar 'just over half a degree'. The sphere relation on the same inputs gives 1864.5457571085 arcseconds, larger by 0.019 arcseconds. The small-angle shortcut gives 1864.5394088122 arcseconds, which is 0.0006809406 % high. Notice the ordering: arctangent below the shortcut, arcsine above it. That is forced by the series expansions and holds for every positive input.

ROUNDING STAGE. Nothing is rounded part-way through. All arithmetic runs at forty significant digits and rounding happens once, at the return boundary, to ten decimal places. The figures in the reading sentence are display roundings of the same unrounded values.

SIGNIFICANT FIGURES. Entirely governed by the inputs. Astronomical diameters and distances are usually known to four to six significant figures, and the Earth–Moon distance varies by seven percent over a month, so treat the trailing digits of a ten-decimal answer as display precision rather than accuracy.

APPROXIMATION REGIME, MEASURED. The small-angle form θ ≈ d/D is shown with its own error rather than assumed. That error is approximately (d/2D)²/3 and it crosses 0.1 % at a size-to-distance ratio of 0.109588322, corresponding to a subtended angle of 6.2726756 degrees. Doubling the ratio roughly quadruples the error. Below that line the shortcut is genuinely fine; above it, use the exact figure.

INVALID DOMAIN. A size or distance of zero or less is rejected. An angle at or above 648000 arcseconds — 180 degrees — is rejected, because nothing subtends half a turn or more when the observer is outside it. And in sphere geometry a size greater than twice the distance places the observer inside the body, where the arcsine has no value; that case is rejected with a message quoting the radius and the distance back. A size exactly twice the distance is accepted and correctly returns 180 degrees, the case of standing on the surface.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NASA JPL Solar System Dynamics, 'Planetary Satellite Physical Parameters'. The Moon (body 301): mean radius 1737.4 ± 0.1 km and GM 4902.800 ± 0.001 km³ s⁻², referenced to ephemeris DE440. Source of the default physical size. Open access. Retrieved 2026-07-29.
2. [2]NASA JPL Solar System Dynamics, 'Astrodynamic Constants'. Astronomical unit = 149 597 870 700 m, attributed to IAU 2012 Resolution B1. Used with the nominal solar radius as the independent cross-check on this page's geometry. Open access. Retrieved 2026-07-29.
3. [3]Mamajek, E.E. et al. (2016). 'IAU 2015 Resolution B3 on Recommended Nominal Conversion Constants for Selected Solar and Planetary Properties.' The Astronomical Journal 152:41. Nominal solar radius = 6.957 × 10⁸ m, exact by definition; nominal terrestrial equatorial radius = 6.3781 × 10⁶ m. With the exact astronomical unit this yields a solar semidiameter of 959.23″, against the traditional almanac value of about 959.6″ — a 0.33″ disagreement recorded rather than resolved. Open access via IOPscience. Retrieved 2026-07-29.
4. [4]Urban, S.E. & Seidelmann, P.K. (eds.) (2013). Explanatory Supplement to the Astronomical Almanac, 3rd edition, University Science Books, §7. Semidiameters and apparent diameters, and the distinction between the geometric and the apparent (irradiation-biased) semidiameter that produces the 959.6″ versus 959.3″ disagreement above. Print / bibliographic reference.