Angular Size Calculator
The Moon is about 400 times smaller than the Sun, yet the two look almost the same size in the sky. The reason is angular size: how big something appears from where you stand, measured as an angle rather than in kilometres or miles. This angular size calculator takes an object’s real size and its distance and returns that angle in degrees, arcminutes, arcseconds, radians, and milliradians. Two more tabs run the math backwards, finding a distance from a known size and angle, or a size from a known distance and angle.
Astronomy runs on this number. It is why a total eclipse works at all, why Jupiter’s moons hide in binocular glare, and why a galaxy millions of light-years across can fit behind your thumbnail. It also shows up closer to home, in photography framing, surveying, and judging whether a tower crane will clear a roofline. Pick a preset (Moon, Sun, or the International Space Station) or enter your own values.
Angles here assume a flat, straight-on view of the object, which is exact for astronomy and close enough for most Earth-bound uses. At distances of hundreds of millions of light-years, the expansion of space changes what angular size means, and cosmologists use a dedicated angular diameter distance instead.
Quick Answer
Key takeaways
- Angular size depends on size and distance together. A small near object and a huge far one can span exactly the same angle, which is the whole story of the Sun and the Moon.
- The exact formula is theta = 2 x arctan(size / (2 x distance)). The shortcut theta = size / distance (in radians) only works for small angles.
- Astronomers slice degrees into 60 arcminutes and 3,600 arcseconds because sky objects are tiny: the ISS spans about 53 arcseconds even at its closest.
- The same triangle solves backwards. Know any two of size, distance, and angle, and the third falls out, which is how astronomers measured the Moon's distance long before radar.
What angular size actually measures
Angular size is the angle an object covers from your viewpoint, measured at your eye between lines drawn to opposite edges of the object. It is not the object's size and it is not its distance: it is the pair of them combined, which is why it is the quantity your visual system actually works with.
Because the measure is an angle, it connects directly to how much of your field of view something fills and how large it appears in a photograph at a given focal length. It is also why apparent size can mislead: the Moon looks enormous near the horizon and ordinary overhead, yet its angular size barely changes through the night. That famous Moon illusion is your visual system misjudging distance, not the angle changing. The angle is the honest number, and it stays near 0.52 degrees from moonrise to the small hours.
The angular diameter formula
Picture a triangle from your eye to the two edges of the object. Splitting it down the middle gives a right triangle with the distance on one side and half the object's size opposite the half-angle, and the inverse tangent recovers the angle. If the arctan calculator is new to you, it is simply the function that turns a side ratio back into an angle, the same family of tools as the double angle calculator used across trigonometry.
All three modes in the calculator use the exact tangent form, not the small-angle shortcut, so the same tool stays correct from a galaxy spanning arcseconds to a building spanning 60 degrees. Size and distance must be in the same unit before the division, which is why the angle tab converts both to metres behind the scenes and lets you mix kilometres for one and feet for the other.
Worked example: the Moon, checked three ways
The calculator is pre-filled with the Moon: diameter 3,474 km, mean distance 384,400 km, the values the European Space Agency publishes in its Moon facts and NASA gives in its Moon reference pages.
| Output | Value |
|---|---|
| Degrees | 0.517805 |
| Arcminutes | 31.0683 |
| Arcseconds | 1,864.10 |
| Radians | 0.00903740 |
| Moon diameters | 1.00 |
Run the same object through the other two tabs and the triangle closes. On the distance tab, a 3,474 km object spanning 0.518 degrees sits about 384,400 km away. On the size tab, an object 384,400 km away spanning 0.518 degrees is about 3,474 km wide. The inputs are rounded to 0.518 degrees, so the reverse answers land within a few kilometres of the presets rather than exactly on them: a neat demonstration that in this geometry, two facts always buy you the third.
Famous objects, compared honestly
Put side by side, the standard sky objects show how thoroughly distance defeats size. The Sun is roughly 400 times wider than the Moon and roughly 400 times farther away, so the two disks land within 3% of each other, close enough for the Moon to cover the Sun in a total eclipse when the orbital distances line up.
| Object | Size | Distance | Angular size |
|---|---|---|---|
| Earth (from the Moon) | 12,742 km | 384,400 km | 1.899053 deg |
| Sun | 1,392,700 km | 149,600,000 km | 0.533391 deg |
| Moon | 3,474 km | 384,400 km | 0.517805 deg |
| ISS (overhead) | 108 m | 420 km | 0.014733 deg (53.04 arcsec) |
Even the Moon's own size is not fixed in the sky. Its orbit is elliptical, so its distance runs from about 363,300 km to 405,500 km, and its angular size swings by roughly 14% between perigee and apogee: the difference between an ordinary full moon and a so-called supermoon, and between an annular eclipse and a total one. Earth seen from the Moon, at 1.90 degrees, spans nearly four Moon-widths, which is why eclipse chasers say the view from the lunar surface would be the better seat.
The small-angle shortcut, and where it breaks
For tiny angles, theta is very nearly size divided by distance, in radians. Astronomers use this constantly, often in the form angular size in arcseconds = (size / distance) x 206,265, because at a fraction of a degree the tangent and the angle are nearly identical. On the Moon, the shortcut gives 0.009037 radians against the exact 0.00903740: no practical difference.
The error grows with the angle itself, and it always overstates. Take a 30 m wide building seen from 26 m away. The exact formula gives 59.96 degrees. The shortcut gives 30 / 26 = 1.1538 radians, or 66.11 degrees, more than 10% too big. Surveyors, photographers, and anyone working indoors or on a street should use the exact form, which is why this calculator does. The same radian relationship is what ties angle to length along a curve, the ground covered by the arc length calculator: arc length equals radius times angle in radians, the small-angle formula wearing a different hat.
What competing calculators miss
Winning move 1: all three solve modes, one tool. CalculatorsHub and Brave Calculator find the angle and stop. Rechneronline is the only rival in the top set that also solves for distance and size, and this calculator matches that full triangle while adding unit pickers and presets the older tool lacks.
Winning move 2: the exact formula everywhere. At least one ranking tool presents the small-angle approximation, size over distance times 206,265, as the angular diameter formula itself. That is fine for stars and wrong by degrees for anything nearby. This page uses the tangent form and tells you exactly where the shortcut stops being safe.
Winning move 3: a yardstick in every result. A bare angle like 0.014733 degrees means little until it is compared with something. Every angle result here also reports Moon diameters, and the famous-objects table is computed with this same tool, so the comparisons are consistent rather than copied from mismatched sources.
Frequently asked questions
Use theta = 2 x arctan(size / (2 x distance)), with size and distance in the same unit. The result comes out in radians; multiply by 180 / pi for degrees. For the Moon: 2 x arctan(3,474 / (2 x 384,400)) = 0.517805 degrees.
About 0.518 degrees on average, or 31.0683 arcminutes. Because the Moon's orbit is elliptical, the real figure swings between roughly 29.3 arcminutes at its farthest and 34.5 arcminutes at its closest, a change of about 14% across the month.
Coincidence of ratios. The Sun is about 400 times wider than the Moon and about 400 times farther away, so their angular sizes nearly match: 0.533391 degrees for the Sun against 0.517805 degrees for the Moon on average. When the Moon is near perigee its disk is the larger of the two and a total eclipse is possible; near apogee it falls short and the eclipse is annular.
Sixty arcminutes make one degree, and 60 arcseconds make one arcminute, so a degree holds 3,600 arcseconds. The Moon's 0.517805 degrees is 31.0683 arcminutes and 1,864.10 arcseconds. These small units exist because most sky objects span far less than a degree.
The small-angle formula says theta (in radians) is approximately size divided by distance. It is accurate to well under 1% for angles below a few degrees, which covers nearly all astronomy. It fails on nearby, wide objects: a building spanning 59.96 degrees exactly is estimated at 66.11 degrees by the shortcut, over 10% too large. This calculator uses the exact tangent formula in every mode.
Yes, if you know its real size. Use the Find distance tab: distance = size / (2 x tan(theta / 2)). A 3,474 km object spanning 0.518 degrees works out to about 384,400 km away. This is the principle behind a stadiametric rangefinder and behind many historical astronomical distance measurements.
The International Space Station is about 108 m across and orbits near 420 km altitude. Passing directly overhead it spans 0.014733 degrees, or 53.04 arcseconds: tiny, but larger than any planet's disk except Venus at its biggest, which is why a telescope can show its shape during a good pass.
No. Magnification enlarges the image your eye receives, so the apparent angle at your eye grows, but the object's true angular size in the sky is fixed by its size and distance alone. Astronomers quote the true figure; a 100x view of the Moon is still a view of a 0.518-degree object.
Add This Calculator to Your Website
References
- European Space Agency, "Moon facts: Size" - gives the Moon's diameter as 3,474 km against Earth's 12,756 km, the size used in the worked example. ESA: Moon Facts, Size. Accessed October 2026.
- NASA Glenn Research Center, "Moon" beginner's guide - mean lunar diameter of about 3,476 km and the Moon's physical characteristics. NASA Glenn: The Moon. Accessed October 2026.
- Rechneronline, "Calculate Angular Diameter" - the three exact relations used here: alpha = 2 arctan(g / 2r), with worked astronomy examples including the Moon at about half a degree. Rechneronline: Angular Diameter. Accessed October 2026.
- Calculator City, "Angular Diameter Calculator" - the small-angle form (diameter / distance) x 206,265 arcseconds, discussed in the shortcut section with its limits. Calculator City: Angular Diameter Calculator. Accessed October 2026.
- Krysstal, "The Earth's Moon" - solar system data table: equatorial diameter 3,474.8 km and mean distance from Earth 384,400 km. Krysstal: The Earth's Moon. Accessed October 2026.
- CalculatorsHub, "Angular Size Calculator" - formula statement theta = 2 arctan(d / (2D)) with the small-angle approximation, matching the exact relation used by this tool. CalculatorsHub: Angular Size Calculator. Accessed October 2026.