Angular Size Calculator (Apparent Diameter)
Calculate angular size from object diameter and distance.
Returns degrees, arcminutes, arcseconds, and radians for telescope and astrophotography planning.
Angular size (also called angular diameter or apparent size) is the angle an object subtends as seen from the observer. It is the answer to “how big does this look from here?” and depends on the object’s actual physical size and how far away it is.
The exact formula:
θ = 2 × arctan(d / 2D)
The small-angle approximation (good for any θ smaller than about 10°):
θ ≈ d / D (in radians, when d is much smaller than D)
Where d is the object’s physical diameter, D is the distance from observer to object, and θ comes out in radians. Converting to other units:
- To degrees: θ × (180 / π) = θ × 57.296
- To arcminutes: θ × 3,437.75
- To arcseconds: θ × 206,265
For astronomy work, arcseconds are usually the right unit because most planets and deep-sky objects subtend less than one arcminute.
The Sun and the Moon — coincidence and total eclipses: By a striking cosmic accident, the Sun and the Moon both appear about half a degree (roughly 31 arcminutes) across from Earth. The Sun is 400 times larger than the Moon, but it sits about 400 times farther away. That near-perfect ratio is what makes total solar eclipses possible. In a few hundred million years, the Moon will have drifted far enough that total eclipses no longer happen — only annular ones.
Worked example, the Moon: The Moon’s diameter is 3,474 km. Its average distance is 384,400 km.
θ ≈ 3,474 / 384,400 = 0.00904 radians θ × 57.296 = 0.518° ≈ 31 arcminutes
Worked example, Jupiter at opposition: Jupiter’s diameter is 139,820 km. At opposition it sits about 628,000,000 km from Earth.
θ ≈ 139,820 / 628,000,000 = 2.226 × 10⁻⁴ radians θ × 206,265 = 45.9 arcseconds
That is why even small telescopes show Jupiter as an actual disk with visible cloud bands, while Mars at the same magnification stays a tiny dot.
Worked example, finding distance from a known size: A galaxy you observed has an angular size of 0.5° and is known to be about 100,000 light-years across. How far is it?
θ (rad) = 0.5 × π / 180 = 0.00873 D = d / θ = 100,000 / 0.00873 ≈ 11.5 million light-years
This is exactly how astronomers estimate distances to galaxies whose physical sizes can be inferred from other clues, called “standard rulers.”
Practical reference values:
| Object | Angular size |
|---|---|
| Sun and Moon (from Earth) | 30 arcminutes (~0.5°) |
| Jupiter at opposition | 45 arcseconds |
| Saturn (rings included, opposition) | 40 arcseconds |
| Mars at perihelic opposition | 25 arcseconds |
| Andromeda Galaxy (M31) | 3° long axis |
| Orion Nebula (M42) | 1° wide |
| A 1-meter object at 1 km | 3.4 arcminutes |
| A 1-meter object at 100 km | 2 arcseconds |
| A penny at arm’s length | about 1° |
Field of view and astrophotography: A telescope eyepiece with a 50° apparent field of view at 200× magnification gives a 0.25° true field. That is enough for the Moon and Sun (just barely), but Andromeda would not fit. Picking eyepiece focal lengths and camera sensor sizes to match what you want to image is exactly an angular-size problem.