Angular Size Calculator
Find how big an object looks in the sky from its true size and distance, with θ = 2·arctan(size ÷ (2·distance)).
Size and distance must share the same unit (km, m, ly…). The ratio is dimensionless, so the angle does not depend on which unit you pick — only on the proportions.
0.5178° · 31.07′ · 1864.1″ (0.0090374 rad)
Angular size is how large an object looks on the sky, set by its true size and distance: θ = 2·arctan(size ÷ (2·distance)). The full Moon (3474 km across, 384,400 km away) spans about 0.52° — roughly 31 arcminutes, or half the width of your little fingernail held at arm’s length.
What angular size means
Angular size (or angular diameter) is the angle an object subtends at your eye — the opening between lines drawn to its two opposite edges. A distant lorry and a nearby coin can look the same size because angular size depends only on the ratio of an object’s true size to its distance, not on either quantity alone. Move twice as far away and the object looks half as big; make it twice as large and it looks twice as big. Astronomers report it in degrees, arcminutes (′), and arcseconds (″) because celestial objects span a huge range of apparent sizes.
size and distance in the same unit; θ in radians, then × 180 ÷ π for degrees, × 60 for arcminutes, × 3600 for arcseconds
Worked example
How big does the full Moon look? Its diameter is about 3474 km and its mean distance is about 384,400 km.
- 1 Write the formula. θ = 2·arctan(size ÷ (2·distance)), with size and distance in the same unit.
- 2 Substitute the values. θ = 2·arctan(3474 ÷ (2 × 384,400)).
- 3 Simplify the ratio. size ÷ (2·distance) = 3474 ÷ 768,800 ≈ 0.004519.
- 4 Take the arctangent. arctan(0.004519) ≈ 0.004519 rad, so θ ≈ 0.009037 rad.
- 5 Convert to degrees and arcminutes. θ ≈ 0.009037 × 180 ÷ π ≈ 0.518°, and 0.518° × 60 ≈ 31.1 arcminutes.
Apparent sizes of familiar objects
Typical angular diameters seen from Earth; planets vary with orbital distance.
| Object | Angular size | Notes |
|---|---|---|
| Full Moon | ≈ 31′ (0.52°) | Varies ≈ 29.4′–33.5′ with the Moon’s distance |
| Sun | ≈ 32′ (0.53°) | Almost identical to the Moon — hence total eclipses |
| Jupiter | ≈ 30″–50″ (≈ 40″) | Largest near opposition when closest to Earth |
| Venus | ≈ 10″–66″ | Swells dramatically as it nears Earth |
| A 1° gap | 2 full Moons side by side | Your little fingernail at arm’s length ≈ 1° |
Why the Sun and Moon look the same size
The Sun is about 400 times wider than the Moon, but it also sits about 400 times farther away. Those two factors nearly cancel, so both span roughly half a degree on the sky. That coincidence is why the Moon can just cover the Sun’s disc during a total solar eclipse — and why, when the Moon is near the far point of its orbit and looks slightly smaller, we get an annular “ring of fire” eclipse instead.
For the small angles common in astronomy, the formula simplifies to the small-angle approximation: θ ≈ size ÷ distance (in radians). At the Moon’s scale this differs from the exact arctangent by only a few parts in a million, so it is used constantly for quick estimates. It breaks down for large angles, where the full θ = 2·arctan(size ÷ (2·distance)) is needed.