Angular Size Calculator Formula
Understand the math behind the angular size calculator. Each variable explained with a worked example.
Formulas Used
Angular Size (radians)
angular_rad = diameter / distanceAngular Size (degrees)
angular_deg = (diameter / distance) * (180 / pi)Angular Size (arcmin)
angular_arcmin = (diameter / distance) * (180 / pi) * 60Angular Size (arcsec)
angular_arcsec = (diameter / distance) * (180 / pi) * 3600Variables
| Variable | Description | Default |
|---|---|---|
diameter | Physical Diameter(km) | 3474 |
distance | Distance to Object(km) | 384400 |
How It Works
Angular Size
The angular size (angular diameter) of an object as seen from a given distance.
Small-Angle Approximation
θ ≈ d / D (radians)
where d = physical diameter, D = distance. Multiply by 180/π for degrees, then by 60 for arcminutes or 3600 for arcseconds.
Worked Example
The Moon: diameter 3 474 km, distance 384 400 km.
- 01θ = 3474 / 384400 = 0.009038 rad
- 02Degrees: 0.009038 × 57.296 = 0.5178°
- 03Arcminutes: 0.5178 × 60 = 31.07'
- 04The Moon is about half a degree across.
Frequently Asked Questions
Why does the Moon appear the same size as the Sun?
The Sun is about 400 times the Moon's diameter and about 400 times farther away, so their angular sizes nearly match (≈0.5°).
When does the small-angle formula fail?
For objects subtending more than a few degrees. Use 2 arctan(d/2D) for the exact result.
What is the angular size of Jupiter from Earth?
At closest approach, about 50 arcseconds. At most, it appears as a tiny disc in binoculars.
Ready to run the numbers?
Open Angular Size Calculator