Small-Angle Formula (Angular Size) Calculator
Find the angular size of a distant object, or work backward to find its physical size or distance, using the small-angle approximation.
d ↑
Angular size grows directly with physical size — a bigger object at the same distance always looks proportionally larger.
D ↑
Angular size shrinks inversely with distance — twice as far away means half the apparent size.
Why 206,265?
206,265 is the number of arcseconds in one radian (180 × 3600 / π), the constant that converts the ratio d/D directly into arcseconds.
Small-angle limit
The approximation assumes d ≪ D (θ well under a few degrees). For large apparent sizes, use exact trigonometry (θ = 2·arctan(d/2D)) instead.
How to Use This Calculator
The small-angle formula links three quantities: how big an object physically is (d), how far away it is (D), and how big it appears in the sky (θ).
1. Pick a length unit
Choose km, AU, light-years, parsecs, or miles — d and D always share the same unit.
2. Set the physical size
Use a preset (planet or Sun diameter) or enter a custom value for d.
3. Set the distance
Use a preset (Moon, 1 AU, nearest star) or enter a custom value for D.
4. Choose what to solve for
Solve for angular size directly, or enter a target θ and solve for the physical size or distance needed.
5. Pick a θ display unit
View the angular size in arcseconds, arcminutes, degrees, or milliarcseconds.
6. Switch to Advanced
See sensitivity to each input and θ converted into every unit at once.
What This Tool Offers
Built around the small-angle formula used throughout astronomy, optics, and surveying to relate apparent size to physical size and distance.
Solve any direction
Find angular size from known geometry, or work backward to the physical size or distance from a target angular size.
Realistic presets
Planet and Sun diameters, plus common astronomical distances from the Moon out to the nearest star.
Flexible length units
Switch between kilometers, astronomical units, light-years, parsecs, and miles for d and D together.
Transparent step-by-step math
Every calculation shows the ratio and the 206,265 conversion explicitly.
Built-in sensitivity check
See exactly how much a ±1% change in size or distance moves the angular size.
Multi-unit angle display
Advanced mode shows θ simultaneously in arcseconds, arcminutes, degrees, and milliarcseconds.
What Is the Small-Angle Formula?
The small-angle formula approximates the apparent (angular) size of an object as seen from a distance, using the fact that for small angles, the arc length of a circle is approximately equal to the straight-line chord. It converts the simple ratio of an object's physical size to its distance directly into an angle, using a fixed constant derived from the geometry of a circle.
Formula and Variables
| Symbol | Meaning | Notes |
|---|---|---|
| θ | Angular (apparent) size | In arcseconds by default; also shown in arcmin, degrees, mas |
| d | Physical linear size of the object | Must use the same unit as D |
| D | Distance from observer to object | Must use the same unit as d |
| 206,265 | Arcseconds per radian | = 180 × 3600 / π |
Because d and D always appear as a ratio, any consistent length unit works for both — the constant 206,265 does the work of converting that dimensionless ratio into an angle in arcseconds.
Worked Example
θ = 206,265 × (3,474 / 384,400)
θ = 206,265 × 0.009036 ≈ 1,864 arcsec ≈ 31.1 arcmin ≈ 0.518° — matching the Moon's real apparent size
Frequently Asked Questions
It relies on the approximation that, for a small angle in radians, the angle itself, its sine, and its tangent are all nearly equal. That approximation degrades as the angle grows — by a few degrees the error becomes noticeable, and the exact relation θ = 2·arctan(d/2D) should be used instead.
No. Angular size describes how large an object looks — the physical angle it spans in the sky. Apparent brightness describes how much light reaches the observer, which follows an inverse-square law with distance and depends on the object's luminosity, not its physical size.
It underlies stellar parallax distance measurements, estimating planet or star diameters from telescope-measured angular sizes, calculating a telescope's angular resolution limits, and converting between angular and physical separations of binary star systems.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| θ | Angular size | arcsecond |
| d | Physical diameter | same units as D |
| D | Distance to object | same units as d |
What it means
The small‑angle formula converts the physical size (d) of an object at distance D into its angular size (θ) in arcseconds, using the factor 206265 (the number of arcseconds in a radian). This is used extensively in astronomy: to calculate the diameter of the Moon, planets, nebulae, and galaxies from their angular sizes. It also applies to telescope resolution and to calculating apparent sizes of objects. Understanding this formula helps astronomers interpret images and plan observations to resolve fine details.
Worked example
| Parameter | Value |
|---|---|
| d (km) | 14,000 |
| D (km) | 778,000,000 |
Common mistakes
- Small‑angle formula: θ (arcsec) = 206265 × (d / D) – where d is linear size, D is distance.
- Constant 206265: The number of arcseconds in 1 radian (180×3600/π).
- Units: d and D must be in the same units (e.g., both km, both AU).
- Valid for small angles: θ must be small (< a few degrees) – for larger angles, use exact trigonometry.
- Applications: Used to compute angular sizes of celestial objects.
Applications
The small‑angle formula, θ (arcsec) = 206265 × (d/D), relates the physical size (d) of an object to its angular size (θ) and distance (D). This is used in astronomy to calculate the physical dimensions of planets, nebulae, galaxies, and other objects from their angular sizes and known distances. It is also used in telescope resolution calculations and in satellite imaging. By applying this formula, astronomers can estimate the true size of celestial bodies, which is crucial for understanding their composition and dynamics. This simple trigonometric relation is a workhorse in observational astronomy and remote sensing.
- Determination of planetary radii and comet nucleus sizes
- Estimation of galaxy and nebula dimensions
- Calculation of spacecraft imaging target sizes
- Resolution and field‑of‑view planning for telescopes
- Education and public outreach (e.g., Moon angular size)
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