8 minute lesson · Mathematics · Original foundation material
Before you start: Ratios; Powers of ten; Degrees and circles.
An object's physical size tells you how wide it is. Its angular size tells you how wide it looks from your position. A small nearby object can cover the same angle as a large distant object.
Start with a radian
An angle in radians equals arc length divided by circle radius: θ = s/r. A complete circle has arc length 2πr, so 360° = 2π radians and 180° = π radians.
To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. One degree contains 60 arcminutes, and one arcminute contains 60 arcseconds.
Use the small-angle rule
For a distant object of transverse width D at distance d, its small angular width is approximately θ = D/d, with θ in radians. D and d must use the same length unit. The approximation improves as D/d becomes smaller.
For a flat object centred perpendicular to the line of sight, the exact angle is 2 arctan(D/(2d)). The simpler ratio works because a narrow viewing triangle has almost the same small-angle behaviour as a short circular arc.
A worked example
A distant object is 900 km wide and 300,000 km away. Estimate its angular width in radians and degrees.
- Both lengths are in kilometres, so their ratio needs no unit conversion.
- Use the small-angle formula: θ ≈ 900/300,000 = 0.003 radians.
- Convert to degrees: 0.003 × 180/π ≈ 0.1719°.
- Round sensibly: the object is about 0.172° wide, or 10.3 arcminutes.
0.003 radians, approximately 0.172°
Check your understanding
If the same small object moves three times farther away, what happens to its angular width?
Show the explanation
It becomes approximately one-third as large because θ ≈ D/d and D has stayed fixed.