Angular Size: The Sky’s Ruler
Geometry
$\theta$
$\theta$ (decimal)
deg
Diameter
Distance
Sky View
- Nearer objects look larger even if they are smaller.
- Different (size, distance) pairs can look the same.
- Exact geometry: $$\theta = 2\tan^{-1}\left(\frac{D}{2d}\right)$$
- $D$ is physical diameter (km). $d$ is distance from the observer to the object (km).
- Small-angle approximation (when $D \ll d$): $$\theta \approx \frac{D}{d}\ \text{(radians)}$$ $$\theta_{\mathrm{deg}} \approx \frac{180}{\pi}\,\frac{D}{d}$$
- Units: internal distance/diameter are always km; $\theta$ may display in $^\circ$ / $^\prime$ / $^{\prime\prime}$.
- Diagram is visually scaled; numeric readouts are authoritative.