Distance & Parallax

Section 3 of 7

Small-Angle Geometry

Part 3: Geometric Foundations — The Small-Angle Approximation

We need a relationship connecting angular size, true size, and distance.

The Geometric Picture

Consider an observer at distance $d$ from a small object of true size $s$ , which subtends an angle $\alpha$ . Drawing the triangle, for small angles the arc subtended by the object’s edge is approximately its true size, and the arc’s radius is the distance.

Basic viewing geometry with observer A at left and object segment BC at right. Rays from A to B and C define an angle alpha, with distance d to the object centerline and object size s shown on the right vertical segment. — Figure 4Angular size comes from geometry: the same physical size s subtends a smaller angle alpha at larger distance d.Fundamentals of Astrophysics (Owocki)

By the definition of angle in radians, $\alpha \approx \text{arc length}/\text{radius} \approx s/d$ . This is the small-angle approximation:

Angular size is inversely proportional to distance: a distant object appears small, a nearby one large. The angle $\alpha$ must be in radians for the formula to work without a conversion factor.

Small-angle approximation

For small angles ( $\alpha \ll 1$ rad), $\sin\alpha \approx \tan\alpha \approx \alpha$ , giving the clean geometric relation $s = \alpha d$ between physical size, angular size, and distance.

Numeric answer

The Moon subtends about $0.5^\circ$ ; its distance is $\approx 3.84 \times 10^{10}$ cm. Estimate its true diameter.

Your answer

The approximation holds while $\alpha \lesssim 0.1$ rad (roughly 5–6 degrees); beyond that, use the exact trigonometry. In astronomy we rarely exceed a degree, so it’s superb for parallax, galaxy sizes, and telescope resolution.