By Kevin Beck
Updated Mar 24, 2022
The Sun is the cornerstone of all life on Earth, yet its apparent size in the sky can be a surprising puzzle. In this concise guide, we break down the geometry behind the Sun’s angular diameter, show you how to calculate it, and compare it to the Moon’s view from Earth.
The Earth orbits the Sun at an average distance of roughly 93 million miles (150 million kilometers). The Sun’s diameter is about 870,000 miles (1.4 million kilometers), nearly 100 times wider than our planet. Light from the Sun takes approximately eight minutes to reach Earth—an instant pause if the Sun were to vanish suddenly.
Angular diameter is the angle an object subtends at an observer’s eye. It is measured in degrees (°) or radians (rad). One full circle contains 360° or 2π rad, so 1 rad ≈ 57.3°.
For example, a dome that stretches from the zenith directly overhead to the horizon on either side subtends a 90° angle (π/2 rad), filling half of your visual field. Angular diameter is not a fixed property of an object; it varies with distance.
The formula for an object’s angular diameter (α) when you know its physical diameter (D) and its distance from you (r) is:
α = 2 arctan (D ⁄ 2r)
Using the Sun’s values (D = 1.4 × 10⁹ m, r = 1.5 × 10¹¹ m):
α = 2 arctan (1.4 × 10⁹ ⁄ (2 × 1.5 × 10¹¹)) = 2 arctan (0.0047) ≈ 2 × 0.270° ≈ 0.54°
Thus the Sun covers about half a degree of sky—roughly 1/360th of the 180° hemisphere above us.
Despite the Moon’s diameter being about 400 times smaller than the Sun’s, it sits roughly 400 times closer to Earth. This geometric coincidence makes the Moon and Sun appear nearly the same size to us, a fact that explains the mechanics of total solar eclipses.
