Planetary Phase Angle Calculator
The phase of a planet, how much of its disc appears sunlit, depends on the angle at the planet between the Sun and the observer. This is the phase angle. Given the three sides of the Sun-planet-observer triangle (the planet's distance from the Sun, its distance from Earth, and the Earth-Sun distance) the law of cosines fixes that angle exactly. From the angle, the illuminated fraction follows directly. This calculator returns both the phase angle in degrees and the illuminated fraction of the disc, useful for planning observations and understanding why Venus shows phases while Jupiter never does.
Phase angle and illuminated fraction formula
cos(alpha) = (r^2 + d^2 - R^2) / (2 * r * d)
alpha = arccos(...) converted to degrees
k = (1 + cos(alpha)) / 2
percent = k * 100
The law of cosines applies to the Sun-planet-observer triangle, taking the angle at the planet (vertex alpha) opposite the Sun-observer side R. The illuminated fraction k follows from the geometry of a sphere lit by a distant source and viewed from the observer direction.
Phase angle facts
- At phase angle 0 degrees the disc is fully lit (full phase); at 180 degrees it is dark (new phase).
- At phase angle 90 degrees exactly half the disc is illuminated, the same as a half moon.
- Inferior planets Mercury and Venus run through a full crescent-to-full cycle of phases.
- Superior planets stay nearly full because Earth orbits inside their orbits.
- The result is unitless in distance: any consistent length unit for r, d, and R gives the same angle.
Phase angle: frequently asked questions
What is the phase angle of a planet?
The phase angle is the angle measured at the planet between the direction to the Sun and the direction to the observer (usually Earth). A phase angle of 0 degrees means the planet is fully lit as seen from the observer (full phase); 180 degrees means it is unlit (new phase). It controls how much of the planet's sunlit hemisphere we can see.
How is the phase angle computed from distances?
The Sun, planet, and observer form a triangle. Given the Sun-planet distance r, the observer-planet distance d (the planet's distance from Earth), and the Sun-observer distance R, the law of cosines gives the phase angle alpha at the planet vertex: cos(alpha) = (r^2 + d^2 - R^2) / (2 r d).
What is the illuminated fraction?
The illuminated fraction k is the portion of the planet's apparent disc that is sunlit, ranging from 0 (new) to 1 (full). It relates to the phase angle by k = (1 + cos(alpha)) / 2. At alpha = 0, k = 1; at alpha = 90 degrees, k = 0.5 (half phase); at alpha = 180 degrees, k = 0.
Which units should I use for the distances?
Use any consistent length unit for all three distances; astronomical units (AU) are convenient for Solar System work. Because the formula uses ratios of squared distances, the result is unitless as long as r, d, and R share the same unit. The phase angle output is in degrees and the illuminated fraction is unitless.
Can superior planets ever appear as a thin crescent?
No. Superior planets (Mars, Jupiter, Saturn and beyond) never show a phase angle larger than a modest maximum because Earth orbits inside their orbits, so they always appear mostly full. Inferior planets (Mercury and Venus) can reach phase angles near 180 degrees and show full crescent-to-full cycles.
Official sources
- NASA Jet Propulsion Laboratory: Solar System Dynamics.
- U.S. Naval Observatory: Astronomical Applications Department.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.