Sun Path and Shading Calculator

How this is calculated

Three pieces of geometry do all the work. Declination for day n of the year is −23.45° × cos(360° × (n + 10) ÷ 365), the form used by PVEducation at the University of New South Wales, and the hour angle is 15° for every hour away from solar noon.

Elevation follows from those two and your latitude: sin(elevation) = sin(declination) × sin(latitude) + cos(declination) × cos(latitude) × cos(hour angle). Bearing comes from the same three numbers through an arc-tangent, clockwise from north, so east is 90° and south is 180°.

Your horizon becomes a second curve on the same axes: one angle per direction, arc-tangent of height divided by distance, joined by straight lines. Only the ratio matters, so feet against feet and meters against meters give the same answer.

Between the five directions the profile runs straight, and north of due east and due west it is treated as clear sky. A treeline that covers three directions needs a height and a distance in all three, or the page will draw a notch that is not there.

The page then steps through every month at five-minute intervals, asks whether the sun sits above or below your profile, and weights each step by clear-sky beam on your tilted plane — Kasten and Young air mass, the 1.353 × 0.7^{AM^0.678} direct-beam form, ten percent added for diffuse.

Everything here is a ratio against a cloudless sky, which is what makes it portable. For absolute yield, take the ratio to the peak sun hours atlas, then size the system with the off-grid solar sizing calculator.

Why due south is the expensive direction

Winter sun lives in a narrow band of bearings. At 40°N in late December it rises at 121°, crosses due south at 26.6° and sets at 239°, spending the strongest hours of that short day within thirty degrees of south.

An obstruction standing higher than the December noon elevation therefore does not clip the day. It deletes the middle of it. The same object east or west shaves the ends, where a long path through the atmosphere has already thinned the light.

Energy lost against a clear horizon, clear-sky beam plus diffuse model, 40°N, array tilted 40° and facing due south. The south columns are that angle typed into the southeast, south and southwest boxes of the calculator above; the east columns into the east and southeast boxes. The page joins the entered directions with straight lines, so the blocked band is wider than the flat section between them.

Horizon height	South side, year	South side, December	East side, year	East side, December
10°	−1%	−5%	−1%	−3%
15°	−3%	−16%	−2%	−6%
20°	−6%	−32%	−4%	−10%
25°	−12%	−63%	−6%	−14%
30°	−22%	−97%	−9%	−17%
35°	−27%	−97%	−12%	−19%

Notice the cliff between 25° and 30°. December noon at 40°N is 26.6°, and once the southern horizon passes that line the month collapses to whatever diffuse light gets through. Two degrees of tree either side of that threshold outweigh any equipment choice.

December noon sun elevation by latitude

This is the number the site question turns on. At the December solstice the noon sun stands 90° minus your latitude minus 23.44°, and the shadow it throws is one divided by the tangent of that angle, times the height of the object. Click a heading to sort; the calculator’s unit selector switches the clearance column.

Geometry only, for the December solstice with a declination of −23.44°. Daylight is the geometric figure for the sun’s center with no refraction correction, so published sunrise and sunset tables run a few minutes longer. Bearings are clockwise from north.

Latitude °N	Near this latitude	Noon sun °	Sunrise bearing	Shadow length	Clear ground for a 30 ft object	Daylight h
25	Key West, Florida	41.6	116	1.1×	34 ft	10.4
28	Tampa, Florida	38.6	117	1.3×	38 ft	10.2
30	Houston, Texas	36.6	117	1.3×	40 ft	10.1
32	Savannah, Georgia	34.6	118	1.5×	44 ft	9.9
34	Los Angeles, California	32.6	119	1.6×	47 ft	9.7
36	Las Vegas · Málaga	30.6	119	1.7×	51 ft	9.6
38	Sacramento · Athens	28.6	120	1.8×	55 ft	9.4
40	Denver · Madrid	26.6	121	2.0×	60 ft	9.2
42	Boston · Rome	24.6	122	2.2×	66 ft	8.9
44	Bend, Oregon · Bordeaux	22.6	124	2.4×	72 ft	8.7
46	Bismarck · Geneva	20.6	125	2.7×	80 ft	8.4
48	Seattle · Munich	18.6	126	3.0×	89 ft	8.2
50	Prague · Frankfurt	16.6	128	3.4×	101 ft	7.9
52	Berlin · Amsterdam	14.6	130	3.9×	116 ft	7.5
54	Belfast · Hamburg	12.6	133	4.5×	135 ft	7.1
56	Copenhagen · Edinburgh	10.6	135	5.4×	161 ft	6.7
58	Juneau · Gothenburg	8.6	139	6.6×	199 ft	6.1
60	Seward, Alaska · Oslo	6.6	143	8.7×	261 ft	5.5

June noon sits 46.9° higher at every latitude, so a site that looks generous in July can be a different site on the shortest day — the month an off-grid bank is sized around, as the winter design month guide argues from the panel’s side.

The tree that grows

A horizon survey is a photograph of one afternoon, and trees are the part of it that moves. Run the numbers again with the treeline twenty years taller and see which side of the December threshold the site lands on. If it is the wrong side, the choice is made now: move the array, take the trees, or plan the generator hours.

• A slope helps or hurts twice: ground falling away to the south lowers the horizon angle and buys you years of tree growth.
• Deciduous cover still costs you in the dark months, because bare branches and trunks pass only part of the beam.
• Anything you cannot cut, you must design around. A neighbor’s ridge is a permanent input.

What a shaded cell does to a string

Cells wired in series all carry the same current, so the string runs at the pace of its weakest member. PVEducation states the extreme case flatly: completely shading one cell drops the whole module’s output to zero, and the power the good cells are still making gets dissipated as heat in the shaded one.

A bypass diode is the release valve. It gives current a path around a group of cells that has stopped pulling its weight, so the module loses that group instead of everything, and the shaded cell stops cooking. PVEducation puts the safe limit near fifteen cells per diode for silicon, which is why a 36-cell module carries two, and why a full-size module today splits its cells into a handful of protected groups rather than one.

Where a shadow is unavoidable and predictable, run the string along it rather than across it, so the loss lands in one string instead of spreading through every string.

Strings against module-level electronics

A plain string inverter or charge controller tracks one maximum power point for the whole string. Under partial shade that single operating point is a compromise, and the array can settle on a local peak that leaves real power behind. Microinverters and DC optimizers give every module its own tracker, so the shadow costs the shaded module rather than its neighbors.

What they cannot do is return sunlight the horizon took. Where the December sun never rises above the treeline, no electronics recover the month; only a different array position, a shorter treeline or more panel does.

Off-grid adds a wrinkle. Module-level electronics put parts on the roof, where repairs happen in winter, against a plain string you can meter with a clamp from the ground. Size the strings themselves with the string sizing calculator.

The survey a phone can do

Stand at the lowest corner of the proposed array, since that is the point that loses light first and everything measured there is conservative for the rest. Use the phone’s compass for the bearing, then hold it edge-on as an inclinometer and read the angle up to the skyline. Eight bearings 45° apart is plenty; sixteen is better where a tree or chimney sits close.

Work to true north rather than magnetic, and cross-check at midday: a vertical stick’s shortest shadow of the day points true north in the northern hemisphere.

The resulting list of angles is a standard input. The European Commission’s PVGIS accepts a plain text file of horizon heights in degrees, one per line, clockwise from north, so the same numbers feed straight into a full irradiation model afterwards.

Quick answers

How much energy does shading actually cost a solar array?

It depends almost entirely on direction. At 40°N a horizon 30° high across the southeast, south and southwest removes 22 percent of the clear-sky year and 97 percent of December; the same angle east and southeast removes 9 percent and 17 percent.

How tall can a tree be before it shades my panels?

Work backwards from the December noon shadow: 2.0 times the object’s height at 40°N, 3.0 times at 48°N, 3.9 times at 52°N, 8.7 times at 60°N.

Does one shaded panel really drag down the whole string?

Series cells share a current, so the worst cell sets the pace. Bypass diodes limit the damage to a group of cells rather than the module or the string.

Do microinverters or DC optimizers fix a shaded site?

They stop one shaded module dragging its neighbors down. They cannot put back sun the horizon has taken, so a bad December stays bad.

How do I survey my horizon without special equipment?

Phone compass for the bearing, phone inclinometer for the angle, taken from the lowest edge of the proposed array, at eight bearings 45° apart.

Sources

PVEducation, University of New South Wales: declination, elevation and azimuth equations, Kasten and Young air mass, the 1.353 × 0.7^{AM^0.678} direct-beam form, bypass diodes and shading. NOAA Global Monitoring Laboratory solar calculator, after Meeus, Astronomical Algorithms, for position checks. Aron P. Dobos, PVWatts Version 5 Manual, National Renewable Energy Laboratory, 2014, Table 6: the 3 percent default shading loss stands for horizon blocking by distant features, and sites shaded by nearby trees or structures need a survey instead. PVGIS user manual, European Commission Joint Research Center, for the uploaded horizon format. Checked September 2026.

Go deeper

The horizon sets what is available; the array and the bank decide what you keep. Run the system through the off-grid solar sizing calculator, set the angle with the tilt angle calculator, look up the local resource in the peak sun hours atlas, and see why the darkest month sets the size in how to size an off-grid solar system.