Sun and Shadow Planner

How the sun's path changes with the season and with latitude

The sun's path is set by two numbers: your latitude, which does not change, and the sun's declination, which does. Declination is the latitude the sun is directly over at noon. According to NOAA's solar calculator glossary, it "varies from -23.44° at the (northern hemisphere) winter solstice, through 0° at the vernal equinox, to +23.44° at the summer solstice." That swing of about 23.44° either way is the tilt of the Earth's axis, and it is the whole reason a patio that bakes in July sits in shade all December.

For anywhere north of the tropics, the sun's height at solar noon follows a simple rule:

noon altitude = 90° − your latitude + the sun's declination

PVEducation, Christiana Honsberg and Stuart Bowden's online solar energy textbook, gives the same rule in its own form. South of the equator, swap the sign of the declination and the sun is to the north at noon instead of the south. Here it is for London, at 51.51° N, and Phoenix, at 33.45° N (coordinates from GeoNames):

The planner shows the same six figures to a tenth of a degree.

It adds the small lift the atmosphere gives a low sun (refraction, modeled as NOAA does), which is why London's midwinter figure is 15.11° rather than the rule's 15.05°. The difference between the two cities is 18 degrees of latitude, and it shows up as 18 degrees of extra sun height in Phoenix at noon on every day of the year.

The season changes the length of the day as well as the height of the sun. In London the planner gives 16 h 39 min of daylight on the June solstice and 7 h 50 min on the December one; in Phoenix, 14 h 22 min and 9 h 56 min. The equinox is not quite 12 hours anywhere (London gets 12 h 10 min) because sunrise and sunset are counted when the top edge of the sun meets the horizon, with 0.833° allowed for the sun's size and refraction, as NOAA's calculator does.

Direction matters as much as height. On the June solstice the sun rises in the northeast in both cities (azimuth 49° in London, 61° in Phoenix) and sets in the northwest, so a wall facing north still gets early morning and late evening sun in summer. In December it rises in the southeast and sets in the southwest. How the sun moves across your garden goes through what that means for each side of a house.

Why shadows are longest in winter

A shadow on flat ground is the height of the thing casting it divided by the tangent of the sun's altitude. The higher the sun, the shorter the shadow, and with the sun below 45° the shadow is longer than the object is tall. That is the "shadow of a 1 ft post" figure in the planner: the ratio holds for any height, in feet or meters.

London's midwinter noon shadow is seven times its midsummer one. The other half of the story is the hour: the sun moves 15° of hour angle every hour, 360° in 24 hours (PVEducation, solar time), and through a summer afternoon it sinks as well as swinging west. In London on 21 June the sun is 46° up from the WSW at 4 pm, so an umbrella canopy 2.3 m (7 ft 7 in) high throws its shade 2.2 m (7 ft 3 in) to the ENE of itself; by 7 pm the sun is 18.5° up in the WNW and the same shade lands 6.9 m (22 ft 6 in) away, probably on the neighbor's lawn. Phoenix in June looks much the same: 43° at 4 pm, 19° at 6 pm. Low evening sun from the west is the shade problem most patios have, and the hardest for an umbrella to reach, which is where a side panel, a fence or a tree earns its place (shade for hot climates covers the options).

What to read off the plan

The plan is drawn from above with your house wall along the top. Set the way the wall faces with the dial (stand with your back to the wall and look out: that direction), and the north arrow and the small sun turn to match. Then set the patio's size, add what casts the shade, and drag it into place. Each object's shadow is worked out point by point: every corner or edge point is pushed along the ground, away from the sun, by its height divided by the tangent of the sun's altitude, and the shadow is the outline around those points.

• Umbrella. The shade sits off to the side of the canopy, away from the sun. The line under the plan says how far, and in which direction to move the umbrella to put the shade on your chairs. Run the day with the play button to see how far it walks. Choosing a patio umbrella covers sizes and tilting canopies.
• Shade sail. Drag the four corners to where you would fix them and give each its own height. The planner draws straight edges between the corners; if your sail's edges are cut with an inward curve, its shade is a little smaller than drawn. See how to install a shade sail.
• Awning. Give its width, how far it projects from the wall, its height at the wall and its pitch. The line under the plan gives how far out from the wall the shade reaches on the ground; when the sun is low in front of the house it says the shadow is on the wall instead. Compare the kinds in types of awnings and what they cost and retractable vs fixed awnings.
• Fence or wall, and tree. A fence's shade is a band on the side away from the sun. A tree is drawn as a solid crown on a trunk; a real canopy lets some light through, so treat its shade as the most you would get.
• The house. With a wall height set, the house shades the patio whenever the sun is behind it. The house is taken to be as wide as the patio.

The figures under the plan give how much of the patio is in shade, in square feet or square meters and as a share, and, with "Measure a seating area" ticked, the share of the rectangle you drag over your table. The shade is measured on a fine grid over the patio, which is accurate to within about half a percent of the patio, and overlapping shadows count once. Everything assumes flat, level ground. To weigh a sail against an awning or a pergola for your own patio, see shade sail vs awning vs pergola, the louvered pergola guide and, for a garden facing the sun all day, the best shade for a south-facing garden. Shading glass rather than a patio is covered in shading windows to cut cooling costs.

How accurate is it?

The sun's position comes from NOAA's solar position equations, the ones behind the NOAA Global Monitoring Laboratory's Solar Calculator, which follow Jean Meeus's Astronomical Algorithms. NOAA says its sunrise and sunset results "are theoretically accurate to within a minute for locations between +/- 72° latitude, and within 10 minutes outside of those latitudes" (calculation details). Here is the planner beside NOAA's own calculator for four cases:

The planner's sunrise, solar noon and sunset agree with NOAA's within a minute in all four, and the same code is checked against the US Naval Observatory's figures for places from Singapore to Tromsø. Clock times use your place's own time zone, daylight saving included. Everything runs in your browser: nothing is sent anywhere, and your last plan is kept only on this device.

Sources: NOAA Solar Calculator, its calculation details and glossary (the sun's position, refraction, the 23.44° declination); PVEducation on the elevation angle and solar time; GeoNames (place coordinates, CC BY 4.0); US Naval Observatory (test figures). Day lengths, sunrise directions and shadow lengths above are the planner's own, from NOAA's equations.