Where the sun is, from two numbers
Solar position for a shading question needs no ephemeris. The declination — how far the sun sits off the celestial equator on a given day — swings between plus and minus 23.45 degrees over the year and is close enough for this purpose from a single sine of the day number. At solar noon the altitude is 90 degrees minus the difference between your latitude and that declination, and that is the whole calculation.
At 40 degrees north on the June solstice the declination is 23.45, so the noon altitude is 90 minus 16.55, which is 73.45 degrees. On the December solstice the declination is minus 23.45 and the altitude is 90 minus 63.45, or 26.55 degrees. The difference between those two is 46.9 degrees, and it is 46.9 degrees at every latitude on earth, because it is simply twice the tilt of the axis. Away from noon the same two numbers plus an hour angle of 15 degrees per hour give the altitude and the azimuth, which is what the hourly rows use.
The shadow, and why the height above the glass matters so much
A horizontal overhang of projection P throws a shadow down the wall of length P times the tangent of the sun altitude, divided by the cosine of how far round the sun has swung from facing the wall square on. At noon on an equator-facing wall that second term is one and it drops out. So a 3 ft overhang at 40 degrees north throws 3 times tan 73.45, which is 10.1 ft of shadow in June, and 3 times tan 26.55, which is 1.5 ft, in December.
| Date at 40°N | Noon altitude | Shadow from a 3 ft projection | 5 ft window, head 1 ft below | Projection for full shade |
|---|---|---|---|---|
| Jun 21 | 73.45° | 10.10 ft | 100% shaded | 1.78 ft |
| Aug 16 | 63.45° | 6.01 ft | 100% shaded | 3.00 ft |
| Sep 21 | 49.80° | 3.55 ft | 51% shaded | 5.07 ft |
| Dec 21 | 26.55° | 1.50 ft | 10% shaded | 12.01 ft |
The last column is the argument. Sizing on the June solstice asks for 1.78 ft of roof. Sizing on the middle of August asks for 3.00 ft, which is 69 percent more. Sizing on the September equinox asks for 5.07, and sizing on December asks for twelve feet of projection over a five foot window, which is not an overhang, it is a carport.
The head clearance is the cruel term. That first foot of shadow lands on the wall above the glass and does nothing. On the December row the whole shadow is 1.5 ft, so after the foot above the head only six inches of it reaches the window — ten percent. Move the same overhang down so it sits three inches above the glass instead of a foot and the December figure goes from 10 percent to 25, and the September figure from 51 percent to 66. That is why a fabric awning bolted just above the head beats an eave with the same projection two feet higher, and it is the cheapest change available on this page.
Late summer is the case, not midsummer
Sizing an overhang on the June solstice is sizing it for the day the sun is highest and therefore the day it is easiest to shade. The hot part of the year runs weeks behind the sun, because the ground and the building take time to warm up. By the middle of August at 40 degrees north the noon sun is down to 63.45 degrees while the afternoons are still at their worst, and the shadow from a 3 ft projection has shrunk from 10.1 ft to 6.01. Run the calculator on day 228 rather than day 172 and the answer it gives is the one you will live with.
The symmetry that makes this awkward is that the sun angle on 16 August is the same as on 26 April, when nobody wants shade. A fixed overhang treats those two days identically. Deciduous planting and a retractable awning both exist because they do not, and that is the honest argument for either of them over more roof.
What it does not cover
Everything above is direct beam on an equator-facing wall. East and west windows are the ones that actually overheat a house, and an overhang does almost nothing for them: the summer sun rises and sets well north of east and west, comes in low and near horizontal, and looks straight under any horizontal projection. Vertical fins, planting, interior blinds and glass with a low solar heat gain coefficient are what work on those elevations. If the answer is going to be the glass, the window film calculator and the window U-factor payback calculator are the pages for it, and window glass area gives you the square footage to feed them.
For a slatted cover rather than a solid one, the pergola shade calculator takes a sun altitude and works out how much gets through the gaps, which is a different mechanism and pairs with this page rather than competing with it: read the altitude off here, put it in there. Panel tilt for the same latitude is on the solar panel tilt calculator, and if the structure doing the shading is a patio cover, its own height and slope are on the patio cover headroom calculator.
Questions people ask
How deep should an overhang be for a south window?
It depends on your latitude, on how far the overhang sits above the glass, and on which date you decide counts. Rather than a rule of thumb, put the numbers in and read the projection the target asks for. At 40 degrees north, a 5 ft window whose head is a foot below the overhang needs about 1.78 ft of projection to be fully shaded at noon on the June solstice, and about 3.00 ft to be fully shaded at noon in the middle of August. That is the size of the disagreement between the two dates people quietly substitute for each other.
Why does my overhang stop working in September?
Because the sun altitude at the equinox is exactly your co-latitude — 50 degrees at 40 north — and a shadow at 50 degrees is barely a third as long as one at 73. September also arrives when the house is still warm and the afternoons still are. The equinox is the hinge of the whole problem: a fixed overhang shades a lot before it and very little after it, and there is no projection that behaves differently in September from March, because the sun is in the same place on both.
Does this work for an east or west window?
No, and it is worth being blunt about it. The calculation assumes the wall faces the equator, so the sun swings symmetrically past it and is high when it is in front. An east or west wall gets the sun low and nearly square on, in the early morning and late afternoon, when the altitude is small and the shadow from a horizontal overhang is correspondingly tiny. West glass on a summer afternoon is the hardest gain in a house to control with any kind of roof, and shading it takes fins, planting, exterior blinds or different glass.
What is the difference between solar time and my clock?
Solar noon is the instant the sun crosses your meridian. Clock noon is a civil convention covering a time zone that can be over a thousand miles wide, so the two can differ by most of an hour from longitude alone, plus up to about a quarter hour from the equation of time as the year turns, plus a whole hour whenever daylight saving is in force. The hourly rows here are in solar time deliberately. To convert roughly, find when the sun is highest on a clear day and call that 12:00.
Does shading the window stop the heat getting in?
It stops the direct beam, which is the large part, and nothing else. Diffuse light from the whole bright sky and light reflected off pale paving right outside still arrive at the glass and still carry the solar heat gain coefficient with them, and on a hazy summer day the diffuse component alone can be a third of the total. The heat figures on this page are direct beam only and should be read as the amount you removed rather than the amount left. They also say nothing about conduction through the frame and glass, which is a separate calculation on a separate page.