Calculate horizontal shadow angle from solar and surface azimuth, or solve vertical shadow angle from HSA and solar altitude in degrees or radians.

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Azimuths run clockwise from north. Surface azimuth is the outward wall normal. Profile-angle modes assume sunlight in front of a vertical facade.

Horizontal Shadow Angle Formula

The following formula is used to calculate the horizontal shadow angle (HSA) from the solar azimuth angle and the surface (facade/window) azimuth angle, using the same azimuth reference and sign convention for both angles.

HSA = ((Azโ‚›แตคโ‚™ - Azsurface + 180)bmod 360) - 180

Variables:

  • HSA is the horizontal shadow angle in degrees (commonly reported in the range โˆ’180ยฐ (included) to 180ยฐ (excluded))
  • Azsun is the solar azimuth angle in degrees (commonly 0โ€“360ยฐ)
  • Azsurface is the azimuth of the outward normal to the facade/window in degrees, measured clockwise from north (0โ€“360ยฐ)

To calculate the horizontal shadow angle, subtract the surface azimuth from the solar azimuth and then normalize (wrap) the result into the range โˆ’180ยฐ (included) to 180ยฐ (excluded). No trigonometry is required for HSA; just keep both azimuths in the same convention (same reference direction and clockwise/counterclockwise sign).

What is a Horizontal Shadow Angle?

The horizontal shadow angle (HSA) is the angle, measured in the horizontal (plan) view, between the direction a surface faces (its azimuth) and the direction to the sun (solar azimuth). In building and solar shading applications, HSA is commonly treated as the azimuth offset between the sun and the facade/window, and it is used when designing vertical fins and other shading devices.

How to Calculate Horizontal Shadow Angle?

The following steps outline how to calculate the Horizontal Shadow Angle.

  1. First, determine the solar azimuth angle (Azsun).
  2. Next, determine the outward surface-normal azimuth angle (Azsurface).
  3. Next, calculate the horizontal shadow angle using HSA = Azsun โˆ’ Azsurface, then wrap the result into the range โˆ’180ยฐ (included) to 180ยฐ (excluded).
  4. Finally, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Solar Azimuth Angle (Azsun) = 225 degrees

Surface Azimuth Angle (Azsurface) = 180 degrees. HSA = 225ยฐ โˆ’ 180ยฐ = 45ยฐ. The sun lies 45ยฐ clockwise from the outward south-facing normal.