Shadow Length Calculator

How shadow length is calculated

The vertical object, its shadow on level ground and the incoming ray form a right triangle. If the sun is α above the horizon, tan α equals the object height divided by the shadow length. Divide the height by tan α to obtain the shadow length.

Example and limits

At a solar elevation of 45°, an object 2 m tall casts a shadow 2 m long. Directly overhead sunlight gives zero shadow length. On or below the horizon, this model does not return a finite length. Sloping ground, tilted objects, nearby obstructions and penumbra are excluded. Near the horizon the result is highly sensitive to small uncertainties in the elevation angle.

Choosing the angle and height

Solar elevation is measured above the horizon. It differs from the zenith angle measured from directly overhead. Do not enter a zenith angle directly as solar elevation. Object height must be the vertical distance above the horizontal plane where the shadow is measured. A sloping length along the terrain is a different input. Check that the selected angle unit matches the number entered.

For example, take a vertical pole 4 m high and solar elevation of 30°. Since tan 30° = 1/√3, the shadow is 4√3 ≈ 6.928203 m. This is the horizontal distance from the pole base to the shadow tip, not the inclined ray length from the pole top to that tip. At the same angle, doubling height doubles shadow length.

The shadow shortens as the sun rises, but the relationship is not linear in angle. Doubling the angle does not always halve the shadow. Near the horizon, a small angle change can produce a large length change. Many displayed digits do not imply equivalent measurement precision.

The calculator does not determine solar elevation from date and location. Obtain that angle separately. It also does not solve shadow direction or mutual shading between nearby objects. On a sloping roof or terrain, the result represents only the ideal level ground geometry.