Enter the height of an object and the angle of the source of light measured from the ground to calculate the shadow length. This calculator can also evaluate the height or angle, given the other variables are known.
- All Physics Calculators
- Angle of Elevation Calculator
- Angle of View Calculator
- Angle of Repose Calculator
- Air Mass Calculator
Shadow Length Formula
The shadow length formula derives from basic right-triangle trigonometry. A vertical object of height H, the ground surface, and the light ray connecting the top of the object to the tip of the shadow form a right triangle. The shadow length L is the adjacent side, the object height H is the opposite side, and the sun elevation angle a is measured between the ground and the incoming light ray.
L = H / tan(a)
Where L is the shadow length, H is the object height, and a is the sun elevation angle (also called solar altitude angle) measured from the horizontal. The formula can be rearranged to solve for height (H = L * tan(a)) or for the sun angle (a = arctan(H / L)).
This formula assumes a flat, level surface and a distant light source like the sun, where rays are effectively parallel. For artificial point-source lighting (such as a streetlamp), the geometry changes because light rays diverge, and the shadow length depends on both the height of the light source and its horizontal distance from the object.
Shadow Ratio Reference Table
The shadow ratio is the multiplier applied to an object’s height to determine its shadow length at a given sun elevation. To find the shadow length, multiply the object’s height by the ratio.
| Sun Elevation | Shadow Ratio (L/H) | Shadow per 1 m Height | Typical Condition |
|---|---|---|---|
| 5 degrees | 11.43 | 11.43 m | Just after sunrise or before sunset |
| 10 degrees | 5.67 | 5.67 m | Early morning or late evening |
| 15 degrees | 3.73 | 3.73 m | Low winter sun at mid-latitudes |
| 20 degrees | 2.75 | 2.75 m | Winter midday at ~47 deg latitude |
| 30 degrees | 1.73 | 1.73 m | Morning/afternoon in temperate zones |
| 45 degrees | 1.00 | 1.00 m | Shadow equals object height |
| 60 degrees | 0.58 | 0.58 m | Summer midday at mid-latitudes |
| 75 degrees | 0.27 | 0.27 m | Near-tropical solar noon |
| 90 degrees | 0.00 | 0.00 m | Sun directly overhead (tropics at solstice) |
At 45 degrees elevation, the shadow length exactly equals the object height. This 1:1 ratio serves as a useful mental benchmark. Below 45 degrees, shadows are longer than the object; above 45 degrees, shadows are shorter.
Solar Noon Elevation by Latitude and Season
The sun’s maximum daily elevation (at solar noon) depends on the observer’s latitude and the time of year. The formula is: solar noon elevation = 90 – |latitude – solar declination|. Solar declination ranges from +23.44 degrees at the June solstice to -23.44 degrees at the December solstice, and is 0 degrees at the equinoxes.
| Latitude | Example City | June Solstice Noon | Equinox Noon | Dec Solstice Noon |
|---|---|---|---|---|
| 0 deg (Equator) | Quito | 66.6 deg | 90.0 deg | 66.6 deg |
| 23.4 deg N | Havana | 90.0 deg | 66.6 deg | 43.2 deg |
| 35 deg N | Tokyo / LA | 78.4 deg | 55.0 deg | 31.6 deg |
| 40 deg N | New York / Madrid | 73.4 deg | 50.0 deg | 26.6 deg |
| 48 deg N | Paris / Seattle | 65.4 deg | 42.0 deg | 18.6 deg |
| 56 deg N | Edinburgh / Moscow | 57.4 deg | 34.0 deg | 10.6 deg |
| 64 deg N | Reykjavik | 49.4 deg | 26.0 deg | 2.6 deg |
At 56 deg N latitude (Edinburgh, Moscow), the winter solstice noon sun sits at just 10.6 degrees, producing shadow ratios above 5x. At the same latitude in summer, noon elevation reaches 57.4 degrees and shadows shrink to about 0.64x the object height. This seasonal swing of nearly 47 degrees in peak sun angle is why shadow behavior varies so dramatically between summer and winter at higher latitudes.
The Shadow Rule for UV Safety
Dermatologists and public health organizations promote the “shadow rule” as a quick way to gauge UV intensity without checking an index or app. If your shadow is shorter than your height, UV radiation is intense and sun protection is strongly recommended. If your shadow is roughly equal to your height (sun at about 45 degrees), moderate caution is appropriate. If your shadow is longer than your height, UV exposure is lower.
The underlying physics is straightforward. When the sun is high (above 45 degrees elevation), UV rays travel through less atmosphere, so less radiation is filtered out before reaching the surface. A shorter shadow directly indicates a steeper sun angle and higher UV dose rate. At most mid-latitude locations, the window where shadows are shorter than body height falls roughly between 10 a.m. and 2 p.m. during spring and summer months, which aligns with peak UV exposure hours identified by skin cancer research.
Practical Applications of Shadow Length
Architecture and Zoning
Many municipalities require shadow impact studies before approving new construction, particularly for buildings above 4 to 6 stories. These studies model how a proposed structure’s shadow sweeps across neighboring properties throughout the day and across seasons. The critical design case is typically the winter solstice, when shadows are longest. Architects use the shadow length formula with the local winter solstice noon elevation to determine minimum setback distances that prevent permanent shading of adjacent buildings, parks, or sidewalks.
Solar Panel Spacing
For ground-mounted or flat-roof solar arrays, panels must be spaced far enough apart that one row does not shade the next. The standard approach calculates row spacing using the shadow length at the winter solstice noon sun angle for the installation’s latitude. Even partial shading on a single cell in a typical 36-cell module can reduce power output by up to 75%, making accurate shadow length calculations directly tied to system efficiency and return on investment.
Photography and Filmmaking
Photographers use shadow length and direction to plan shoots around specific lighting conditions. The “golden hour” near sunrise and sunset produces shadow ratios above 5x, creating long, dramatic shadows favored in landscape and portrait work. Knowing the exact shadow ratio at a planned shoot time lets a photographer predict where shadows will fall in a scene and position subjects accordingly.
Landscaping and Garden Planning
A fence, wall, or tree casts a shadow zone where shade-tolerant plants should be placed. Calculating the shadow length at summer and winter solstice noon gives the minimum and maximum shadow footprint. For example, a 2 m fence at 40 deg N latitude casts a noon shadow of about 1.0 m in June but roughly 4.0 m in December. That 3 m difference defines a transitional planting zone that receives full sun in summer but deep shade in winter.
Forensics and Geolocation
Open-source intelligence (OSINT) analysts and forensic investigators use shadows in photographs and satellite imagery to estimate the time a photo was taken or to verify the claimed location. By measuring the shadow length of an object with known height in an image, analysts can back-calculate the sun elevation angle, then cross-reference that angle against solar position data for the claimed date and coordinates. A mismatch between the calculated sun angle and the expected angle for that time and place can reveal that an image was taken at a different time or location than stated.
Shadow Length Throughout the Day
Shadow length changes continuously as the sun moves across the sky. The table below shows approximate shadow lengths for a 1.7 m tall person (average adult height) at 40 deg N latitude on the equinox and summer solstice.
| Solar Time | Sun Elev. (Equinox) | Shadow (Equinox) | Sun Elev. (June Solstice) | Shadow (June Solstice) |
|---|---|---|---|---|
| 7:00 AM | 12 deg | 8.0 m | 25 deg | 3.6 m |
| 8:00 AM | 23 deg | 4.0 m | 36 deg | 2.3 m |
| 9:00 AM | 33 deg | 2.6 m | 48 deg | 1.5 m |
| 10:00 AM | 41 deg | 2.0 m | 58 deg | 1.1 m |
| 12:00 PM | 50 deg | 1.4 m | 73 deg | 0.5 m |
| 2:00 PM | 41 deg | 2.0 m | 58 deg | 1.1 m |
| 4:00 PM | 23 deg | 4.0 m | 36 deg | 2.3 m |
| 5:00 PM | 12 deg | 8.0 m | 25 deg | 3.6 m |
On the equinox at 40 deg N, an average adult’s shadow shrinks from about 8 m at 7 AM to 1.4 m at noon, then stretches back out symmetrically into the afternoon. On the summer solstice at the same latitude, the noon shadow drops to just 0.5 m because the sun reaches 73.4 degrees elevation.
Limitations and Edge Cases
The standard formula L = H / tan(a) assumes flat terrain, a single directional light source, and a vertical object. On sloped ground, the effective angle changes: an uphill slope shortens the shadow while a downhill slope lengthens it. For objects that are not vertical (such as a leaning pole or a person bending), the height H must be replaced with the vertical component of the object. At extremely low sun angles (below about 2 degrees), atmospheric refraction bends light enough to introduce noticeable error, and shadows become so long that minor ground undulations distort them significantly.
For artificial point-source lights like streetlamps, the shadow formula changes entirely. If the light is at height H_L and the object of height H_O is at horizontal distance d from the light, the shadow length is: L = (H_O * d) / (H_L – H_O). This produces longer shadows as the object approaches the light’s height and theoretically infinite shadows when the object and light are the same height.
