Shadow Geometry for Lighting

TL;DR: To find a building's shadow length, multiply its height by the cotangent of the sun’s angle above the horizon ($L=h\cdot \cot (\theta )$).

The Geometry of Light

In Station 1, we asked how geometry shapes the spaces we inhabit. We moved through the ergonomic design of our desks in the previous station, and now we zoom out to the city skyline. When we place a building, we aren't just placing a structure; we are placing a giant light-blocker. Understanding shadow geometry isn't just for artists sketching a sunset; it is a fundamental requirement for urban planners who need to ensure that a new skyscraper doesn't leave the local park in total darkness for half the year.

Shadows are simply the projection of an object onto a surface. When the sun is directly overhead, the shadow is nonexistent or tucked neatly beneath the building. But as the sun moves across the sky, that shadow stretches like a rubber band. The length of that shadow depends entirely on the height of the building and the angle of the sun, which we call the solar altitude.

Calculating the Projection

Think of a building as the vertical side of a right-angled triangle. The shadow is the base of that triangle, and the sunlight acts as the hypotenuse. If we know the height of the building ($h$) and the solar altitude ($\theta$), we can use basic trigonometry to find the length of the shadow ($L$).

Because the sun is so far away, the rays of light hitting the top of the building and the ground are essentially parallel. This creates a predictable relationship:

$$\tan (\theta )=\frac{\text{opposite}}{\text{adjacent}}=\frac{h}{L}$$

To find the length of the shadow, we rearrange this to solve for $L$:

$$L=\frac{h}{\tan (\theta )}$$

Since $\frac{1}{\tan (\theta )}$ is the same as $\cot (\theta )$, we can say $L=h\cdot \cot (\theta )$. If a building is 100 meters tall and the sun is at a 30-degree angle, the math tells us the shadow will stretch approximately 173 meters. This calculation allows architects to simulate the "shadow footprint" of a building at any time of day or any season of the year.

Applying Shadow Logic to Urban Planning

Urban planning is a game of compromise. If you build too close to a public space, you risk creating a "canyon effect" where the ground level remains in shadow, cooling the microclimate and discouraging foot traffic. By applying the projection formulas we just explored, planners use sun path diagrams to map out exactly where shadows will fall.

This isn't just about light; it's about thermodynamics. In dense cities, shadows dictate heating costs for nearby buildings and the viability of rooftop gardens. By adjusting the orientation or the height of a structure, planners can use geometry to "steer" the shadow away from sensitive areas. It is a precise, invisible choreography of light and stone.

**A simple script to calculate shadow length**
import math

def calculate_shadow(height, angle_degrees):
    angle_radians = math.radians(angle_degrees)
    # L = h / tan(theta)
    return height / math.tan(angle_radians)

**Example: 100m building at 30 degrees**
print(calculate_shadow(100, 30))

We have moved from the personal scale of our chairs to the massive scale of our cities. Now that we can predict where shadows fall, we must consider how to navigate through the city itself. In our next station, we will take this knowledge of space and geometry and apply it to moving objects, exploring how to calculate the most efficient routes for travel. We are moving from static structures to the fluid patterns of motion.

The length of a shadow is determined by the ratio of an object's height to the tangent of the sun's angle, allowing us to mathematically predict the impact of any structure on its environment.