Sun Paths and Shadow Maths
Calculating the perfect cucumber arrangement
My cucumber seedlings have turned out to be an interesting experiment. I transplanted about half of them into larger pots while the rest remained in smaller containers, all still growing indoors. The difference is pretty striking: the plants in the bigger pots are growing much faster. For a plant to grow taller, make sure it can grow deeper!
I’m growing the pumpkins in the vacant lot because their vines and fruits are massive. But with still-limited water access there, growing cucumbers would be tricky. Fortunately, I have access to a 4x4-foot garden bed right next to my apartment building. It’s part of a community garden managed by Good Trouble Church. The space is called Red Shed Village, and it houses three people in brightly painted tiny sheds as part of a transition from homelessness to stable housing. The residents have been kind and helpful from the start—willing to converse with me about vegetables, pointing to the right contacts for the garden, showing me how to access water, and wishing me lots of luck.
A 4x4 bed isn’t huge, but it’s enough to work with. I plan to grow the cucumbers upward on trellises, spacing each plant about a foot apart. I have eight cucumbers and three basil, and I’d like all of them to thrive. It seems straightforward: I build two trellis walls with four cucumbers each, and plant the basil in a row between them.
But then comes the challenge: sunlight and shadows.
Basil loves full sun, and so do cucumbers. I also don’t want to cast unwanted shadows on my neighbors’ garden beds—especially as the new gardener on the block. So where should I place the trellises? Along the north and south edges? East and west? Maybe an L-shape?
To figure this out, I need to understand how the sun moves across the sky. In summer, the sun rises in the northeast and sets in the northwest. Throughout the day however, it shines from the southern part of the sky. That means if I place the trellises on the east and west sides of the beds, in the morning and evening they’ll cast shadows on my own plants. Worse, during peak sun hours (around 11am to 1pm), the northernmost cucumbers would get less light than the rest.
So, for the cucumbers, placing the trellises on the north and south sides seems smartest for sun exposure. That layout also leaves room in the center for the basil row. But, I worry, if the sun shines from the south, will a trellis on the north side cast a shadow on my neighbors’ beds, just four feet away to the north, northeast, and northwest? Will a trellis on the south side shade my own basil?
Time to bring in the math.
The Shadow Problem
Here’s the set-up: I have a 5-foot-high trellis (I’m using just one for the calculation, since both will cast the same length shadow). With the sun shining from the south, the trellis will cast a shadow to the north. I want to know how long it will be.
A right triangle is formed: the trellis is the vertical side (5 feet tall), the shadow is the horizontal side (unknown = ‘s’), and the beam of sunlight from the top of the trellis to the end of the shadow is the diagonal hypotenuse (length unknown), which forms an angle to the ground. Since I don’t know the hypotenuse’s length, I can’t use the Pythagorean Theorem to solve for the missing shadow length. But, if I can figure out the angle to the ground that it creates, I can use right-angle trigonometry.
Specifically, I can use the tangent function from SOH-CAH-TOA ratios, where:
Tangent (of the angle) = Opposite (length) / Adjacent (length)
In this case:
Opposite side = trellis height = 5 feet
Adjacent side = shadow length (what I want to find) = s feet
Angle = sun’s elevation at a particular time
All I need now is the angle of the sun in Baltimore during peak sunlight hours in June. The website SunEarthTools.com calculates solar path chards that show the sun’s elevation and azimuth (its compass direction) throughout the day. I entered June 1 as the day to calculate specifically, shown in the chart by the yellow curve.
Reading the Sun Chart
According to the chart, before 9am and after 5pm, the sun shines from the north—so shadows fall southward. No problem there. At 9am and 5pm, the sun shines directly due east and west respectively, so shadows cast by trellises aligned east-west would be minimal. The most important time to consider is when the sun is highest and strongest in the sky—when its rays shine from due south and cast their most direct shadows northward. According to the chart, on June 1st that moment happens at 1pm, when the sun’s azimuth is 0° (it’s directly south) and its elevation—the angle its rays form to the ground—is about 73°.
I’ll use that angle in the tangent formula.
Calculating the Shadow
So at 1pm, the shadow from the trellis will stretch about 1.5 feet to the north.
Since my neighbor’s bed begins 4 feet away, this shadow won’t even come close to reaching it. My basil, planted in the center of the bed, should also be fine—so long as I keep it at least 1.5 feet north of the southern trellis. To be safe, I’ll skew it slightly north of center instead of placing it dead in the middle.
Other Times of the Day
Just to be thorough, I’ll also check a few more times when the sun is strong, but its azimuth (compass direction) is angled. These shadows will be longer, but also diagonal and more diffuse—not as dense or opaque. So I don’t think I need to worry but since I already have my formula derived, why not check?
Angled shadow calculations using the tangent formula and azimuth data from the solar chart:
11 am: 59° elevation → 3.00 ft shadow, northwest (295°)
12 pm: 68° elevation → 2.02 ft shadow, northwest (320°)
2 pm: 68° elevation → 2.02 ft shadow, northeast (30°)
3 pm: 60° elevation → 2.89 ft shadow, northeast (65°)
All of these shadows stay within the 4-foot margin, and because they are angled they would be less obstructive anyway.
Final Plan
So I’m good to go! I’ll place one trellis on the north edge and one on the south, giving my cucumbers plenty of full sun. I’ll plant my basil slightly north of center to make sure it avoids even the brief 1.5-foot shadow cast at midday.
As someone new to this type of planting, it feels empowering—and comforting—to be able to mathematically calculate the most ideal placements. I can’t help but think: if math were taught this way to students, with real-life, testable applications, it might actually come alive. One fun experiment could be to go out and measure the length of the shadow once the cucumbers have grown in, testing the calculations in action. Students could document any variability they see and describe any ways the real shadows differ from their imagined ones. How many people can say they learned trigonometry while growing food for their community?
As for me, the math says I’m in great shape. I’ve got enough space in my 4x4 bed, and everything’s ready. So next up, planting the cucumbers!
Until then, grow forth and prosper!