Settling Disagreements with Trigonometry: Don’t Take Sides, Solve Them

Photo of the island of Cayo Norte from Zoni Beach in Culebra, Puerto Rico. How far away is Cayo Norte?

“You think I can swim out to that island over there?”

My friend, Roman, pointing to the island of Cayo Norte (pictured above), whimsically asked the group if he could traverse the 3 miles of ocean water to the next island. Or so he thought it was 3 miles.

Another friend responded, “Yeah sure, it looks only half a mile or so away.” A frustrating debate ensued about the actual distance to the island. As the banter increased, we became determined to figure out who’s guess was closest. But how can we, sunbathing on a beach with no cell signal, actually find out who was right?

Enter trigonometry.

We reasoned that if we can form a triangle with two points on our beach and the third point on Cayo Norte, we can use trigonometry to solve all the triangle’s angles and side lengths with only a few measurements from our beach. By doing so, we find our distance to the island of Cayo Norte and see who was right.

Figure 1: Triangulation method to find distance to the island of Cayo Norte. Left: Photos from point A depicting the angle between sides B and C. Middle: Satellite image of Zoni Beach and Cayo Norte with a triangle overlaid connecting the points of interest. We want to solve for Side A (Sa), the distance between our beach setup and a point on Cayo Norte. Right: Photos from point B depicting the angle between sides A and C.

Making the Measurements

As depicted in Figure 1 (middle), we labeled point B as the location of our beach chairs and point A some arbitrary distance down the beach. We agreed to label point C as the lowest saddle point amongst Cayo Norte’s rocky hills. We chose some driftwood as a straightedge to carve the three sides of the triangle in the sand from points A and B (see Figure 1, left and right).

Next, we made three measurements: (1) the angle θA (Figure 1, left), (2) the angle θB (Figure 1, right), and (3) the side SC, the distance between points A and B. To measure the two angles, we took a photo of the lines carved into the sand and used our phone’s built-in photo cropper to find the angle. We determined the length of side SC by multiplying the number of steps between points A and B by our estimated stride length.

We measured:

θA = 100°

θB = 74°

SC = 268.5 meters (~0.167 miles)

Solving the Triangle

With these three measurements, we could now solve the entire triangle! Since the angles of a triangle add to 180 degrees, we know the angle θC=180-θA-θB. Then we used the Law of Sines to find the two missing side lengths.

Specifically, to find side SB, the desired distance between our beach chairs and Cayo Norte, we calculated

To complete the triangle, we similarly found side SA:

We plugged our measurements into the equations to find:

θC = 6°

SA = 2,529 meters (~1.57 miles) *our measurement of interest*

SB = 2,469 meters (~1.53 miles)

We determined that the distance between us (point B) and Cayo Norte (point C) is approximately 2,529 meters, or 1.57 miles!

Limitations of Our Beach Surveying

But how accurate is this method? While it all sounds good in theory, drawing lines in the sand and estimating stride length is definitely not very precise. Once we got back to our Airbnb with internet connection, we used Google Maps to find the true distance of 1,810 meters (~1.12 miles) between our beach chairs (point B) and Cayo Norte (point C), meaning we have an error of 39.7%.

Although our result could rule out the more extreme guesses, we could have still increased our accuracy in a number of ways while still only using our beach tools. We recognize the accuracy of this triangulation method is significantly affected by Abbe error, meaning a small error in an angular measurement (i.e., measuring our θA and θB) magnifies over distance. For example, if we would have measured θA = 99° and θB = 73° — just 1 degree less than our real measurement — our result would have been 1,906 meters, or an error of just 5.3%. Thus, fixing either θA or θB to be 90° with a manufactured object (e.g., the frame of a beach chair) and averaging θA and θB across each friend’s individual measurement would have greatly reduced error. Additionally, making point C a well-defined structure on the shoreline and measuring SC with a consistent tool (e.g., a section of a rope) instead of highly variable stride lengths would have provided more reliable estimates.

All told, it was fun to solve a seemingly impossible task — measuring a distance over an ocean — with tools you bring to a beach.

Note: When I first wrote this article, I surprised myself at how boring it was. So I spiced it up a bit (read embellished) in the article titled, “A New Angle on the Legend of Cocaine Island.“