One-Upping the Stand Up Mathematician?

In October of 2021, Matt Parker - number 3 on my list of YouTube man crushes - and Hannah Fry released a video on YouTube demonstrating how to measure the radius of the Earth using a tall building and protractor. Their attempt was moderately successful despite facing several obstacles. On a recent family vacation, I realized I had the perfect opportunity to make a more accurate measurement. Keep reading to see how our estimates compare.

Hannah and Matt

The process outlined in the video requires an observer to climb to a high point and measure the angle of declination down to the horizon. Using your elevation, the angle of declination, and some surprisingly elementary trigonometry, you can calculate the radius of the Earth. This method was devised by 10th Century mathematician al-Biruni. Using his method, al-Biruni was able to calculate the radius with surprising accuracy.

Hannah and Matt had several difficulties to overcome in their experiment. The main problem was getting their measuring equipment through building security. Additionally, Matt insisted on measuring the height of The Shard through trigonometry.

The duo's results are:

  • Height of observation deck: 263m (Actual: 244 m)
  • Angle of Declination: 1.5 degrees
  • Radius of Earth: 875 km (Actual: 6371'ish km)

Visiting the Perfect Location

Visiting the perfect location for this experiment was a lucky coincidence for me. There was a mountain with high elevation located close to the sea on our vacation itinerary: Haleakala. The lack of building security is a plus.

Haleakala Visitors Center sign. Elevation 9740ft or 2969 m.
Photo Credit: Emily Sears

Standing on the top of a volcano was the one experience I wanted out of the trip. We took a tour bus up for sunrise. The pickup time was 2:30 am, which sucked. The rest of the trip was amazing.

There was a healthy crowd at the Visitor Center. Fortunately, everyone was respectful. That is not always a guarantee, but the Aloha Sprit was with us tourists. The light of sunrise played nicely with the cloud layer below. I was able to poke around the crowd to get a few shots. The thinner air at the top made moving around more difficult. I hiked up to the higher observation area, and had to stop every few minutes to catch my breath.

Silhouettes of people watching a sunrise over Haleakala.
Silhouette of the Crowd. Photo Credit: Chris Sears

The sun rises of a cloud layer.
The view from the top. Photo Credit: Chris Sears

The clouds enhanced the natural beauty, but they worked against my secondary objective for this trip. Finding the horizon turned out to be a problem. The clouds started to break enough to see the ocean. If we had stayed for an extra half hour or so, the view of the horizon would have been clearer. With the tour bus loading, I made the best measurement I could.

A measurement of 0.95 degrees seemed good enough for me. I remembered that Hannah and Matt should have measured around 0.5 degrees from The Shard. A larger angle is expected from a higher height. I was happy with the result. My family was happy I stopped talking about math for the rest of the morning.

View of the Pacific Ocean from Haleakala. Clouds obscure the horizon.
The view of the horizon. Photo Credit: Chris Sears

A digital level reads 0.95 degrees.
0.95 degrees was the best I was going to get. Photo Credit: Connor Sears


After returning to the mainland, it was time to crunch the numbers. Here are the results:

  • Height of observation: 2960 m (Rounded to three significant digits and adjusted for standing below the Visitor Center)
  • Angle of declination: 0.95 degrees
  • Radius of Earth: 21500 km (Actual: 6371'ish km)
Well, that is disappointing. Matt and Hannah were off by 86.3% of the actual value. I was off by 238%. Looks like I'll be eating the humble pi tonight.

It is possible to work backward to find the proper angle measurement for a height of 2.96 km. I should have measured 1.75 degrees. This is one time where aiming for the clouds was the wrong thing to do.

The video below shows how I made the computations.

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