Measure the Earth’s Radius! (with this one complicated trick) 

Are you saying that the Shard security did not let you in with weapons of maths instruction?

"The thing is, we got a number" This is the professionalism we subscribe for.

Ah yes, the age old joke: “Two mathematicians go to the shard with a fancy protractor and a laser-inclinometer…”

“You know what Hannah, it’s a small world” Couldn’t stop laughing xD

"anything between 10 and a million" the Parker radius, everybody

"Oi you got a license for that?" "It's a protractor" "Exactly, obviously a weapon"

This demonstration is so convincing, I've gone from being a *'Flat Earther'* to a *"Small Earther'*

I love how Hannah started all serious and embarassed by Matt's antics and got progressively more Parkerish as the video progressed.

Matt Parker is such a maths geek his body is metric.

Matt: "I've got this really fun idea on how we can do publicity for your book!" Hannah: *sigh* "Can't you just mention it on a video or something?"

The fact that a Matt cubit is almost exactly half a meter made me actually laugh out loud. That’s awesome.

this reminds me of the time when we were calculating the speed of muons using two scintilator detectors. But we used rulers to measure the distance between those so we got end result of c+-c. Which is technically correct answer for any question regarding speed asked ever.

“Well we’re not going to just calculate the sine of the angle. I built my very own lookup table last night just for that!”

Matt and Hannah have such great chemistry together on screen. It’s wonderful to see them working together! Especially seeing Hannah get fed up with Matt’s shenanigans.

25:30 I want to say, with the actual value of the heigh, they get a an Earth radius of 708 km. Even smaller

Matt is the perfect foil for Hannah. I love it every time these two collaborate.

"it's actually professor Fry" Best part.

As Matt keeps stacking scope-creep onto the required tasks in this project, I can sense Hannah's "I did not sign up for all this" energy.

This is the kind of doable experiment that bridges the relevance gap kids experience when learning math(s) in school. It's also a great grounds for relating sensitivity of the outcome to the accuracy of your tools and measurements.

Matt and Hannah seemed like a great team as they Shard their experiment but then a protractor argument let to them going off at a tangent, which is never a good sine. At least they realised the magnitude of their error.

Hannah: This is going to be straightforward, half an hour at the most. Matt: Hold my beer

You got a loicense for that giant protractor? -Shard Security

The angle you would have to observe to get the correct radius from 300 meter would have been arccos(6371/6371.3) =0.556 degrees, and to get the radius to be equal to 6322 you would need to observe an angle of 0.558 from 300 meters. So you need a very accurate protractor to get good results.

Hannah: "We're going to go up the shard with an electrical inclinometer and then do one calculation after looking some stuff up on Wikipedia." Matt holding his makeshift plumb bob inclinometer and cubit shoes: "We are not!"

I imagine Matt's "between 10 and 10 million" pessimism compared to Hannah's "within 1000" can be explained by the fact he has estimated pi to a wild variety of decimal places over the years

They measured the Parker height of the Shard, the Parker angle to the horizon, the Parker radius and circumference of Earth. Bravo. Well done, Matt and Hannah. And, can we give an Honorable Mention to the Parker cubit? They really did a lot of the footwork in all of this.

This was once a question in one of our trigonometry tests during high school! We were all so baffled by it, that it became a joke for the class: "I've bought a pound of tomatoes. What is the radius of the earth?"

I'm a sheetmetal mechanic. When I was first learning the trade I worked with on of our companies best mechanics. He could estimate the circumference of a vessel within a few inches by standing a short distance from it (20-30ft or so) and gauging the angle by his outstretched arms. He told me that he learned math while in university in Mexico where he was a top student in the math dept. I think it would be fantastic to have a course all in the "hands on" mathematics. There really are tons of material out there. Some of which is being forgotten.

This is why I love Matt Parker. The hard work, dedication, and attention to detail is all there!...unlike the correct answer, but expectations were set. I nominate that this non-negative, _well within_ an order of magnitude radius of the Earth be called a Parker Earth Radius.

I find it creepy and depressing that they wouldn't let you go to the top with an inclinometer and an attractive, home-made astrolabe.

The lack of uncertainty values and reporting agreement between measured values is *exactly* how you can tell an experiment was done by mathematicians and not scientists lol Loved this

Eratosthenes did the calculation much earlier, and it was quite accurate. He noted the angles of shadows in two cities on the Summer Solstice.

"We're off by an order of magnitude" As an astrophysicist, I approve

It's obviously 1 in natural units.

Going through Teacher's College now here in Canada, and I can't help but be inspired to want to try this with a class of Grade 9 students. Absolutely awesome!

If Hannah was my math teacher I might have actually paid attention in class.

Given that the "correct" angle should be 0.5 degrees, coupled with the stated height of 243 meters to get 6382 km , and 0.4 degrees would result in 9971 km, either protractor would have give a way off result, but it was fun as always, a nice lesson in propagation of error 😃

Matt Parker has been so active lately! He must really want his million subscriber goal lol

Whenever I want to work out the circumference of the Earth I just pace out the distance from Alexandria to Syene.

This was so much fun to watch. The outcome didn't even matter. I learned something and laughed so much

Watching Hannah losing the will to live and appreciating the geekiness simultaneously is a treat it itself

The trig battle at 17:34 is the closest we'll get to a real-life wizard duel.

You two have such great banter. Very enjoyable!

Hannah and Matt make awesome mathematics communicator's, Matt's can-do attitude and Hannah's down to earth humor is such a beautiful combination 🥰

"If we round this to a thousand its the same order of magnitude" Can't believe I've never tried this argument on my math teacher, clearly the difference between a pleb like me and a pro.

Matt’s almost child-like enthusiasm (I mean that as a compliment), combined with Hannah’s calm, slightly exasperated, yet kind, patience, make for a great team! (“Yeah yeah yeah, I want to measure the height of the shard!!!”…. (Rolls eyes).. “Ohhh, OK then……..”

Honestly, one of Matt's best. Loved this video.

This was a blast to watch honestly. Matt's enthusiasm is absolutely contagious.

Well, that's how you get extraordinary precision: Put the difference of 1 and the cosine of a very small angle measured with your phone manually aligned to a brochure into the denominator of your equation.

I admire Matt's insistence to replicate al-Biruni's calculation with analog tools. Sometimes it's just so much more satisfying to use the simple tools to craft the answer.

Regardless of the Small World result, you did get an R value and did show the Earth is not flat! Shockingly, a lot of people in the 21st century need to be convinced of that.

I admire his dedication to the measurements. A+++

At every single step, Hannah was like "what if we did this part the easy way" and at every single step Matt went "NO! WE DO THIS LIKE ANCIENT GREEKS!"

Hannah: Do I need to be in the street with you? Matt: Yes, I need someone to apologize for me while I count.

One of the funniest videos I've seen in quite some time, and it's about math 🙂 I remember in high school, I think we used Pythagoras' formula to calculate the height of a hill using a map for the distance, and the distance from your eye to your thumb. We figured it out my ourselves, and we were super proud 🙂

That was way too much fun! Thank you both!

Getting his forearm length surgically altered is serious commitment

"Can you come and look at the angle, Dr. Fry?" "I just need to come around the ... it's actually professor Fry." FLEX ON 'EM HANNAH

I loved this video! Fun, educational and funny. I love the chemistry between them too. One doesn't need a calculator to figure out how they ended up with an order of magnitude error. For the small H/R and C, we can approximate: R/(R+H) ~ 1-H/R and cosC ~ 1 - C^2/2, so the equation becomes 2*H/R = C^2. 2*H/R ~ 2*0.3/6000 ~ 1/10000. Square root of that: C ~ 0.01 rad or ~0.57° They measured 1.5° - almost 3 times more. That values squared is 9, which gives an order of magnitude error. Detecting the small angle using the mobile phone was the culprit. And the error was magnified by the nonlinear cos function.

What a delightful video! Forget the subject matter, that must have been the most entertaining video I've seen for a long time. And I do love Hannah Fry's English accent offsetting - by which I mean complementing! - Matt Parker's Strine (I mean Australian accent).

Did Matt say "Degree to Disdegree" when argue about SIN vs TAN?

If you two wants to collab each time one has a new book out, I am all in favour of you both writing more books.

A question for Matt; when you were on the street measuring the angle to the Shard viewing platform, were you aiming for the ceiling of the viewing platform, the floor of the platform, or some bit in between approximating your height standing on the viewing platform? All three positions could yield slightly different angles and therefore derive different values for H and also give you different margins of error.

This is one of the most joyously wholesome videos I've ever seen.

At the observation platform, one of you should have stepped, say, 10m back from the window. Then have the other person, still at the window, under your direction, move their finger (on the glass) up or down until it lines up with the horizon. The measure how much lower their finger is compared to your eye height. From that you could calculate the dip angle of the horizon.

Using the small angle approximation (which is definitely valid here), when θ is expressed in radians, the radius of the earth is given by R = 2H/θ², which shows why having only three times the correct angle causes a change of an order of magnitude.

This was hilarious, and demonstrates the humor in math. Awesome video!

This is exactly the content I needed right now. Thank you for this amazing video!

Also for the record, correcting the height to the observation platform doesn't salvage the result. Changing the angle to 0.5 absolutely does (doing both gets about 6460 km). If your weapons hadn't been confiscated, you probably would have gotten pretty close.

"Mathematicians Acting Stupidly for Fun!" Definitely buying Hannah's book, so I can be a real mathematician just like you two.

19:46 Hannah: „m please - thank you.“ so british… the units not the politeness. I love that woman

They should try this again with absolutely old school techniques. Like string to measure the distance between A&B, a mount for the astrolabe, and an actual mountain

I love it when Hannah and Matt make videos together. They have such a unique chemistry and they always nerd out when doing these videos 😁

Can we call 2x Matt’s arm the “Parker Metre?” We can have a whole Parker Unit System with that!

This was pure comic genius. Loved it!

So fun episode with Matt and Hannah together. I hope you make more episodes, and that Matt will make a guest appearance at some point on Curious Cases

After crunching some numbers, I'm super impressed with Al-Biruni's result! According to Wikipedia, Al-Biruni used a dip angle of 34 arc minutes. This means the mountain that he measured the dip angle from would have been about 312 m above sea level (remarkably close to the height of the Shard if you ignore refraction). By the same logic, you should have measured about 30 arc minutes at the observation deck. Accuracy of measuring the dip has a huge impact on the accuracy of the result. With your measurements, if C is off by a ±0.25° (even with every other measurement being perfect) your error bound goes to about ± the radius of the earth.

I've been playing around with the numbers a bit on my end, and you'd actually need to measure to hundredths of a degree from the Shard to get anything close to an accurate result - it seems like 0.52° would get something OK (from your height measurement of 263m which you explained isn't quite accurate anyway), while only a tiiiiny deviation to 0.56° already makes you off by more than a thousand kilometers ; so from that alone, imagine having 200% error with your 1.5° measurement, and it turns out being "only" an order of magnitude off as a result is actually pretty impressive, hahaha! PS: If you wanted to keep only one significant figure, 0.5° would yield 6900km radius, which isn't too bad! :-) EDIT: Also, considering they probably didn't have such precise protractors a thousand years ago, I think this experiment is best done with actual mountains - if the mountain is 2.6km tall, one might assume you could knock a decimal off the protractor measurement to get a similar precision radius measurement? EDIT2: Again not quite apparently - tried the 2630m mountain, 1.65° gets a relatively accurate measurement while 1.6° is roughly 400km off, so I'm guessing they probably had a protractor that was precise to at least a 20th of a degree, which is impressive in its own right :-)

I love all the videos with you and Hannah.

Such great chemistry between you two, it's amazing. Please do more collabs together :-)

These two have SO MUCH FUN together. It's always entertaining to watch them dig into some math quirks.

You are basically doing 1 minus almost 1. Meaning you're dividing by a number super close to 0. I'd say even arcminutes are significant here.

Watching you guys together is an absolute riot

The buildup for this gets increasingly amusing, good stuff (as usual).

"Call it 'c'! [...] Yeah, you're looking at it." :D Really appreciated that joke.

They have good chemistry together I always like these two together. And honestly it seems like Matt took every opportunity to do things the long way lol

I hope you continue to plan and prepare as little as possible. I love how you respond to situations like the great protractor confiscation lol.

Pure gold. Thank you guys!

I am disappointed that Matt didn't build a wheel on a stick and counted the number of revolutions and then multiplied by its circunference to get the total distance. He could've made a bell that would ding at each revolution and count the number of bells without looking at the wheel!

The real lesson here is that every measurement has an error. In this case any small error is the measurement of angles creates a much larger error in the final result.

You both are so funny! I can’t wait to pick up Hannah’s book!

Bloody brilliant! You guys are fantastic

Of course, The Shard security detail has a "geometry bin". Doesn't everyone?

I demand to see a remake with yourself and Dr Fry where you use an actual mountain and the giant protractor.

This was really entertaining, you two make a good maths duo

hannah is a total stud, nice work guys this was really a fun video. hope you got more than one cocktail

I have never smiled that much during a math lesson. That was so much fun. We need more of you two interacting on math questions

Three Parker videos in two days? Is this what maths heaven is like?

Love all your videos. They are truly amazing

The energy between them is amazing.