Last April, renowned mathematician Timothy Gowers presented this puzzle on Twitter. I replied. I don’t think I’ve ever had as many as 160,000 views on Twitter for anything I’ve put there, but if someone is after exposure, putting a comment on a big name’s post is one way to go about it. It looks like I had about 1% as many views as the main post did, which still amounts to a lot of views.
I did bring the question up in class, starting by asking the students what was wrong with the question. They quickly spotted the grammatical error, that it needs a verb and should be, “How Many Degrees Are Inside the Following Shapes?” Even with the grammar fixed up, though, the question is poorly put. What does it mean for a degree to be inside a shape?
We shouldn’t just complain about bad grammar and poorly worded questions, though. The writer is trying to convey some idea, and we should try to help them out. Communication is a two-way activity. The writer should try to convey an idea to the reader as clearly as he can, but for communication, the reader has to pay attention and to help out when the writer’s skill fails. In teaching this is important. If I explain something and the student doesn’t understand, he should ask a question. But if he didn’t understand, then he probably won’t be able to ask a very good question, either. He may not even know how to start except by babbling some of the words that the teacher used. The teacher should take that babbling to mean “Teacher, I’m very confused,” and repeat the explanation in case it went too fast, or try explaining it in a different way. It is crucial to not take the question literally, or to ask the student to explain it more, if the words are so confused that the question doesn’t make sense.
Sometimes a poorly written question has a single answer. An example would be “How many sides are inside a regular triangle?” The question is either “How many sides does a triangle have?” or “How many sides does a triangle with three equal sides have?”, and in either case the answer is “Three”.
The “How many degrees?” question has a couple of different reasonable answers, though, plus a foolish one— and it is the foolish one that perhaps was intended. Commenter Dan Johnsson says it in a short tweet that will be helpful to mathematicians, but probably not most readers. I’ll show you anyway.
Zero is the cop-out answer, the easiest one to argue for and explain, but unlikely to be what the questioner was trying to get at. A circle has zero degrees because it has no straight sides that could meet together to make an angle. A triangle has three sides that we can call A, B, and C, so there will be angles the meeting points AB, BC, and CA. Those add up to 180 degrees, however you draw a triangle (even if it has one angle bigger than 90 degrees). A square has four 90 degree angles, which add up to 360 degrees, and a rhombus— a special kind of four-sided shape— also has 4 angles that add up to 360 degrees. A pentagon has 5 angles, that add up to 540 degrees. A circle, however, has no angles. Thus, its angles add up to zero.1
This feels like cheating because it twists the question. It’s similar to saying we have a “category error”, where we use adjectives that don’t apply to a category. An example is to ask, “What is the IQ of a rainbow?”, and to give the answer “Zero— because rainbows don’t have IQ’s.” So if you like the 0-degree answer, you should perhaps answer “Asking how many degrees are in the angles of a circle is foolish, because a circle doesn’t have angles.”
The other good answer (since the answer of 0 does have a good argument for it even if it’s a bit like a trick) is that the number of degrees in a circle is infinity. Or, rather, if you tried to count the number of angles inside a circle you’d go on forever, since infinity is more like a description of a process than it’s like a number.2 That’s because the number of degrees grows from the 180 of a triangle to the 360 of a quadrilateral to the 540 of a pentagon and beyond. In fact, the number of degrees in an n-sided polygon is 180 (n-2) degrees. To check, see that 180 (3 - 2) = 180 for triangles, 180(4 - 2) = 360 for quadrilaterals, 180(5 - 2) = 540 for pentagons, and so forth— for example, a 12-sided shape would have 180(12 - 2), which is 1,800 degrees. The answer of “infinity” is nice because it’s unsettling to have polygons that look more and more like circles have an increasing number of degrees, just to have the number drop to 0 at the ultimate polygon, the circle.3
There is another plausible answer besides 0 and infinity. That’s 360 degrees. You may have memorized this in school: “A circle has 360 degrees”. That’s true, but it has a problem I’ll come back to in a paragraph or two.
After going through triangles, squares, and pentagons, I asked my students what they thought was the number of degrees inside a circle. The most popular answer was infinity, but some said zero. I asked them to argue for their answers, and then I asked if anyone could come up with an argument for 360 degrees. Someone did, rather unhappily. It was good for them: we should learn to figure out what reasoning people might be using to get to a false conclusion.
360 Degrees
When you think of 360 degrees, you think of a sweep of an angle all the way around from where you start, forming a circle. When you get back to where you started, that’s a complete circle. Or think of a protractor. It is a semicircle showing 180 degrees. Putting two of them together, you get a circle. This is probably the answer the teacher intended, since students memorize that there are 360 degrees going all the way around the circle.
The reason 360 degrees is a bad answer is that although it takes 360 degrees to sweep around a rotating horizontal line from the center of a circle to the edge back to where you started, the same is true for a square, or for other shapes. Look at Figure 1’s circle and square. Both start the angle with a horizontal line from the center to the edge. Then, we can rotate that line to sweep out bigger and bigger portions of the shape. After 360 degrees of angle, we’ve swept through the entire shape.
A protractor is made round just for convenience, so if you are drawing lines for the side of an angle they’ll be the same length. There’s no reason angle lines have to have the same length. Figure 2 shows a conventional round protractor and an unconventional rectangular one. Both protractors show how to draw angles of various sizes and you could use either one to draw a line out from the vertex point to create a 45-degree angle.
So 360 degrees is a bad answer. It doesn’t differentiate a circle from other shapes in terms of how many degrees have to be rotated through to get back to the start. A. Klarke Heinecke (@a_klarke) puts it this way:
There is a difference between rotational and enclosed angles. Each enclosed angle is measured as a rotation from a baseline of one linear side, then summed for a polygon. A rotational angle, the 360 degrees of a circle, is measured from an arbitrary radial line from the center of a plane shape, curved or polygonal.
Imagine my surprise on googling the question “How many degrees are in a circle” to find that the leading answer on the Web is 360 degrees.
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How about AI? Using Google on Chrome:
I tried the free version of ChatGPT, but the answer was the same.
Even ChatGPT math solver gave 360 degrees.
The lesson is that web searches and AI are unreliable. If you ask a math question that is asked in a grade school class, you’ll get a grade-school-level answer.
If you’ve read this far, you probably like math, but in case you don’t, I have a reward for your persistence. The funniest comment in the Gowers Twitter post on “How many degrees inside a circle is by Vladimir Sadov (@Veillantif):
They did not say Celsius or Fahrenheit, and most importantly they did not say how many degrees outside.
Footnotes
I am speaking of “interior angles” here. That’s the kind of angle we talk about in ordinary life. There are also “exterior angles”. If I wrote out the definition, you wouldn’t understand it, so I’ll just say that the exterior angles are alpha and beta in Figure f-1, two footnotes down. I dislike exterior angles, but one of Euclid’s most important theorems is the Exterior Angle Theorem. See the Wikipedia article, or see Proposition 16 on Professor Joyce’s wonderful website. When someone talks about an “angle” of a polygon without the qualifier “exterior”, it’s always the interior angle.
If you treat infinity like a number, you get weird results such as that since 2*infinity = 3*infinity, it must be that 2 = 3. We talk about it like a number, but it’s really a process. Then number of positive numbers is infinite because if you started counting them, you’d go on forever. (The same is true of points on a circle, even though that infinity is bigger in a certain sense).
For a simple proof that the formula 180(n-2) = degrees is correct, see “Interior Angles in a Polygon”. This applies even if the shape isn’t convex, which means it bends in somewhere like the pentagon in Figure x. As I was writing this, I wondered, and came up with that pentagon as a counterexample. I constructed it to have a 270 degree angle in the middle, and two 90-degree angles at the square corners. That would make 540 degrees, and if we then add the two acute angles at the pointy corners, we’d have over 540 degrees for the pentagon. Googling, though, I found the formula is supposed to apply, and I found a convincing proof. Divide my pentagon into three triangles as shown, and each triangle will have 180 degrees, so when they’re added up, we get 3*180 = 540. How can that be, given what I have just explained about 270 + 90 + 90 + two more angles > 540? The resolution is simple. I made an arithmetic error. 270 + 90 + 90 = 450, not 540, so we do have room for the two pointy angles.
Many exciting results in mathematics are obtained by arithmetic errors. Around 2007 I visited Nuffield College, Oxford, for the year. I came up with a theoretical argument for concerning how a weighted sum of estimates would give different accuracy depending on the weights (for example, if you are looking for an average July 18 temperature in Bloomington, and you decide to weight years ending in 5 more heavily). This was not supposed to happen. I coded an example on the computer to verify my idea, and it did. I talked with Professor David Cox about it. After lunch in the college, it’s a custom to go down to the senior common room to have coffee and talk. We went down, and he kindly listened to me. Cox was very old, and was perhaps the top scholar in the College, more famous than the late James Mirrlees who a Nobel Prize. He did not see an error in my explanation— at the coffee-after-lunch level, at least. But then I check my code again, and found a flaw. My theory was wrong. I told Sir David, and he nodded with equanimity. He knew that many exciting results in mathematics are obtained by arithmetic errors.
Of course, if you put a dot inside (i.e., not touching any side) any shape and extend a ray from it, you can spin it around 360 degrees. The problem as posited is that what's being asked is implicitly inconsistent. Interior angles, sure: 180 (sides-2) and the reason for that is obvious by inspection. Circle: trick part, which queers the entire question. It doesn't test mathematical ability, other than having the skill to call foul on it. [I wrote something else, but I don't know who the audience is here....]