Estimating the Circumference of a Circle

Circumference of a Circle

How long is the circumference of a circle? Unlike polygons, we can't just measure with our ruler. You could use a piece of string to wrap around a circle, mark the length then stretch it flat to get a pretty good estimate. But it's hard to do that on a piece of paper. We could also just look up the answer. But before we look it up, I'd like todo a little geometry exercise. One of the things that happens in astronomy (and much of science) is that we often have to find a way to bound our estimates. That means we try to find some pair of numbers that describe the largest and smallest the real answer might be. Sometimes we really can't figure out the exact answer. If that happens, it might still be useful to have not just an estimate, but also know if our estimate is off by a little or off by a lot.

So what we'll do is find a way to estimate using a very old technique of trying to find the limit. In the end, we'll make an estimate based on our approximations.

Tools for the Task:

• printed copy of Perimeter.pdf (attached below).
• ruler, preferably metric (so you can more easily do the measurements)
• pencil (of course!

First lets start with a circle with an inscribed square. That means we make the square as big as possible while still fitting inside the circle. Here is what this looks like (ignore the dashed lines, that are just guides for later):

We can use the perimeter of the square as an estimate of the perimeter of the circle. You're probably thinking its not a very good estimate and you'd be right. But it also forms a lower bound. That is, the perimeter of the circle is certainly larger than the perimeter of the square. We could have used an polygon inside the circle, but I chose a square. We'll change that in a minute, but first....

Lets draw a square around the circle so the circles is inscribed in the square. That looks like this:

Again, we can use the perimeter of this square as an estimate for the perimeter of the circle. Again, it's not a very good estimate. But this square provides an upper bound for the perimeter of the circle; that is, the perimeter of the circle is certainly smaller than the perimeter of the square.

Upper and lower bounds appear all over the place in daily life. When you drive down the road and you pass another car, you automatically know an upper bound for the speed of the other car---it's your speed. The other car must be travelling slower than you or you couldn't pass it. When another car passes you, you automatically know a lower bound for the speed of the other car---again, it's your speed. The other car must be travelling faster than you or it couldn't pass you. You can probably think of other examples.

Back to our circle. We can do better by turning our squares into octagons. First lets do this for the inner square. To convert it into an octagon, we will keep the four corners we had and add four more vertices (this is the plural form of vertex which is just the point where the two sides meet). Those dashed guide lines show us where we want to put the new vertices; right at the half-way points between the old vertices like this:

If we measure the perimeter of this octagon, then we have a better estimate of the perimeter of our circle. It's still too small, but not as bad as the first try. So our new estimate forms a better lower bound that the first estimate. Its a better lower bound because we know it is closer to the correct answer without going over.

Let's do the same thing now for our outer square, converting it into an octagon. That looks like this:

At this point, you know the drill---measure this perimeter and we get a better upper bound on the perimeter of the circle. It's better because is closer to the real perimeter without going under.

Okay, we could repeat this again, using a decahexagon which is the fancy name for a 16-sided regular polygon. But that not only gets tedious, but it's hard to measure accurately with just a ruler. So instead, we'll take our upper bound and our lower bound and average them and call that our best estimate.

If you print out the file Perimeter.pdf (attached below) you can do the exercise yourself. The starting image looks like this:

You'll have to convert the squares to octagons then measure. If you use a metric ruler, you can probably measure each side to the nearest 1/2 mm. The circle has a diameter of 120 mm. I measured the inner octagon as 368 mm (8 x 46 mm) and the outer octagon as 396 mm (8 x 49.5 mm). The average of these two gives 382 mm.

Okay, so that's nice for our circle. What if I have a different circle? If the circle is twice the diameter, what happens to the octagons? They end up twice as big, too. So if we look at the ratio of the perimeter to the diameter, we have a way of converting the diameter of any circle into a perimeter. For our estimate, this ratio is 382/120. That's a pretty awkward number, but its a bit more than 3 (use a calculator and you get 3.18).¹ An even better "simple" estimate is 22/7 which is about 3.14 which is what we would get if we worked very hard and split our octagons into decahexagons.

I know that in your math classes, you've talked about using an estimate of about 3 for the ratio of a circle's perimeter to it's diameter. Now you can explain where that "magic number" came from and you can tell people you know how to get an even better estimate.

NB: Parents, if you are feeling ambitious, you can download the ZIP archive below which contains high-quality PDFs of all of the above images as well as a drawing of the decahexagon arrangement that you can use to do your own estimating.  The PS102 mathematics text has a section on geometric constructions, but I'm not sure if they will actually do this in class.  I've taken a somewhat different approach than the classical methods which emphasize constructions using only a straight-edge and a compass because we're interested in making a measurement which is much less abstract.

Footnotes

¹ This is an error of only just over 1% which is quite good. If you are willing to be very careful and actually construct the decahexagon, you can get an error of much less than 1%. I tried and was able to get an error of about 0.4%. But that's an exercise for the adults, not the children....

Stop the World I Want to Get Off!

Around the Earth in 24 hours

Now that we know how to estimate the circumference of a circle from its radius we can go back to what we really want to know, if we travel around the earth in 24 hours, how far do we go and how fast? Okay, get out your calculators (or you pencils and paper)....

The Earth is about 12,750 km in diameter. We'll round that up to 13,000 km and multiply by 3 to get the circumference as 39,000 km. It takes you 24 hours to travel that distance. So we divide by 24 hours to get just over 1,600 kph (1,000 mph)! That's faster than the speed of sound! It's a good thing the Earth also drags the atmosphere along with it, too.

Actually, that's the speed at the equator. We're pretty far north compared to the equator so it turns out we' "only" travelling about 1,200 kph (760 mph). That works out to be about 1 km every 3 seconds. Quick question: how fast would you be travelling if you were standing at the North Pole? At the South Pole?

Around the Sun in 365 (and 1/4) Days

Now let's repeat our estimate for our annual trip around the Sun. The distance to the sun is 150 million kilometers. So the trip around the Sun is almost 900 million kilometers. Let's estimate the year as 360 days just because we can do the math easier:

900,000,000 km / 360 days = 2,500,000 km/day.

Okay, there's 24 hours in a day, so since we're estimating, 2,400,000 km/day is pretty close and it divides by 24 easily, so we'll just "round" down a little to get 100,000 kph. Yep, you read that correctly, 100,000 kilometer every hour. That works out to almost 28 kps (kilometers per second).

Around the Galaxy in 250 Million Years (More or Less)

You might have thought we were done. Heh, heh. The Sun is not sitting still, it's orbiting the galactic center. It takes a very long time to go around. People have not been around on Earth long enough to have witnessed a single full orbit, but we can still measure the distance to the galactic center and how fast we're moving. The distance to the center of our galaxy is about 28,000 light-years. That makes the diameter of our orbit about 56,000 light-years. Since it takes 250,000,000 years to cover "onlhy" 56,000 light-years, that sounds pretty slow compared to all the other speeds. Get your calculator and you find that works out to 0.000672 light-years per year. But...

A light-year is the distance traveled by light¹ in one year. And light travels fast. 300,000 kps (that's kilometers per second) fast. It takes light a bit over one second to get to the moon. In that same second it goes around the earth 7-1/2 times.² And there are a lot of seconds in a year: (60 sec/min) x (60 min/hour) x (24 hour/day) x (365 days/year) gives 31,536,000 seconds/year. The official definition of a light-year is a little different and it works out to be 9,460,730,472,580,800 meters.³ Call it 9.5 trillion kilometers.

Okay, at the risk of being boring, we do the math again, and just get out your calculator:

(9,500,000,000,000 km/ly) x (3 x 56,000 ly) / (250,000,000 year) = 6,384,000,000 km/year

Convert it to kps and we get just over 200 kps!

Stop!

Okay, so we're spinning around the Earth at a "mere" 1200 kph taking a whole 3-seconds to travel one kilometer. Then the Earth is whipping around the Sun at nearly 28 kilometers every second. And if that wasn't enough, the whole Solar System is whizzing around the galactic center at over 200 kilometers every second. Whew, that's fast. But wait! Our galaxy isn't sitting still. We're currently on a collision course with M31, the Andromeda galaxy speeding toward one another at about 130 kps⁴. And our entire local group of galaxies is heading in the general direction of galaxies in the constellation Virgo at a speed of about 5,000 kps!.⁵

Footnotes

¹ Technically, its the distance traveled by light in a vacuum. That may sound picky, but light slows down when it has to travel through matter, even the air. It can slow down a lot in certain types of naturally occuring crystals and glasses. In fact, your eyes depend on this effect in order to focus!  Light coming in at an angle to a medium where its speed changes results in the light bending. The same thing happens to water waves when, for example, the cross from a deep to a shallow area (or vice-versa).  Also, while we talk about "light," the speed really applies to any form of electromagnetic radiation, including things like radio and X-rays.

² The numbers I used here are not consisten with our earlier estimate of the size of the earth but used the slightly more accurate number of 40,000 km for the radius.

³ http://en.wikipedia.org/wiki/Light_year. The IAU uses 365.25 days/year for their calculation and a more exact speed of light number.

⁴ See http://www.cita.utoronto.ca/~dubinski/tails/node11.html.

⁵ See http://adsabs.harvard.edu/abs/1985vcg..work..391D