Circles: Radius, Diameter and Circumference Understanding the relationship between the radius diameter and circumference of a circle.
Circles: Radius, Diameter and Circumference
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- The circle is arguably the most fundamental shape in our
- universe, whether you look at the shapes of orbits of
- planets, whether you look at wheels, whether you look at
- things on kind of a molecular level.
- The circle just keeps showing up over and
- over and over again.
- So it's probably worthwhile for us to understand some of the
- properties of the circle.
- So the first thing when people kind of discovered the circle,
- and you just have a look at the moon to see a circle, but the
- first time they said well, what are the properties
- of any circle?
- So the first one they might want to say is well, a circle
- is all of the points that are equal distant from the
- center of the circle.
- All of these points along the edge are equal distant from
- that center right there.
- So one of the first things someone might want to ask is
- what is that distance, that equal distance that everything
- is from the center?
- Right there.
- We call that the radius of the circle.
- It's just the distance from the center out to the edge.
- If that radius is 3 centimeters, then this radius
- is going to be 3 centimeters.
- And this radius is going to be 3 centimeters.
- It's never going to change.
- By definition, a circle is all of the points that are equal
- distant from the center point.
- And that distance is the radius.
- Now the next most interesting thing about that, people might
- say well, how fat is the circle?
- How wide is it along its widest point?
- Or if you just want to cut it along its widest point, what
- is that distance right there?
- And it doesn't have to be just right there, I could have just
- as easily cut it along its widest point right there.
- I just wouldn't be cutting it like some place like that
- because that wouldn't be along its widest point.
- There's multiple places where I could cut it
- along its widest point.
- Well, we just saw the radius and we see that widest point
- goes through the center and just keeps going.
- So it's essentially two radii.
- You got one radius there and then you have another
- radius over there.
- We call this distance along the widest point of the
- circle, the diameter.
- So that is the diameter of the circle.
- It has a very easy relationship with the radius.
- The diameter is equal to two times the radius.
- Now, the next most interesting thing that you might be
- wondering about a circle is how far is it around the circle?
- So if you were to get your tape measure out and you were to
- measure around the circle like that, what's that distance?
- We call that word the circumference of the circle.
- Now, we know how the diameter and the radius relates, but how
- does the circumference relate to, say, the diameter.
- And if you're not really used to the diameter, it's very
- easy to figure out how it relates to the radius.
- Well, many thousands of years ago, people took their tape
- measures out and they keep measuring circumferences
- and radiuses.
- And let's say when their tape measures weren't so good,
- let's say they measured the circumference of the circle
- and they would get well, it looks like it's about 3.
- And then they measure the radius of the circle right here
- or the diameter of that circle, and they'd say oh, the diameter
- looks like it's about 1.
- So they would say -- let me write this down.
- So we're worried about the ratio -- let me
- write it like this.
- The ratio of the circumference to the diameter.
- So let's say that somebody had some circle over here -- let's
- say they had this circle, and the first time with not that
- good of a tape measure, they measured around the circle
- and they said hey, it's roughly equal to 3 meters
- when I go around it.
- And when I measure the diameter of the circle,
- it's roughly equal to 1.
- OK, that's interesting.
- Maybe the ratio of the circumference of
- the diameter's 3.
- So maybe the circumference is always three
- times the diameter.
- Well that was just for this circle, but let's say they
- measured some other circle here.
- It's like this -- I drew it smaller.
- Let's say that on this circle they measured around it and
- they found out that the circumference is 6 centimeters,
- roughly -- we have a bad tape measure right then.
- Then they find out that the diameter is
- roughly 2 centimeters.
- And once again, the ratio of the circumference of the
- diameter was roughly 3.
- OK, this is a neat property of circles.
- Maybe the ratio of the circumference to the diameters
- always fixed for any circle.
- So they said let me study this further.
- So they got better tape measures.
- When they got better tape measures, they measured hey,
- my diameter's definitely 1.
- They say my diameter's definitely 1, but when I
- measure my circumference a little bit, I realize
- it's closer to 3.1.
- And the same thing with this over here.
- They notice that this ratio is closer to 3.1.
- Then they kept measuring it better and better and better,
- and then they realized that they were getting this number,
- they just kept measuring it better and better and they were
- getting this number 3.14159.
- And they just kept adding digits and it would
- never repeat.
- It was a strange fascinating metaphysical number
- that kept showing up.
- So since this number was so fundamental to our universe,
- because the circle is so fundamental to our universe,
- and it just showed up for every circle.
- The ratio of the circumference of the diameter was this
- kind of magical number, they gave it a name.
- They called it pi, or you could just give it the Latin or the
- Greek letter pi -- just like that.
- That represents this number which is arguably the most
- fascinating number in our universe.
- It first shows up as the ratio of the circumference to the
- diameter, but you're going to learn as you go through your
- mathematical journey, that it shows up everywhere.
- It's one of these fundamental things about the universe that
- just makes you think that there's some order to it.
- But anyway, how can we use this in I guess
- our basic mathematics?
- So we know, or I'm telling you, that the ratio of the
- circumference to the diameter -- when I say the ratio,
- literally I'm just saying if you divide the circumference by
- the diameter, you're going to get pi.
- Pi is just this number.
- I could write 3.14159 and just keep going on and on and on,
- but that would be a waste of space and it would just be hard
- to deal with, so people just write this Greek
- letter pi there.
- So, how can we relate this?
- We can multiply both sides of this by the diameter and we
- could say that the circumference is equal to pi
- times the diameter.
- Or since the diameter is equal to 2 times the radius, we could
- say that the circumference is equal to pi times 2
- times the radius.
- Or the form that you're most likely to see it,
- it's equal to 2 pi r.
- So let's see if we can apply that to some problems.
- So let's say I have a circle just like that, and I were to
- tell you it has a radius -- it's radius right there is 3.
- So, 3 -- let me write this down -- so the radius is equal to 3.
- Maybe it's 3 meters -- put some units in there.
- What is the circumference of the circle?
- The circumference is equal to 2 times pi times the radius.
- So it's going to be equal to 2 times pi times the radius,
- times 3 meters, which is equal to 6 meters times
- pi or 6 pi meters.
- 6 pi meters.
- Now I could multiply this out.
- Remember pi is just a number.
- Pi is 3.14159 going on and on and on.
- So if I multiply 6 times that, maybe I'll get 18 point
- something something something.
- If you have your calculator you might want to do it, but for
- simplicity people just tend to leave our numbers
- in terms of pi.
- Now I don't know what this is if you multiply 6 times
- 3.14159, I don't know if you get something close to 19 or
- 18, maybe it's approximately 18 point something
- something something.
- I don't have my calculator in front of me.
- But instead of writing that number, you just
- write 6 pi there.
- Actually, I think it wouldn't quite cross the
- threshold to 19 yet.
- Now, let's ask another question.
- What is the diameter of the circle?
- Well if this radius is 3, the diameter is just twice that.
- So it's just going to be 3 times 2 or 3 plus 3, which
- is equal to 6 meters.
- So the circumference is 6 pi meters, the diameter is 6
- meters, the radius is 3 meters.
- Now let's go the other way.
- Let's say I have another circle.
- Let's say I have another circle here.
- And I were to tell you that its circumference is equal
- to 10 meters -- that's the circumference of the circle.
- If you were to put a tape measure to go around it and
- someone were to ask you what is the diameter of the circle?
- Well, we know that the diameter times pi, we know that pi times
- the diameter is equal to the circumference; is
- equal to 10 meters.
- So to solve for this we would just divide both sides
- of this equation by pi.
- The diameter would equal 10 meters over pi or
- 10 over pi meters.
- And that is just a number.
- If you have your calculator, you could actually divide 10
- divided by 3.14159, you're going to get 3 point something
- something something meters.
- I can't do it in my head.
- But this is just a number.
- But for simplicity we often just leave it that way.
- Now what is the radius?
- Well, the radius is equal to 1/2 the diameter.
- So this whole distance right here is 10 over pi meters.
- If we just 1/2 of that, if we just want the radius, we
- just multiply it times 1/2.
- So you have 1/2 times 10 over pi, which is equal to 1/2 times
- 10, or you just divide the numerator and the
- denominator by 2.
- You get 5 there, so you get 5 over pi.
- So the radius over here is 5 over pi.
- Nothing super fancy about this.
- I think the thing that confuses people the most is to just
- realize that pi is a number.
- Pi is just 3.14159 and it just keeps going on and on and on.
- There's actually thousands of books written about pi, so
- it's not like -- I don't know if there's thousands, I'm
- exaggerating, but you could write books about this number.
- But it's just a number.
- It's a very special number, and if you wanted to write it in a
- way that you're used to writing numbers, you could literally
- just multiply this out.
- But most the time people just realize they like leaving
- things in terms of pi.
- Anyway, I'll leave you there.
- In the next video we'll figure out the area of a circle.
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