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## 7th grade

### Unit 6: Lesson 1

Area and circumference of circles- Geometry FAQ
- Radius, diameter, circumference & π
- Labeling parts of a circle
- Radius, diameter, & circumference
- Radius and diameter
- Radius & diameter from circumference
- Relating circumference and area
- Circumference of a circle
- Area of a circle
- Area of a circle
- Partial circle area and arc length
- Circumference of parts of circles
- Area of parts of circles
- Circumference review
- Area of circles review

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# Relating circumference and area

Sal uses formulas and a specific example to see how area and circumference relate..

## Want to join the conversation?

- I am really, really confused. Where does the 2*pi*r come from? Can someone explain to me how this works??(15 votes)
- Pi is the ratio of circumference over diameter. 2 and r come from the formula of diameter.

You have to find circumference of a circle. Pi comes here because of its ratio. 2 and r comes because it equals the diameter.

So pi times 2 times r is basically circumference over diameter times diameter which gives circumference.

So that is where 2*pi*r comes from.

Hope this helps.(12 votes)

- At0:38Sal mentions that from Circumference you can figure out Radius; and then from Radius you can figure out Area. What does he mean by that?(17 votes)
- Circumference is 2 times pi times radius. That how you figure radius from circumference. Area is pi times radius squared. This is how you figure out Area from Radius.

Hope this helps(4 votes)

- what is the circumference of a circle and area of a circle?(6 votes)
- In geometry, the circumference (from Latin circumferences, meaning "carrying around") of a circle is the (linear) distance around it. That is, the circumference would be the length of the circle The area of a circle is pi (approximately 3.14) times the radius of the circle squared. The circumference is pi times the diameter if it were opened up and straightened out to a line segment.

The circumference of a circle can be found by multiplying pi ( π = 3.14 ) by the diameter of the circle. If a circle has a diameter of 4, its circumference is 3.14*4=12.56. If you know the radius, the diameter is twice as large.(8 votes)

- Wait, where did 2 pi r come from?I just don't understand pretty much anything in this video.I need help!(2 votes)
- 2 pi r is the way to find the circumference when you have the radius.(15 votes)

- how do i find the diameter of a circle when all i have is circumference(8 votes)
- If you measure a number of different circle circumferences and compare them to their diameters you will always find that there is a constant relationship between the two elements [C - circumference and d-diameter] which we call π. This can be expressed as C/d = π. C & d are in a constant proportion to each other (if you increase d then C will also increase (that constant proportion is π). If you know two of the values in this equation then with some algebraic manipulation you can rearrange the formula to give you the answer you require. If you have C and want to find d; then multiply out d whilst dividing by π to get C/π = d.(3 votes)

- At3:32, how is 2 pi squared equal to 4 pi squared?

Also, how is pi divided by pi squared 1 over pi? Why doesn't he include the 4?(8 votes) - What would the area of a circle be if the circumference is 3.14 units?(5 votes)
- right at the beginning, he says the circumference=6π, but I'm trying to solve a problem that has circumference=8.45. There's no pi at the end. How does that change how I solve the problem?(4 votes)
- When the circumference is just a number like that, it means they solved for pi. If you were to divide by pi, you'd get something like 6π.(2 votes)

- I'm a little confused about the relationship between the circumference and the radius. I though that you could find it by dividing it by pi and then two but the I'm having a hard time catching on. Can I keep doing it that way or have I been doing it wrong all along?(2 votes)
- You are sort of correct, if C = 2πr, then r = C/(2π). You may not want to do r = C/π/2 depending on which calculator you are using because this is ambiguous and some calculators will be correct and some will be incorrect. So you could do C/2 = and then divide answer by π. Or do C/π = and then divide the answer by 2. Or just C/(2π) with parentheses in the denominator.(4 votes)

- Could someone please explain the whole 'pi * pi squared' part at3:40? I'm not quite getting it. Does it influence the outcome because the pi squared is in the denominator, or not? I'm pretty sure I've already learned the answer to my question, but I'm having a momentary lapse of memory on the subject. :/ Thanks.(2 votes)
- Usually Sal goes through all the steps but in this case he took a few shortcuts. Here's how I figured it.

A = PI r^2

A = PI(C/2PI)^2 <- put the formula for radius into the formula for area.

A = PI(C^2/2^2 PI^2) <- note that both 2 and PI are squared in this step

A = PI(C^2/4PI^2)

A = PI(C^2)/4PI^2

A = C^2/4PI <- at this step it seems like some kind of magic happened to get rid of the squared PI in the denominator but think of it like this:

It's basically PI/PI^2. What if we used regular numbers instead of PI? Something like ...

4/4^2. So now that's 4/16 which reduces to 1/4. The exponent has vanished! Basically multiplying any number by its reciprocal cancels that number out to make 1 and if there's an exponent it will reduce the exponent by 1. Yeah, I bet someone could explain this better than I did. :)(3 votes)

## Video transcript

- [Instructor] So we have a circle here, let's say that we know
that its circumference is equal to six pi, I'll write it, units. Whatever our units happen to be. Let's see if we can figure out, given that its circumference
is 6 pi of these units, what is the area going to be equal to? Pause this video and see
if you can figure it out on your own. And first, think about
if you could figure out the area for this particular circle, and then let's see if we can come up with a formula for
given any circumference, can we figure out the area and vice versa. Alright now let's work
through this together. The key here is to realize that from circumference
you can figure out radius and then from radius
you can figure out area. So we know that circumference,
which is six pi, so we know six pi is equal
to two pi times r, radius. And so what is the radius going to be? So the radius, we're
talking about that distance, well we can divide both sides by two pi, so let's do that. If we divide both sides by two pi, to solve for r, what are we left with? Well we have an r on the right hand side, we have r is equal to pi divided by pi, that's just one. Six divided by two is three. So we get that our
radius, right over here, is equal to three units. And then we can use the fact that are is equal to pi, r squared, to figure out the area. This is going to be equal
to pi times three squared. Oh and you have to
write parenthesis there. Pi, times three, squared,
which is of course going to be equal to nine pi. So for this particular example, when the circumference is six pi units, we're able to figure out that the area, this is actually going to
be nine pi square units, or I could write units squared. Cause we're squaring the radius. The radius is three units, so you square that, you
get the units squared. Now let's see if we can come
up with a general formula. So we know that circumference
is equal to two pi r. And we know that area is
equal to pi r squared. Can we come up with an expression or a formula that relates directly between circumference and area? And I'll give you a hint, solve for, you could solve for r right over here, and substitute back into
this equation, or vice versa. Pause the video, see if you can do that. Alright, so let's do it over here. Let's solve for r. If we divide both sides by two pi, do it in another color, so if we divide both sides by two pi, and this is exactly what we did up here, what are we left with? We're left with on the right hand side, r is equal to c the
circumference divided by two pi, the radius is equal to the
circumference over two pi. And so when we're figuring out the area, area, remember, is equal to
pi times our radius squared. But we know that our
radius could be written as circumference divided by two pi. So instead of radius I'll write
circumference over two pi. Remember, we want to relate between area and circumference. And so what is this going to be equal to? We get area is equal to pi
times circumference squared, over two pi squared is four pi squared. Now let's see, we have a pi, if we multiplied this out,
we'd have a pi in the numerator and a pi in the denominator, or two pi's in the
denominator being multiplied. So pi divided by pi squared, is just one over pi, and
so there you have it. Area is equal to circumference
squared divided by four pi. Let me write that down. So this is a neat, you don't
tend to learn this formula, but it's cool that we
were able to derive it. Area is equal to circumference
squared over four pi. And we can go the other way around. Given an area, how do we
figure out circumference? Well you could just put
the numbers in here, or you could just solve for c. Let's multiply both sides by four pi. Let's multiply both sides by four pi, and if we do that, what do we get? We would get four pi
times the area is equal to our circumference squared and then to solve for the circumference we just take the square
root of both sides. So you would get the square
root of four pi times the area is equal to r circumference. And you could simplify this
a little bit if you wanted. You could take the four
out of the radical, but this is pretty neat, how you can relate circumference and area.