### What is the relationship between cylinder volume and circumference? | Wyzant Ask An Expert

2. Brief description of the lesson To help students realise there is a relationship between volume of the cylinder and the radius, keeping the height constant. they can solve different types of problems they meet in the future. Two cylinders may have same volume and different circumference of the base, because they may have different heights. Think about two. Volume of Cone and Cylinder Picture a rectangle divided into two right The volume relationship between these cones and cylinders with equal bases and.

So that gives us centimeters cubed. Or pi cubic centimeters. Remember, pi is just a number. We write it as pi, because it's kind of a crazy, irrational number, that if you were to write it, you could never completely write pi.

So we just leave it as pi. But if you wanted to figure it out, you can get a calculator. And this would be 3. So it would be close to cubic centimeters. Now, how would we find the surface area? How would we find the surface area of this figure over here?

Well, part of the surface area of the two surfaces, the top and the bottom. So that would be part of the surface area. And then the bottom over here would also be part of the surface area. So if we're trying to find the surface area, it's definitely going to have both of these areas here. So it's going to have the 16 pi centimeters squared twice. This is 16 pi.

This is 16 pi square centimeters. So it's going to have 2 times 16 pi centimeters squared. I'll keep the units still. So that covers the top and the bottom of our soda can. And now we have to figure out the surface area of this thing that goes around. And the way I imagine it is, imagine if you're trying to wrap this thing with wrapping paper. So let me just draw a little dotted line here. So imagine if you were to cut it just like that.

## What is the relationship between cylinder volume and circumference?

Cut the side of the soda can. And if you were to unwind this thing that goes around it, what would you have. Well, you would have something. You would end up with a sheet of paper where this length right over here is the same thing as this length over here.

And then it would be completely unwound. And then these two ends-- let me do it in magenta-- these two ends used to touch each other. And-- I'm going to do it in a color that I haven't used yet, I'll do it in pink-- these two ends used to touch each other when it was all rolled together.

And they used to touch each other right over there. So the length of this side and that side is going to be the same thing as the height of my cylinder. So this is going to be 8 centimeters.

And then this over here is also going to be 8 centimeters. And so the question we need to ask ourselves is, what is going to be this dimension right over here. And remember, that dimension is essentially, how far did we go around the cylinder.

Well, if you think about it, that's going to be the exact same thing as the circumference of either the top or the bottom of the cylinder. So what is the circumference?

The circumference of this circle right over here, which is the same thing as the circumference of that circle over there, it is 2 times the radius times pi.

## Cylinder volume & surface area

Or 2 pi times the radius. So this distance right over here is the circumference of either the top or the bottom of the cylinder.

It's going to be 8 pi centimeters. So if you want to find the surface area of just the wrapping, just the part that goes around the cylinder, not the top or the bottom, when you unwind it, it's going to look like this rectangle. And so its area, the area of just that part, is going to be equal to 8 centimeters times 8 pi centimeters. So let me do it this way. It's going to be 8 centimeters times 8 pi centimeters. And that's equal to 64 pi. You have your pi centimeters squared.

So when you want the surface area of the whole thing, you have the top, you have the bottom, we already threw those there. And then you want to find the area of the thing around. The figure below shows a rectangle "split" along a diagonal, demonstrating that the rectangle can be thought of as two equal right triangles joined together.

The areas of rectangles and right triangles are proportional to one another: In a similar way, the volumes of a cone and a cylinder that have identical bases and heights are proportional. If a cone and a cylinder have bases shown in color with equal areas, and both have identical heights, then the volume of the cone is one-third the volume of the cylinder.

### Cylinder volume & surface area (video) | Khan Academy

Imagine turning the cone in the figure upside down, with its point downward. If the cone were hollow with its top open, it could be filled with a liquid just like an ice cream cone. One would have to fill and pour the contents of the cone into the cylinder three times in order to fill up the cylinder. The figure above also illustrates the terms height and radius for a cone and a cylinder. The base of the cone is a circle of radius r. The height of the cone is the length h of the straight line from the cone's tip to the center of its circular base.

Both ends of a cylinder are circles, each of radius r. The height of the cylinder is the length h between the centers of the two ends.

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The volume relationship between these cones and cylinders with equal bases and heights can be expressed mathematically. The volume of an object is the amount of space enclosed within it. For example, the volume of a cube is the area of one side times its height.