Cylinder volume & surface area (video) | Khan Academy
How to derive the formula for the surface area of a cylinder. The width is the height h of the cylinder, and the length is the distance around the end circles. This is the circumference Cylinder relation to a prism · Cylinder as the locus of a line. Since a cylinder is closely related to a prism, the formulas for their surface areas Find the lateral surface area of a cylinder with a base radius of 3 inches and a. In this lesson, you'll review the definition of a cylinder and learn how to find its surface area. Then, you The sum of the areas of the two circles and the rectangle is: By combining like-terms, we can simplify this equation to.
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.
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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.
Circle Diameter to Area Calculator
Let's solve for r. Source In order to visualize the shape of the side of the can unroll the label. Notice the label is a rectangle. Source Roll the label back up. Notice that the width of the label is actually the circumference of the can. Source Put it all together and the surface area of a cylinder is the area of 2 circles plus the area of 1 rectangle!
How to Find the Radius of a Cylinder When Given the Volume and Height | Sciencing
Source Math Made Easy! Tip Admittedly, the formula for the surface area of a cylinder isn't too pretty. So, let's try to break the formula apart into understandable pieces. A good math tip is to try to visualize the geometrical shape with an object with which you are already familiar. What objects in your home are cylinders?
I know in my pantry I have a lot of cylinders - better known as canned goods. Let's examine a can. A can is made up of a top and bottom and a side that curves around. 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. 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. 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. We just figured that out. So it's going to be plus 64 pi centimeters squared. And now we just have to calculate it.