in this video we're going to work on calculating volume for different shapes so let's start with this problem what is the volume of a cylinder that has a radius of 4 inches and the height of 6 inches so let's begin with a picture so let's say this is the cylinder that we're dealing with and so the radius of the cylinder is four inches and the height is six inches so what is the volume of a cylinder the volume of a cylinder is the base times the height and the base is the area of this circle which is the same as the area of the circle on the bottom the area of a circle is pi r squared so the volume of the cylinder is pi r squared times the height so all we need to do is plug in the values that we have into this equation so the radius is four inches and the height is six now four squared that's four times four which is sixteen and sixteen times six we know that ten times six is sixty six times six is thirty-six sixty plus thirty-six is ninety-six so sixteen times six is 96 so our answer is 96 pi cubic inches now if you want to get the decimal value here's what's going to be i'm going to use the pi value in my calculator which is 3.14159 with some other numbers added to it so let's take that multiply it by 96 and so i got 301.6 cubic inches now let's say if you just use 3.14 times 96 then you should get an answer that's close to it as well 301.44 so depending on if you use 3.14 or a more exact value of pi your answer will differ a little number two the surface area of a sphere is 256 pi square feet what is the volume of the sphere in cubic feet so let's draw a picture and so this is the radius of the sphere which is what we need to find now the formula to calculate the volume of a sphere it's four thirds pi r cube and the surface area is 4 pi r squared so we're given the surface area and we need to calculate the radius once we have the radius we can then plug it into this formula to calculate the volume so let's go ahead and begin so the surface area is 256 pi and let's solve for r so what we need to do is we need to divide both sides by 4 pi on the right these two will cancel on the left we could cancel pi and so 256 divided by 4 is 64. so we have 64 is equal to r squared and now we can take the square root of both sides so the square root of 64 is eight so the radius is eight feet so now that we have the radius we can now calculate the volume so the volume is going to be four over three pi times r cubed but r is eight so eight raised to the third power that's eight times eight times eight and so that's going to be 512. so 512 times 4 divided by 3 so that's 2048 over 3 times pi so this is our answer in cubic feet that's the exact answer 2048 pi over 3. now to get the decimal value let's multiply pi which is one five 3.14159 times twenty forty eight divided by three and so you should get two thousand one hundred forty four point sixty six but i'm going to round it to 0.7 and the units cubic feet and so that's the volume of this sphere that's how you could find it if you're given the surface area of a sphere now let's move on to the third question what is the volume of a cone with a radius of five centimeters and a height of eight centimeters so let's begin with a picture so here we have a cone and here the radius is five centimeters and then the height which is basically eight centimeters it's this distance here so with this information given the radius and the height what is the volume of the cone all we need to do is use this formula the volume of a cone is one third of that of a cylinder so it's one third pi r squared times the height so we have the radius which is 5 and we know the height which is 8. 5 squared or 5 times 5 is 25 and what is 25 times eight so if you had eight quarters how much money do you have eight quarters is basically two dollars so 25 times eight is 200 and so our answer is going to be 200 pi over three and we're dealing with centimeters and so this is going to be cubic centimeters the units for volume is always cubic units cubic centimeters cubic feet cubic inches or something like that so now let's convert this into a decimal so 200 pi divided by 3 is the same as 209.44 if you round it cubic centimeters and so you can report your answer like this if you want to or like this number four what is the volume of a rectangular prism with dimensions four feet by five feet by six feet so here's the rectangular prism it looks like a box and so that's how we can draw to find the volume you just need to multiply the length by the width by the height and so this problem is not going to be too difficult because we have the length and the width and the height it really doesn't matter what order we multiply those three numbers so it's just going to be 4 feet times 5 feet times 6 feet and four times five is twenty and twenty times six if you have six twenty dollar bills you have a hundred twenty dollars and since we multiplied feet by feet by feet it's going to be cubic feet or feet raised to the third power and so that's the volume of this particular rectangular prism what about this problem the length of the longest diagonal in the cube is 12.1243 centimeters what is the volume of this cube in cubic centimeters so if we're given the diagonal length of a cube the longest diagonal how can we use that to calculate the volume of a cube so let's call this point a b and c so the distance between a and b that's one diagonal and the distance between a and c is the longest diagonal which is what we're trying to find now for a cube all sides are the same so we can call this x x and x so if we focus on let's say the blue triangle notice that we can come up with this equation let's call the hypotenuse of that triangle l so l squared is equal to x squared plus x squared now if we focus on let's say the green triangle let me draw that better notice that we can use the pythagorean theorem again which is a squared plus b squared equals c squared but here we have l and x and let's call the longest diagonal d which is what we're trying to find let me use a different color for that so we could say that d squared the hypotenuse for the green triangle is l squared plus x squared so now because l squared is x squared plus x squared i'm going to replace l squared in the second equation so i have d squared is x squared plus x squared plus another x squared so d squared is equal to one x squared plus one x squared plus one x squared is three x squared now if we take the square root of both sides the length of the diagonal of a cube is the square root of three times the side length and so that's the formula that we need in this example now we mentioned that the volume of a rectangular prism is the length times the width times the height in this case for the cube the length and the width and the height they're all x so x times x times x so the volume of a cube is x cubed so let's write that equation here we're going to use it later so first we need to calculate x using this formula and so we have the length of the diagonal which is 12.12436 centimeters and that's equal to the square root of three times x so let's divide both sides by the square root of three if you divide twelve point one two four three six by the square root of three you get seven technically like seven point zero zero zero zero zero two five one but for the most part that's about seven so x is seven now we can use this equation so the volume is going to be seven centimeters raised to the third power so seven to the third power or seven times seven times seven that's 343 so our final answer is 343 cubic centimeters and so now you know how to find the volume of common shapes like cubes rectangular prisms cones cylinders and spheres and so that's it for this video thanks for watching you