Geometry Part 27 Volume and Surface Area of Cubes
hello stranger let's start in this video we are going to talk about the
volume
and surface
area
of a cube the technical name of a cube is hexahedron it is a spatial figure where all faces are congruent squares it is really a special case of a box or a rectangular prism that we talked about in the other videos so if we want to talk about the volume
of a cube that has dimensions s by s by s we can think of multiplying the length by the width by the height just like we do in a normal box that will give us s times s times s or s cubed for the volume
we can also see that each side of the cube is going to have an area
of s squared but we have six different sides so the surface
area
is going to be six s squared so lets use these formulas in this example we are being asked to find the volume
and surface
area
of a cube and it is labeled to have dimension five feet which since we are told it is a cube means what ue has dimensions of five feet along each edge so let's ng to use the formula for the volume
of a cube you could use the formula for the volume
of a box length x width x height it will work the same but in the In the case of a cube, all we have to do is take that dimension five and cover it up, raise it to the third power, so the volume
will be 5 to the third power, which will be 125 cubic units, in this case, cubic feet, so there is the volume
of that cube now let's find the surface
area
so again the surface
area
is six times the area
of one side because all six sides have the same area
the area
of one side is five squared or twenty-five so the surface
area
is six times 25 which is going to be 150 what would the the units be ok since we're talking about area
it's always going to be square feet so square feet 150 square feet let's do some work backwards, what if I told you that a cube has a volume
of 1000 cubic meters and I ask you to do it? find its surface
area
so we'll still need the volume
and surface
area
formulas but we'll use them a little differently so if I tell you the volume
is 1000 cubic meters then I'll give you the v in the equation of the volume
and I'm going to ask you to find s so that 1000 is equal to s cubed so we're looking for a number that you multiply by itself three times to give you a thousand a way to find it or at least approximate it is just start counting cubes
one cubed is one two cubed is eight three cubed is 27 and so eventually you're going to get to 10 cubed which is a thousand and then you know that s equals 10.![geometry part 27 volume and surface area of cubes]()
What would the units be? If the
volume
is in cubic meters then the length of an edge will be in meters it is a linear measurement so s will be 10 meters you can also do this on your calculator calculators usually have the ability to take cube roots , so what you're going to want to look for on your calculator would be something like xth root button I'm looking for the cube root of 1000 so I would press 3 and the n I would have to press on my calculator the x the root button is actually on top from another function so I have to press second then the x button then type thousand then press equals and then you'll get 10 that way too. ok now that we have found that s is 10 we are not done yet because we need to find the surface
area
so let's plug in the surface
area
formula to have uppercase s the surface
area
equals 6 by small s squared or 10 squared so it will be 6 times 100 which is 600 and the units will be square meters this time because area
is always in square units let's look at one more cube question that says if you double the length of the edge of a cube how many times greater is the volume
of the new cube than the volume
of the original how has the surface
area
changed in this case i am going to keep the dimensions of my cube as s when double the dimensions now i get a new cube i'll just draw a little sketch of a larger l cube here my largest cube has dimension 2s so what would its volume
be well its volume
would be whatever the length of the side raised to the third power what if means that that means two s times two s times two s so that would be two you can rearrange the Commutative Property of Multiplication tells us we can do 2 times 2 times 2 times s times s times s so it will be 2 times 2 times 2 is 8 s to the third power notice that that is 8 times larger than the volume
of this one this one had a volume
of s cubed this one has a volume
of eight s cubed so it is eight times larger so the answer to the first part
is eight times now we were also asked how the surface
area
would change so come on look at the formula for surface
area
the surface
area
of the smallest cube is going to be six s squared according to the formula for surface
area
of the largest cube we need to figure out by substituting 2s in place of s so that the surface
area
is going to be 6 times 2s squared well that's 6 times 2s time s 2s which is 6 times 2 times 2 times s times s going to be, let's look at 24 s squared so how do you comp for this the surface
area
of the smaller cube was 6s squared the surface
area
of this new larger cube is 24s squared if you want to know how many times bigger it is to divide what is 24s squared divided by 6s squared that just goes to be 4. so the surface
area
is 4 times larger when you double each of the dimensions I hope you found this video helpful if you did please give it a thumbs up it will help other students find the video![geometry part 27 volume and surface area of cubes]()
