Geometry Part 27 Volume and Surface Area of Cubes

Hello, I'm missing, let's start in this video. We're going to talk about the

volume

and

surface

area

of ​​a cube. The technical name for a cube is a hexahedron. It is a space figure where all faces are congruent squares. It's really a special case of a box or rectangular prism that we talked about in the other videos. So if we want to talk about the

volume

of a cube with dimensions s by s by s, we can think of multiplying the length times the width times the height just like in a normal box that gives us s times s times s or s squared as

volume

, we can also observe that each side of the cube has an

area

of ​​s squared, but we have six different sides.

The

area

will be 6 s squared, so we use these formulas in this example. We are asked to find the

volume

and

surface

area

of ​​a cube and it is marked with a dimension of 5 feet, which, since we have been told it is a cube, means that it has dimensions of five feet on each edge, so we are goi to use the formula for the

volume

of a cube you could use the formula for the

volume

of a box length by width by height it will work the same but in the case of a cube we just need to take this dimension five and cube it raise it to the cube, so the

volume

will be 5 to the cube, which will be 125 cubic units in this case cubic feet, so there's the

volume

of this cube, now let's find the

surface

area

again the

surface

area

is 6 times that like a side because all six sides have the same

area

, the

area

of ​​a side is five squares or twenty-five, so the

surface

area

is six times 25, which is 150 which would be good units since we're about Speaking of

area

, it will always be square units, so square feet equals 150 square feet.

Let's work backwards a little, what if I told you that a cube has a

volume

of 1000 cubic meters and I asked you to find its

surface

area

, so we still need the

volume

and

surface

area

formulas, but we're going to use them a little differently. So if I tell you the

volume

is 1000 cubic meters, I'll give you the v in the

volume

equation and ask you to find s such that 1000 equals s in cubic. So we're looking for a number that you multiply by itself three times to give you a thousand one way to find these or at least about to start listing

cubes

.

One to the third is one two to the third is eight three to the third is 27 and so eventually you get to 10 cubic which is a thousand and then you know that s equals 10. What would the units be good if the

volume

is in cubic meters then the length of an edge in it will be meters It's 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 you should look for something like xth on your calculator root key I'm looking for the cube root of 1000 so I would press 3 and the n I would have to press the x on my calculator , the root key is actually above another function so the second thing I have to do is press then the x through key then type in thousand and then press the same and then you get 10 that way too okay now that we Once we figured out that s is 10, we're not quite done because we need to find the

surface

area

, so we're going to plug into the formula for the

surface

area

so we capitalized s, which equals the

surface

area

6 times little s for Squared or 10 squared, that's 6 times 100, which is 600, and the units this time will be square meters because

area

is always in square units.

Let's look at another question about

cubes

that says if you double the length of the edge of a cube how many times larger is the

volume

of the new cube than the

volume

of the original how has the

surface

area

changed in this case i'll give the dimensions of my cube just keep it like s when i double dimensions i will get a new cube now i will just draw a little sketch of a l Bigger cube here my bigger cube has dimension 2s so what would be its

volume

, its

volume

would be independent of the Length of side to the third power.

What does that mean that two s times two s times two s means 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 that becomes 2 times 2 times 2 is 8 s to the power of 3. Notice that that's 8 times the

volume

of this here this had a

volume

of s cubic meters this here had a

volume

of 8 s cubic cubic so it's 8 times the

volume

, so is the answer to the first

part

eight times. They also asked us how the

surface

would change, so let's look at the formula for the

surface

.

The

surface

area

of ​​the smaller cube will be six s squared, according to the formula the

surface

area

for the larger cube we need to find out by substituting 2s instead of s, so the

surface

area

will be 6 times 2s squared, so that's 6 times 2s time s 2s, that's 6 times 2 times 2 times s times s, which will be 24 s squared. How does this compare the

surface

of the smaller cube at 6s squared, the

surface

of this new larger cube at 24s squared if you want to know how many times larger it is when you divide it which is 24 squared divided by 6 Squared, that's 4.

So the

surface

area

is 4 times larger if you double each of the dimensions. I hope you found this video helpful, if please give a thumbs up so other students can find the video