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### Course: 6th grade (Illustrative Mathematics) > Unit 1

Lesson 13: Lesson 18: Surface area of a cube# Evaluating expressions with variables: cubes

Discover how to calculate the total surface area of cube-shaped containers with varying side lengths. Learn the formula 6x², where x represents the side length of a cube. Apply this formula to find the surface area of multiple cubes, and add the results to determine the combined surface area that needs painting. Created by Sal Khan.

## Want to join the conversation?

- What does ^ mean ?(136 votes)
- The carat is normally used to indicate an exponent. For instance, x^2 is x to the power of 2, or x squared. x^3 is x to the power of 3 or x cubed.(166 votes)

- Because my region, I usually use the sign "," (coma) to denote decimal numbers. I can't figure out if the dot confuses me because I'm unfamiliarizated with it, or it really can be pretty problematic. For instance, does anyone else confuses the expression that shown in the video "1.5²" with "1 times 5²"?

By the way, I'm sorry for my terrible english grammar.(73 votes)- Andres,

Your English grammar is almost perfect.

It is quite understandable that the "." (period) causes confusion in regions that use the "," (comma) as a decimal point. If I tried to do math using the "," (comma) as the decimal point instead of the "." (period), I would be very confused.

I try to use the "*" (asterisk) as a multiplication symbol because it is less likely to cause confusion. You can also use the "•" (dot) for multiplication, but it is harder to type on most keyboards.(91 votes)

- so how come he didnt make it 2 to the third power(16 votes)
- Sal didn't make it 2^3 because the formula states that to find the surface area of a cube, you do 6(x^2). Now think about it, let's say that you have a regular, 2D square. The side lengths are all 6cm. The formula to find the area is x^2. You would get 6^2=36cm^2. You would not, however, do x^3. The formula to finding the surface area of a cube is only slightly different than that of finding the area of a square.(35 votes)

- How does 1.5 x 2 = 2.5. I'm confuse can you please explain more.(18 votes)
- 1.5*2 is not 1.5 squared. 1.5*1.5 is 1.5 sqaured(7 votes)

- Which word phrase could correspond to the variable expressions n/8?(12 votes)
- The quotient of n divided by 8 = n/8

One-eighth of n = n/8

n things shared among 8 people = n/8

how many 8's in n = n/8

ratio of n and 8 = n/8

All of the above phrases correspond to n/8

8 times a number is n. what is the number?

All the above correspond to n/8(21 votes)

- Why is the formula 6x^2, rather than 11x^2 or 7x^2?(8 votes)
- x^2 gives the area of one side of a cube. The 6 is in there because a cube has 6 sides.(20 votes)

- Before allowing the video to play I made cubes a, and b selecting the length of one side of each, squaring it, adding them together, then multiplying by six. Thus my equation was

a^2 + b^2 (6)

My question is, will doing it this way cause me problems later on down the road?(10 votes)- This will not work... You are missing a set of parentheses. Your version only multiplies b^2 with the 6. You need to write it as: 6(a^2 + b^2) or (a^2 + b^2) 6 or 6a^2 + 6b^2.(15 votes)

- Isn't The Measurement In Cenimeters(5 votes)
- Or often in Geometry, you will just see the measurement as "units" where you normally see cm, ft, etc.(15 votes)

- How to solve simultaneous equations with two positive signs e.g. 9a+b=13 3a+b=20 Or.

5a+b=27

a+b=7(7 votes)- You can use elimination method because its easier , so you use (9a+b=13) - (3a+b=20) so your answer will going to be 6a = 7 then switch the 6 to the other side of the equal sign and you got a= -7/6, then you got the answer for a, substitute a=-7/6 into (3a+b=20) so you multiply the 3a by -7/6 so the answer is -3.5 , switch the -3.5 to the other side and you get b=20+3.5 and b=23.5, so the answer for a is -7/6 and for b is 23.5 . For the 5a+b=27 and a+b=7 i'll give you the answer (a equals to 5 and b equals to 2) so try solve it :)(8 votes)

- why did he draw cubes that was confusing me the most(9 votes)
- bro your a god how do you not know(3 votes)

## Video transcript

The surface area of a
cube is equal to the sum of the areas of its six sides. Let's just visualize that. I like to visualize things. So if that's the cube,
we can see three sides. Three sides are facing us. But then if it was
transparent, we see that there are actually
six sides of a cube. So there's this one--
one, two, three in front-- and then one--
this is the bottom. This is in the back, and
this is also in the back. So you have three
sides of the cube. So I believe what
they're saying. The surface area of a
cube with side length x-- so if this is
x, if this is x, if this is x-- is given by
the expression 6x squared. That also makes sense. The area of each side is going
to be x times x is x squared, and there's six of them. So it's going to be 6x squared. Jolene has two
cube-shaped containers that she wants to paint. One cube has side length 2. So this is one cube
right over here. I'll do my best to draw it. So this right over
here has side length 2, so that's its dimensions. The other cube has
side length 1.5. So the other cube is going
to be a little bit smaller. It has side length 1.5. So it's 1.5 by 1.5 by 1.5. What is the total surface
area that she has to paint? Well, we know that the
surface area of each cube is going to be 6x
squared, where x is the dimensions of that cube. So the surface area of
this cube right over here is going to be 6. And now-- let me do it in
that color of that cube-- it's going to be
6 times x, where x is the dimension of the cube. And then the cube all
has the same dimensions, so its length, width, and
depth is all the same. So for this cube,
the surface area is going to be 6
times 2 squared. And then the surface
area of this cube is going to be 6
times 1.5 squared. And if we want the total
surface area she has to paint, it's going to be the
sum of the two cubes. So we're just going to
add these two things. And so if we were to compute
this first one right over here, this is going to be 6 times 4. This is 24. And this one right
over here, this is going to be a
little bit hairier. Let's see. 15 times 15 is 225. So 1.5 times 1.5 is 2.25. So 1.5 squared is 2.25. And 2.25 times 6-- so let
me just multiply that out. 2.25 times 6. Let's see. We're going to have
6 times 5 is 30. 6 times 2 is 12, plus 3 is 15. 6 times 2 is 12, plus 1 is 13. I have two numbers behind
the decimal-- 13.5. So it's going to be 13.5. And if I add these
two together, this is going to be equal to
the total surface area that she has got to paint, is
going to be 37.5 square-- well, I guess they're not
giving us the units. Well, 37.5 is going to be the
total area of square units of whatever the
units happen to be.