Evaluating expressions with variables: cubes
About this video
A cube has six square faces, so its surface area is 6x^2, where x is the side length. The video uses this expression to find the surface areas of Jolene's two cubes, with sides 2 and 1.5 (24 and 13.5), and adds them to get 37.5 square units to paint.
You'll be able to
- explain why the surface area of a cube is 6x^2.
- evaluate 6x^2 for whole number and decimal side lengths, squaring before multiplying by 6.
- add surface areas to find a total area to paint.
Key words
- surface area
- The total area of all the outside faces of a 3D shape.
- cube
- A 3D shape with six square faces that are all the same size.
- squared
- A number times itself; x^2 means x times x.
- square units
- The units used to measure area, like square centimetres.
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.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.