The given figure is made by placing a cube of side length 13 centimeters on top of another cube of side length 18 centimeters. Find its surface area.
The surface area is the sum of all the areas that cover the surface of an object. But what does that mean?
You could think about it like the space that you would paint if you would paint this object. Also, if you imagine picking it up, it’s all the places that you could touch the object. Our small cube has a side length of 13 centimeters. Because we know that it’s a cube, we can also say that each face is a square. And we know to find the area of a square, we take the side length and we square that value. The area of every face of the small cube is 13 squared. The question is, “How many faces of this small cube are showing?”
Here’s the right side. And opposite the right side is a left face that we can’t see but is part of this cube. We have a front face. And opposite the front side is a back side that we can’t see that will also be 13 squared. The surface area of the part that’s made up of this small cube would be equal to five times 13 squared. We say five times 13 squared because we need to add 13 squared to itself five times. So we simplify it and say five times 13 squared.
We won’t consider the bottom of this cube. We wouldn’t consider that sixth face. And that’s because these cubes together make up one figure. So we can’t separate the top from the bottom. We couldn’t paint underneath there because it’s attached. We’ll only be interested in this part, but we’ll come back to that a little bit later.
Let’s now consider the large cube surface area. It has a side length of 18, which means the area of each face will be 18 squared. The area of the front is 18 squared. And that means the part opposite the front, the back side, would also be 18 squared. The right face is 18 squared. And that means we’ll have a left side with area 18 squared.
At this point, it’s really tempting to think that we’ve covered every surface. However, we can’t forget the bottom of this figure, the square base, that would also measure 18 squared. So the large cube surface area would be five times 18 squared. This is because we have five faces that measure 18 squared each and we’re adding them all together. We can simplify that process by multiplying 18 squared by five. Now we’re ready to consider the area of the part shaded in pink.
How would we calculate this surface? We have the square that’s the top of our large cube, a square that has a side length of 18, with the smaller square stacked on top. We know that that smaller square has a side length of 13. To find the area of this extra piece, we’ll find the area of the top of the large cube, 18 squared. Then, we’ll find the area of the smaller square, 13 squared. And we’ll subtract. We’ll take away that small square from the large square. When we subtract 13 squared from 18 squared, we get 155. This additional piece has an area of 155.
To find the total surface area, we’ll need to add all these figures together. The surface area will equal five times 13 squared plus five times 18 squared plus 155. Five times 13 squared equals 845. Five times 18 squared equals 1620. And we bring down the 155. The surface area is equal to 2620. We’ll need to find the units. Our initial measurements were given in centimeters. And that means our surface area is a measure of centimeters squared. The surface area of this figure is 2620 centimeters squared.