What is the difference in surface area between cube with an edge length of 16 inches and a cube with an edge length of five inches?
Before we start, a few things we should know. We’re dealing with surface area. Surface area of a cube is the area of one side, so side squared, multiplied by six because a cube is made of six equal faces. One of our cubes has a side length of 16 inches. That’s the larger square. So we’ll say the surface area of the larger equals six times 16 squared.
Now the smaller, surface area of the smaller is equal to six times five squared. We want to take the difference of these two. Six times 16 squared minus six times five squared. You notice that both of these terms are being multiplied by six. So we can undistribute that six. Our new equation would say six times 16 squared minus five squared.
Here’s where a big temptation will come in. You might want to do this. You might want to subtract five from 16 and then square it. However, this is not true. We do not distribute exponents over addition and subtraction. And you’ll see why. 16 squared equals 256. Five squared equals 25. We now have six times 256 minus 25. 256 minus 25 equals 231. Six times 231equals 1386. Remember we’re dealing with surface area. And that means our unit is inches squared.
Let’s go back to this distribution of exponents idea and see why it doesn’t work. If you had begun by subtracting five from 16, your new statement would say 11 squared times six. 11 squared equals 121. Multiply that value by six and you get 726. It’s not the same thing. I wanna show you what this looks like with an image. Our small square is five by five. Our large square is 16. That means we have a partial length of 11 on two sides and then a length of 16 on the other two sides.
The difference in area is how much bigger the blue square is than the yellow square. If we break that difference up into these three spaces, we have a square that measures 11 by 11, a rectangle that measures 11 by five, and another square that measures 11 by five. See this difference of 11 squared? It only accounts for this tiny corner of difference. We have this 11 squared equal to 121, 11 times five equals 55, and 11 times five equals 55.
If you add up these three values, it equals 231. And that’s why we have to take the full value of 16 squared, 256, the full value of 16 squared and then subtract the corner, 25, from that value. After we find the surface area difference, and the final answer is 1386 inches squared.