Express the edge length of a cube as a function of the cube's diagonal length D Then express the surface area and volume of the cube as a function of diagonal length. So the first thing we need to realize is we need a cube. So I drew a cube and we're talking about the diagonal of the whole cube. Well, I need to have some relationships between that diagonal of the cube and a side of the cube. So the first thing I'm going to realize is that if I take a side of the cube and a diagonal of just one of the faces of the cube and then the diagonal of the whole cube, I actually have a triangle and it is a right triangle. The right angle is occurring right here. It is rotated, but it's still a right angle. So we know that we can use our Pythagorean theorem that says a ^2 + b ^2 = C ^2. Now we want to figure out how to have the diagonal with just the side. We don't want the diagonal of a face. So we're going to look at just a face of the cube and realize that that diagonal here is really just side. Actually, let's call this F instead of D. So the face, the diagonal of a face is really side and side and it forms another right triangle. So we can look at this and realize that side squared plus side squared is going to equal the F ^2. Here's the F. So side squared plus side squared equal F ^2 or two side squared equals F ^2. If we wanted to solve for F, we would get F equaling F sqrt 2. So that was just one face of the cube. Now if we look at that that other right triangle, remember these are both right triangles. So we can use the Pythagorean theorem. We know that side squared plus the diagonal of the face squared is going to equal the diagonal of the cube squared. So s ^2 + F ^2 equal d ^2 s ^2 plus instead of F we're going to put in what it equaled that we found back here S root 2 ^2. So S root 2 ^2 is going to be two S squareds, or we're going to get a total of 3 S squareds equaling our diagonal. We wanted to solve for S. We wanted it in terms of a side of the cube. So I'm going to divide by three and I'm going to take the square root. So we get s ^2. I left my there s ^2 equaling d ^2 / 3. And then when we square root it, we get S equal D over root 3. We're never going to leave a radical in the denominator. So we're going to multiply by root 3 over root 3 to get S equal D root 3 / 3. So that's the side length. Then it asks us to find the surface area. Well, the surface area is made-up of six faces, and we can find the area of one face by side squared. So six faces times the area of one of the faces side squared. 6 * d root 3 / 3 ^2 is going to simplify to 2D squared. Then it asks for the volume, and the volume of a cube is just the side cubed. So we take our side that we found before D root 3 / 3 and we cube it. When we simplify that out, we get D cubed root 3 / d If we go back to the previous page, we can see that is indeed the answers we got.