# Math Help - Cube woes

1. ## Cube woes

Hi guys. I'm having trouble with this math problem. I have a cube that I know the side lengths of. It is orientated so that one of the corners is at a right angles to the ground. Two opposite corners of the cube are labelled O (highest corner) and P (lowest corner). The line OP is perpendicular to the ground. The cube is submerged so that the middle of the cube is at ground level. What is the shape of the intersection with the ground? What area does this intersection make and what shadow does it make?

After visualising the problem, I've come to the conclusion that this will be a hexagon shape. After experimenting with a die I don't think that this hexagon intersection will be a perfect (same-sided) one. I'm really struggling to find the area of this intersection though. Any help appreciated!

2. It is, actually - it's a regular hexagon.

Raise it a bit, or lower it a bit, so that the ground cuts through the three corners adjacent to either the top or the bottom, and you can see through symmetry that this is now an equilateral triangle. Half way between those two equilateral triangles, by symmetry again, must be a hexagon.

I may not have explained it very well!

3. Originally Posted by hellohello123
Hi guys. I'm having trouble with this math problem. I have a cube that I know the side lengths of. It is orientated so that one of the corners is at a right angles to the ground. Two opposite corners of the cube are labelled O (highest corner) and P (lowest corner). The line OP is perpendicular to the ground. The cube is submerged so that the middle of the cube is at ground level. What is the shape of the intersection with the ground? What area does this intersection make and what shadow does it make?

After visualising the problem, I've come to the conclusion that this will be a hexagon shape. After experimenting with a die I don't think that this hexagon intersection will be a perfect (same-sided) one. I'm really struggling to find the area of this intersection though. Any help appreciated!
In fact it is a regular hexagon. Each edge of the hexagon joins the midpoints of two adjacent edges of the cube. So if the cube has sides of length $a$ then the hexagon will have sides of length $a/\sqrt2$. By splitting the hexagon into six equilateral triangles you can then find the area of the hexagon.

There's a good picture of the hexagonal cross-section of the cube here (scroll about halfway down the page).

The shape of the shadow will depend on where the sun is. If it is immediately overhead then the shadow will be another regular hexagon.

4. Thanks fellas