Question 30. A hole in the shape of a hemisphere is carved into each face of a cube. The holes are identical and centered at the center of each face. The holes touch their neighbors at only one point. The cube has a side of two. What is the diameter of each hole? Let's take a look at our cube. First thing we can mark down is the centers on the faces of the cube. We know that they are on the center. We can connect them on the inside with a diagonal line between the two centers of the two faces. Next we can draw two segments like so, which make it up to the midpoint. And since the cube has a side of two, each of these values will be one. If we shift this upwards and have it meet the centers, we see that we have made a right triangle with the hypotenuse that is equal to two radii of the two spheres, which are identical. So this will also be equal to the diameter of a single hemisphere. We know that to get the hypotenuse of a triangle, we do a squared plus b squared equals c squared. In this case, both sides of the triangle are one. So one squared plus one squared equals c squared. So one plus one, two equals c squared. So the hypotenuse, as well as the diameter of a hemisphere, will be equal to two equals c squared. To get this, we just have to take the square root of both sides, and this gives us our answer, which will be c, the square root of two.

cube

hemispherical holes

diameter

right triangle

geometry