So, for example, say there's a cube with side length L, and you drill a cylinder with a diameter larger than L through the cube, what volume remains?
[Math] How to find the volume of a cube remaining after drilling a cylinder with a diameter larger than the cube’s side length
geometry
Best Answer
Due to symmetry, you can restrict your calculations to one quadrant. In your figure for example, look at the square from $0$ to $10$. Take the area of the piece outside the circle, multiply by the height of the cube, and by $4$ (you have 4 identical pieces), and you get the volume.
The only issue is how to calculate the area of the small piece. Let's name the corners of the rectangle in the first quadrant $OABC$, with $O$ at the origin, and $A$ along the horizontal axis, and let's call $D$ and $E$ the intersection of the circle with this square. the area you require is then the area of the square minus the area inside the circle. We divide this inside area into three pieces. You have two right angle triangles $OAD$ and $OEC$ of equal area. You know the hypotenuse (the radius of the circle) and one of the sides (the side of the rectangle in the first quadrant = half of the side of your original cube). You can use Pythagoras' theorem to calculate the other side, then the area. You can also calculate the angle $\angle AOD$ fromthe same triangle, using $\arccos$ function. The only piece left is a sector of a circle. The area is radius squared, multiplied by the $\angle DOE$. Since you calculated $\angle AOD$, you get $$\angle DOE=\pi/2-2\angle AOD$$
You now have all the information to finish the problem