:
Volume of a right square pyramid with base length 
and height 
Y by
A sphere with radius 
is centered at the apex of a right square pyramid with base length 
and height 
The volume of the space where the sphere overlaps with the pyramid can be expressed in the form 
Find 
is centered at the apex of a right square pyramid with base length 
and height 
The volume of the space where the sphere overlaps with the pyramid can be expressed in the form 
Find 
This post has been edited 6 times. Last edited by ReticulatedPython, Mar 18, 2025, 7:04 PM
Thus the answer is 
, alright this problem is kind of trivial knowing properties of 3d graphs cough cough thanks "Three Year Mathcounts Marathon" First we can just draw a sphere on a pyramid and then..but the key point is to understand that this is a symmetrical shape which will be split into 4 different sections, and then we can make a 3d coordinate grid which makes the question much easier, we have to find a part of the spheres area which is called a spherical cap. Then you can substitute the radius and height of the sphere, which is we know that it is 1 h and 1 r, we get something like this
so the answer is 
so we can easily see that the right square pyramid at the apex will make a shape that will take up 1/2 of the spheres area which is another way to go but................. I guess both work? 
.
lines from the center to the vertices, this creates 
equal pyramids that are as described.
