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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 
unit can be expressed in the form 
units
, where 
and 
are integers. What is the value of 
?
. Which area is correct, because both areas come to the right answer of 
.
. Which area is correct, because both areas come to the right answer of .
is correct
is correct. your mistake in the first sol is that the side length of the equilateral triangle is 
, not 
:
Volume of a right square pyramid with base length 
and height 
triangle with legs of length 
,
by the properties of 
triangles. Solving this, we get 
.
.
.
one with legs 
and 
and one with legs 
and 
so 
We get 
from Expression from 1 and 2. So 
Expression 3 gives us that the side of the triangle is 
so the area is 
and 