Is there a formula for calculating the volume when you slice through a sphere at any given point Please keep the answer simple?

Answer 1

While there are formulae that can be used, they are certainly not simple and there is pretty much no way around this fact.

One such way to calculate this volume is to first work out the volume of the sphere, then the volume of the section you are removing the "top" of, and subtracting these volumes to determine the volumes of the large and small sections. This assumes that the slice is one plane (i.e. you took one slash with a straight edge in a perfectly straight line).

Assuming the sphere is centered at the origin (x=0, y=0, z=0) with radius R, and that the slicing plane is perpendicular to the x axis, slicing somewhere between x=-R and x=+R, then the volume of the "truncated" sphere is PI R2x + 2/3 PI R3 - 1/3 PI x3. This answer is the "simple" answer, for the case of only one plane, and it applies to all cases due to symmetry.

Deriviation, with most intermediate steps removed...

Start with a circle centered at the origin, with radius R. This is the projection of the sphere into two-space, or the slice of the sphere by the plane z=0. Draw a right triangle from the origin to any point (x,y) with distance R, i.e. on the circle.

Note that R2 = x2 + y2. Going back to the three-dimensional image of the sphere, note that y is the radius of a circle perpendicular to the x axis, and that it represents the slicing plane.

The area of that circle is PI (R2 - x2).

The volume of the truncated portion of the sphere to the left of x, i.e. between -R and x is the definite integral from -R to x of PI (R2 - x2) dx. This is a first year calculus problem involving stacking circles of known area with thickness dx, as dx approaches zero, to determine the volume.

Calculus majors will note that this is a trivial problem, and no one should need any further steps. The only thing to watch is the sign of x3 when x is negative.

Trigonometry majors, with all due respect, will probably not understand any of the further steps, so they are not presented. Besides; the original question was categorized as trigonometry, and it requested a "simple" answer, which has been provided above.