[Math] How to calculate the volume and surface area of a sphere cap which has been cut by a plane parallel to its axis

geometryspheres

Suppose I have a spherical cap, defined by known angle $\alpha$ and radius $R$, as shown in the left hand portion of the attached figure. Here is an image:

I am working on a project where the sphere cap is cut into two parts, by a plane parallel to the axis of symmetry of the sphere cap, at a distance $d$ from the axis – see right hand portion of the above figure.

If I then remove the region which is shaded the lighter colour blue (the $x>d$ region), how can I calculate the volume of the remaining darker blue region? Likewise, how about the surface area of the curved portion of the remaining region?

Many thanks in advance

(Image source: http://mathworld.wolfram.com/SphericalCap.html)

Best Answer

As you suggested in your comment, the volume can be calculated as follows: $$V=\int_{-a}^{d}A(x)\,dx=\frac{1}{2}\int_{-a}^{d}r(x)^2(\theta(x)-\sin\theta(x))\,dx, $$ where $r(x)$ is the radius of the intersection of the sphere with the plane with the given $x$-coordinate, and $\theta(x)$ is the angle corresponding to the cross-sectional circular segment. You can find out $r(x)$ from the equation $$r(x)^2=y^2+z^2, $$ where $x,y,z$ satisfy the equation of the sphere. Once you know $r(x)$, you can use a formula from the MathWorld page to calculate $\theta(x)$. I am deliberately giving you just hints for now, as you will remember the answer better if you work for it, but I intend to turn this into a full answer later. Good luck!

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[Math] Calculate “Volume” and “Surface Area” of a voxel-based sphere

That's very easy, it's a function of an XYZ loop....

You would be actually doing a marching cubes kind of logic, and an ISOsurface logic for a spherical volume.

The first result is this programming query (in metacode):

var cubecount: ForLoop(x,y,z){ If space(x,y,z) corresponds to DistanceFunction(x,y,z) == 1; Then cubecount +=1:

question 2 is the same using >=1

For a sphere on zero of radius 1. (distancefunction is xposvectormagnitude, same thing if the sphere is origin is 0; otherwise the sphere pos equation is for the xyz loop is:

This code makes a 3d printable off axis voxel ball if you want:

num = 45.9874;

for (x =[-num-5:num+5]) 
{
    for (y =[-num-5:num+5]) 
    {    
        for (z =[-num-5:num+5]) 
        {    
              if ((x+1.24)*(x+1.24)+(y+1.66)*(y+1.66)+(z+1.88)*(z+1.88)<=num)  
                translate([x,y,z])    
                cube(1);


        }


    }
}

https://www.thingiverse.com/thing:3075040

[Math] How to calculate the surface area of a spherical cap

The following is due to Archimedes of Syracuse (287 BC - 212 BC).

First, remeber that the lateral surface area of a right conical frustum is $$\pi(r_1+r_2)S=2\pi mS\tag 1$$ where $r_1$ is radius of the base, $r_2$ is the radius of the top, $m=\frac{r_1+r_2}2$, the "mid-radius", and $S$ is the lateral (slant) height.

Now, consider the following figure.

Here a horizontal slice of the black sphere has been replaced by a red frustum. Also there is a blue cylinder containing the sphere. Consider the blue slice of the cylinder matching the frustum in its height and matching the sphere in its radius. We want to show that the lateral surface area of the blue cylindrical slice equals the lateral surface area of the red frustum.

The lateral surface area of the blue cylindrical slice is$$2\pi rd.$$

How come that $2\pi rd=2\pi mS$? The red filled triangle is similar to the triangle $MNC$. So $$\frac{m}{r'}=\frac d S,$$ from where $$d=\frac{mS}{r'}.$$ Substituting this into the formula for the lateral cylindrical surface, we get $$2\pi \frac{r}{r'}mS.$$ Compare this to $(1).$

Then think: If the spherical slice is very thin then $r'$ is very close to $r$ and the lateral surface area of the frustum is very close to the lateral surface area of the spherical slice... This is all independent from the actual position of the slice we chose.

Then slice the whole spherical cap. The lateral surface area of the slices will equal the corresponding lateral surface area of the cylinder.

Calculus is needed only if the argumentation above seems to be shaky. But the genius did not need calculus.

Best Answer

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[Math] Calculate “Volume” and “Surface Area” of a voxel-based sphere

[Math] How to calculate the surface area of a spherical cap

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