Calculating Sphere Volume Cut by 2 Planes & Angle "e" & Distance "a

This can be calculated by setting up a double integral with the limits of integration determined by the intersection of the two planes.
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I want to calculate the volume of a sphere cut by two arbitrary plane. There is a intersection angle between these two planes, which is not 90 degrees. One of these two planes is fixed and located on plane "x-o-y", and the other is perpendicular to plane "x-o-z" and moves the distance "a" from the original point.
How to establish the relationship between the volume of a sphere cut by these two arbitrary plane and the distance "a"?
I set up the double integral as follows. The "e" is the intersection angle between these two planes, and the "r" is the radius of this sphere, both of which can be assumed to be a value.
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To establish the relationship between the volume of a sphere cut by these two arbitrary planes and the distance "a", we can use a double integral to calculate the volume of the intersection region.

First, we can define the sphere as a function in spherical coordinates, with the center at the origin and the radius "r". This can be written as:
f(θ,ϕ) = r

Next, we can define the two planes as functions in Cartesian coordinates. The fixed plane on the x-o-y plane can be written as:
g(x,y) = 0

The moving plane, perpendicular to the x-o-z plane and distance "a" from the origin, can be written as:
h(x,z) = a

To find the intersection region, we can set the two plane functions equal to each other and solve for the values of x and y. This will give us the limits of integration for the double integral.

x = a
y = √(r^2 - a^2)

Now, we can set up the double integral using these limits of integration and the function of the sphere in spherical coordinates.

V = ∫∫f(θ,ϕ) dA
V = ∫∫r sin(θ) dθdϕ
V = r ∫∫sin(θ) dθdϕ

Using the limits of integration for x and y, we can rewrite the double integral in terms of θ and ϕ.

V = r ∫∫sin(θ) dθdϕ
V = r ∫0^π/2∫0^2π sin(θ) dθdϕ

Now, we can solve the double integral and substitute the value for θ.

V = r ∫0^π/2∫0^2π sin(θ) dθdϕ
V = r ∫0^2πdϕ
V = 2πr

Finally, we can substitute the value for "a" into the equation to get the final relationship between the volume of the intersection region and the distance "a".

V = 2πr(a)

In conclusion, the relationship between the volume of a sphere cut by two arbitrary planes and the distance "a" is given by the equation V = 2πr(a), where "r" is
 

FAQ: Calculating Sphere Volume Cut by 2 Planes & Angle "e" & Distance "a

How do you calculate the volume of a sphere cut by two planes at a given angle and distance?

To calculate the volume of a sphere that has been cut by two planes at a given angle and distance, you will need to use the formula for the volume of a frustum (a portion of a sphere between two parallel planes). This formula is V = (πh/3)(R^2 + r^2 + Rr), where h is the distance between the planes, R is the radius of the larger base, and r is the radius of the smaller base. You will also need to calculate the height (h) of the frustum using trigonometry and the given angle (e).

How does the angle "e" affect the volume of the sphere cut by two planes?

The angle "e" will affect the volume of the sphere cut by two planes by changing the height (h) of the frustum. The larger the angle, the shorter the height will be, resulting in a smaller volume. Conversely, a smaller angle will result in a taller height and a larger volume. This is because the angle affects the shape and size of the frustum, which directly impacts its volume.

Can you provide an example of calculating the volume of a sphere cut by two planes at a given angle and distance?

Yes, suppose we have a sphere with a radius of 5 units, cut by two planes at a distance of 4 units and an angle of 45 degrees. To calculate the volume, we first need to find the height (h) of the frustum. Using trigonometry, we can determine that h = 4cos(45) = 2.83 units. Plugging this and the given values into the formula, we get V = (π*2.83/3)(5^2 + 2.83^2 + 5*2.83) ≈ 85.09 cubic units.

How does the distance "a" between the two planes affect the volume of the sphere cut by two planes?

The distance "a" between the two planes will affect the volume of the sphere cut by two planes by changing the height (h) of the frustum. The larger the distance, the taller the height will be, resulting in a larger volume. Conversely, a smaller distance will result in a shorter height and a smaller volume. This is because the distance affects the shape and size of the frustum, which directly impacts its volume.

Can the formula for calculating the volume of a sphere cut by two planes be applied to any shape?

No, the formula for calculating the volume of a sphere cut by two planes can only be applied to frustum-shaped objects, where the shape is a portion of a sphere between two parallel planes. Other shapes will have different formulas for calculating their volumes based on their unique dimensions and properties.

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