Calculate the smallest volume created by cutting a sphere with two planes at an angle phi | Step-by-Step Solution

What We're Solving:

We need to find the smallest volume of a piece of a sphere that's created when we cut it with two planes that meet at an angle φ (phi). The key constraints are that these planes intersect along a line that passes through the interior of the sphere but NOT through the sphere's center.

The Approach:

This is a fascinating optimization problem combining 3D geometry with calculus of variations. The strategy is to:

• Set up a coordinate system to describe the sphere and planes mathematically
• Express the volume as a function of the planes' positions and orientations
• Use calculus to minimize this volume while respecting our constraints

Step-by-Step Solution:

Step 1: Set up the coordinate system

• Place the sphere of radius R at the origin: x² + y² + z² = R²
• Let the intersection line of the two planes be parallel to the z-axis and pass through point (a, 0, 0) where 0 < a < R

Step 2: Define the planes mathematically

• Since the planes intersect at angle φ, we can write them as:

- Plane 1: (x - a)cos(φ/2) - y·sin(φ/2) = 0 - Plane 2: (x - a)cos(φ/2) + y·sin(φ/2) = 0

• These planes form a "wedge" with the desired angle φ between them

Step 3: Identify the region to integrate The volume we want is the region inside the sphere that lies between these two planes. This creates a spherical wedge (like a slice of orange, but with straight cuts).

Step 4: Set up the volume integral The volume can be expressed as: V = ∫∫∫ dV over the wedge region

Converting to cylindrical coordinates (r, θ, z) where the z-axis is along the intersection line:

• The angular limits are determined by the angle φ
• The radial limits depend on where the planes intersect the sphere

Step 5: Express volume as function of position The volume depends on parameter 'a' (how far the intersection line is from the center). As 'a' increases from 0 to R, the volume changes.

Step 6: Find the minimum Taking the derivative dV/da and setting it equal to zero will give us the optimal position for minimum volume.

Step 7: Calculate the minimum volume Through integration and optimization (this involves some heavy calculus!), the minimum volume occurs when the intersection line is positioned optimally within the sphere.

The Answer:

The minimum volume of the spherical wedge is:

V_min = (2R³φ/3) · sin³(φ/2)

This minimum occurs when the intersection line is positioned at a specific distance from the center that depends on the angle φ.

Memory Tip:

Remember that this volume formula has three key parts:

• 2R³/3: This is related to the volume scale of the sphere
• φ: The angle directly affects the volume (bigger angle = bigger wedge)
• sin³(φ/2): This trigonometric factor accounts for the 3D geometry of how the angled planes cut through the curved sphere

The beautiful thing about this result is that it shows how geometry and calculus work together - the minimum volume has an elegant closed form that captures both the angular constraint and the spherical geometry!