Spherical Zone Calculator

What Is a Spherical Zone Calculator?

A spherical zone is the curved band of a sphere's surface caught between two parallel cutting planes — imagine a slice out of the middle of an orange peel. Remarkably, thanks to Archimedes' Hat-Box Theorem, the curved surface area only depends on the sphere's radius and the distance between the planes — not on where the slice is positioned. This calculator finds both the curved surface area and the enclosed volume.

How to Use This Spherical Zone Calculator

1. Measure the sphere's radius R.
2. Measure how far each cutting plane sits from the sphere's center, as signed distances along the axis (d1 for the first plane, d2 for the second, with d2 greater than d1).
3. Enter R, d1, and d2 and click Calculate to get the zone height, curved surface area, and volume.

Formula

• Zone Height h = d2 − d1
• Curved Surface Area = 2 × π × R × h (Archimedes' Hat-Box Theorem — independent of position!)
• Volume = (π × h / 6) × (3r1² + 3r2² + h²), where r1, r2 are the circle radii at each plane

Worked Example

A sphere of radius 10 cm is sliced by two planes at distances 2 cm and 6 cm from the center. Zone Height = 6 − 2 = 4 cm. Curved Surface Area = 2π(10)(4) ≈ 251.33 cm² — the same as any other 4 cm-tall band on this sphere, no matter where it's cut.