Calculators
Spherical Zone Calculator
Calculate spherical zone volume, curved surface area, and cut-face areas between two parallel planes inside a sphere.
Measure the solid region between two parallel sphere cuts.
A spherical zone lies between two parallel planes cutting a sphere of radius R. Enter signed plane positions z₁ < z₂ measured from the center along the same axis, each between −R and R. Height h = z₂ − z₁. Cut radii are r₁ = √((R − z₁)(R + z₁)) and r₂ = √((R − z₂)(R + z₂)). Curved area = 2πRh; volume = πh(3r₁² + 3r₂² + h²)/6. Total closed surface area adds the two circular cut faces πr₁² + πr₂². For R = 5 and cuts at −2 and 2, height is 4, both cut radii are √21, volume is 284π/3 ≈ 297.404104540 cubic units, and curved area is 40π. Cuts at −R and R describe the full sphere; a cut at a pole gives a spherical cap. This assumes a solid sphere, parallel planes, and consistent length units. Radius must be from 1e-50 to 1e50. Scientific notation is accepted; results use browser floating-point arithmetic rounded to 12 significant digits. All calculations stay in your browser.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
How are the plane positions measured?
Both positions are signed distances from the sphere center along one axis. Negative and positive positions lie on opposite sides of the center. Enter the lower position first; both must lie within the sphere.
What is the difference between curved and total area?
Curved area is the spherical band only: 2πRh. Total closed area also includes the two flat circular cut faces. At a pole the corresponding cut face has zero area.
Can I calculate a cap or a complete sphere?
Yes. Set one cut at a pole for a cap, or use −R and R for the full sphere. Distinct cut positions are required.
Which units can I use?
Use one length unit for every input. Areas are in square units and volume is in cubic units. Scientific notation is supported; results are rounded to 12 significant digits.
Are the measurements uploaded?
No. Calculation and copying run locally in your browser.