Calculators
Spherical Shell Calculator
Calculate spherical shell material volume, wall thickness, and inner, outer, and total surface areas from two concentric radii.
Calculate the material in a hollow sphere.
A spherical shell is the material between two concentric spheres. Enter outer radius R and inner radius r in the same unit, with R > r ≥ 0. Material volume = (4π/3)(R − r)(R² + Rr + r²), wall thickness = R − r, outer area = 4πR², inner area = 4πr², and total area = 4π(R² + r²). The factored volume formula reduces cancellation for thin shells. For R = 5 and r = 3, material volume is 392π/3 ≈ 410.501440069 cubic units, thickness is 2 units, and total surface area is 136π ≈ 427.256600888 square units. The cavity volume is excluded. An inner radius of zero gives a solid sphere. Divide diameters by two before entering them. Positive radii must be from 1e-50 to 1e50. Results use browser floating-point arithmetic rounded to 12 significant digits; extremely close radii may round to the same value and be rejected. All calculations stay in your browser.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What is the spherical shell volume formula?
Material volume is (4π/3)(R³ − r³). This calculator uses the equivalent factored form (4π/3)(R − r)(R² + Rr + r²) to reduce subtraction error for thin shells.
Does this include the hollow cavity?
No. The inner sphere volume is subtracted from the outer sphere volume, leaving only the shell material.
What does total surface area include?
Total surface area adds the outer and inner spherical surfaces: 4πR² + 4πr². It is not just the exterior area.
Can I enter zero for the inner radius?
Yes. An inner radius of zero gives a solid sphere, with zero inner surface area. The outer radius must be positive and greater than the inner radius.
Should I enter radii or diameters?
Enter radii measured from the shared center. Divide diameters by two. Use consistent length units; volume is in cubic units and area is in square units.
What inputs are supported?
Positive radii from 1e-50 to 1e50 and a zero inner radius are supported, including scientific notation. Results are rounded to 12 significant digits and calculated entirely in your browser.