Spherical Zone Calculator
Find the volume and surface of a spherical zone — the band of a ball between two parallel planes — from the two base radii and the height, from a centered cut of a known sphere, or from the sphere, one base, and the height. The formulas are V = (1/6)πh(3a² + 3b² + h²) and zone area 2πRh.
Zone 31.4159265359 · total S = 109.955742876
- The unique sphere through both circles has R² = a² + d² with d = (a² − b² − h²) / (2h) = 3, so R = 5.
- Volume V = (1/6)πh(3a² + 3b² + h²) = (1/6) × 3.14159265359 × 1 × 76 = 39.7935069455.
- Zone (the band) = 2πRh = 31.4159265359 — it depends on R and h, not on a and b separately. Total including both lids = zone + πa² + πb² = 109.955742876.
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How to Use the Spherical Zone Calculator
From two base radii and a height the page recovers the sphere and the volume. From a centered cut it uses a = b = √(R² − (h/2)²). From the sphere, one base, and the height it recovers the other radius.
- Choose From a, b, and h; From centered R and h; or From R, a, and h.
- Enter non-negative lengths. Height cannot exceed the diameter 2R. A base radius cannot exceed R.
- Read the volume, the band 2πRh, the total including both lids, and the recovered R or b.
One radius 0 is a cap, on the Spherical Cap Calculator (V = (1/3)πh²(3R − h)). The full ball belongs on the Sphere Calculator (V = (4/3)πR³). A dome of height R belongs on the Hemisphere Calculator. Each circular cut alone belongs on the Circle Calculator. Converting cubic units belongs on the Volume Converter.
Volume and the 2πRh Band
A spherical zone is the part of a ball between two parallel planes. The cuts are circles of radii a and b. The perpendicular distance between the planes is h. Volume uses only those three lengths: V = (1/6)πh(3a² + 3b² + h²). The same numbers determine a unique sphere. From the center, the signed distances d and d + h to the two planes satisfy R² = a² + d² = b² + (d + h)², so d = (a² − b² − h²) / (2h) and R = √(a² + d²).
The curved band — the zone surface, not counting the lids — is 2πRh. It depends on the sphere and the height, not on where the band sits. Slide the same thickness h up or down the ball and the orange-peel area stays 2πRh. The two lids πa² and πb² do change, so the total S = 2πRh + πa² + πb² depends on the cut. A cap is the special zone with one radius 0; then V reduces to (1/3)πh²(3R − h) and total surface to 2πRh + πa².
Spherical Zone Calculator Examples
Radii 4 and 3, height 1
d = (16 − 9 − 1) / 2 = 3, so R = 5. V = 38π/3 ≈ 39.7935069455. Zone = 10π ≈ 31.4159265359. Total S = 35π ≈ 109.955742876. The 3-4-5 right triangle from the center to the first rim is the same triangle the Spherical Cap Calculator uses for R = 5, a = 4.
Centered R = 5, h = 2
a = b = √(25 − 1) = √24 ≈ 4.89897948557. V = 148π/3 ≈ 154.985237577. Zone = 20π ≈ 62.8318530718, the same band area as the default cap of height 2 on this sphere — a different solid, same peel.
b = 0, a = 4, h = 2
R = 5 again. V = 52π/3 ≈ 54.4542726622, the Spherical Cap Calculator default. Zone = 20π. Total S = 36π ≈ 113.097335529, one lid gone.
h = 2R = 10 on R = 5
a = b = 0. V = 500π/3 ≈ 523.598775598, the whole ball. Zone = 100π, the whole skin. Both lids vanish.
R = 5, a = 4, h = 7
Start at the parallel of radius 4 (distance +3 from the center) and step a height 7 toward the opposite pole. You meet a parallel of radius 3 on the far side (distance −4). V = 434π/3 ≈ 454.483737219. Zone = 70π ≈ 219.911485751. This is a different pair of planes than the first example (planes at +3 and +4). A sphere has two parallels of radius 3; h = 1 uses the near one and h = 7 uses the far one.
Height past the diameter
R = 5 and h = 11 is impossible: the ball is only 10 units across. The calculator rejects it.
Why the Band Ignores a and b
Archimedes’ hat-box theorem: a zone of height h on a sphere of radius R has area 2πRh, equal to the lateral area of a cylinder of radius R and the same height. The two base radii cancel when you integrate the sphere’s slant. That is why this page’s default (a = 4, b = 3, h = 1, R = 5) and a centered band of the same R and h have the same peel 10π, but different volumes and different lids.
A cap is a zone with one lid gone. Do not use (4/3)πR³ unless h really is 2R. Do not use a cylinder πa²h unless the sides are straight. And do not treat two different radii at height 0 as a zone — that is two circles in one plane, not a slice of a ball.
Common Spherical Zone Mistakes
Using (1/3)πh²(3R − h) for two bases
That is a cap, on the Spherical Cap Calculator, and only when one radius is 0. Two lids need (1/6)πh(3a² + 3b² + h²).
Using (4/3)πR³ for a band
That is the full ball, on the Sphere Calculator. Use the zone formula unless h = 2R and both radii vanish.
Using a cylinder πa²h for the volume
A zone’s sides bulge with the sphere. The cylinder of radius a underestimates or overestimates unless a = b = R (an equatorial sliver of height 0).
Thinking the band area needs a and b
Zone = 2πRh. The lids need a and b; the peel does not.
Letting h exceed 2R
A zone cannot be taller than the sphere. Check 0 ≤ h ≤ 2R before substituting.
Converting liters on this page
V is already in the cube of your input unit. Use the Volume Converter for liters and gallons.
Spherical Zone Calculator FAQs
What is a spherical zone?
The band of a ball between two parallel planes, also called a spherical segment of two bases. It has two circular lids of radii a and b and height h.
What is the volume of a spherical zone?
V = (1/6)πh(3a² + 3b² + h²). For a = 4, b = 3, and h = 1, V = 38π/3 ≈ 39.7935069455 cubic units.
What is the surface area of a spherical zone?
The curved band is 2πRh. The total including both lids is 2πRh + πa² + πb². For the 4-3-1 zone on R = 5 those are 10π ≈ 31.416 and 35π ≈ 109.956.
Is a spherical cap a zone?
Yes — the special case with one radius 0. Use the Spherical Cap Calculator when that is all you need. This page also covers two lids.
Does the band area depend on where you cut?
No. For fixed R and h the peel is 2πRh everywhere on the sphere. Volume and the lids do change with a and b.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
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Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
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Disclaimer
This calculator is intended for general educational and practical use. Review important results before relying on them in professional work.