Calculate Spherical Segment Volume, Lateral Surface Area & Total Area Online

What Is the Spherical Segment Calculator?

The Spherical Segment Calculator computes the volume, lateral (zonal) surface area, total surface area, and sphere radius of a spherical segment -- the region of a sphere between two parallel cutting planes. It generalizes the spherical cap, which uses only one cutting plane and an apex. The calculator accepts the two cross-section radii ($a$, $b$) and the perpendicular distance ($h$) between the planes, then derives all other quantities automatically. For simpler cases, try our Sphere Calculator or Spherical Cap Calculator.

Spherical Segment Formulas

A spherical segment is fully characterized by the upper circle radius $a$, lower circle radius $b$, and the perpendicular distance $h$ between the two parallel planes. The parent sphere radius $R$ is derived from those three inputs:

$$z_2 = \frac{1}{2}\left(\frac{b^2 - a^2}{h} + h\right)$$

$$R = \sqrt{a^2 + z_2^2}$$

$$V = \frac{\pi h}{6}(3a^2 + 3b^2 + h^2)$$

$$S_{\text{lat}} = 2\pi R h$$

$$S_{\text{total}} = 2\pi R h + \pi a^2 + \pi b^2$$

Where:

$V$ is the volume of the spherical segment (the band between the two planes)
$S_{\text{lat}}$ is the lateral (zonal) curved surface area, excluding both flat disks
$S_{\text{total}}$ is the total surface area: lateral band plus both flat circular disks
$R$ is the full sphere radius (derived from $a$, $b$, $h$)
$a$ is the radius of the upper circular cross-section
$b$ is the radius of the lower circular cross-section
$h$ is the perpendicular distance between the two cutting planes
$z_2$ is the signed height of the upper plane above the sphere's center

How to Use the Calculator

Select the quantity you want to solve for. Enter the top radius $a$, bottom radius $b$, and segment height $h$. Choose appropriate units and the result appears in real time with a complete step-by-step breakdown. The derived sphere radius is always shown as supplementary information.

Example Calculation

Problem: A spherical tank of unknown radius is sliced between two parallel cuts. The smaller cross-section has radius $a = 3$ m, the larger has radius $b = 4$ m, and the perpendicular distance between the cuts is $h = 1$ m. Find the segment volume and the parent sphere radius.

Solution: Select "Solve for Volume (V)", enter $a = 3$ m, $b = 4$ m, $h = 1$ m:

$$z_2 = \frac{1}{2}\left(\frac{16 - 9}{1} + 1\right) = \frac{7 + 1}{2} = 4 \text{ m}$$

$$R = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \text{ m}$$

$$V = \frac{\pi \times 1}{6}(3 \times 9 + 3 \times 16 + 1) = \frac{\pi}{6}(27 + 48 + 1) = \frac{76\pi}{6} = \frac{38\pi}{3} \approx 39.79 \text{ m}^3$$

Key Concepts

Archimedes' hat-box theorem: The zonal area $S_{\text{lat}} = 2\pi R h$ depends only on $R$ and $h$, not on the band's position along the sphere. Two equal-width parallel slices have equal curved surface area regardless of where they are cut.
Closed-form volume: $V = (\pi h/6)(3a^2 + 3b^2 + h^2)$ does not require $R$ -- it works directly from the three measurable inputs.
Degeneracy to cap: When $a = 0$, the segment becomes a spherical cap (the top circle collapses to the apex).

Applications

Spherical tanks: Liquid volume between two measured depth markers in a spherical storage tank
Architecture: Dome roof with a circular skylight cut out -- volume is enclosed air space, total surface area is paint plus glazing area
Planetary geology: Caldera, crater rim, or polar ice cap modeled as a sphere slice
Marine engineering: Ballast volume in a partially flooded spherical buoy between two waterline marks

Frequently Asked Questions

How do you calculate the volume of a spherical segment?

Use $V = (\pi h / 6)(3a^2 + 3b^2 + h^2)$, where $a$ is the upper circle radius, $b$ is the lower circle radius, and $h$ is the perpendicular distance between the planes. The full sphere radius is not required.

What is the difference between a spherical segment and a spherical cap?

A spherical cap has one cutting plane and an apex (pointy top). A spherical segment has two parallel cutting planes and no apex -- it is a band sliced between two depths. A cap is the special case of a segment where one circle radius is zero.

How do you find the sphere radius from a spherical segment?

Compute $z_2 = ((b^2 - a^2)/h + h)/2$, then $R = \sqrt{a^2 + z_2^2}$. Both $(a, z_2)$ and $(b, z_1)$ where $z_1 = z_2 - h$ must give the same $R$ as a consistency check.

What is the zonal area of a sphere?

The zonal area is the curved surface of a spherical segment between two parallel planes. It equals $S_{\text{lat}} = 2\pi R h$ -- Archimedes' hat-box theorem says this depends only on $R$ and $h$, not on the band's latitude.

Can $a$ and $b$ be equal?

Yes. When $a = b$, the two cutting planes are equidistant from the sphere center ($z_1 = -z_2 = -h/2$). This creates a symmetric barrel-like band centered on the equator.

Does the spherical segment volume formula require the sphere radius?

No. $V = (\pi h / 6)(3a^2 + 3b^2 + h^2)$ is closed-form in $(a, b, h)$ alone. The sphere radius $R$ is only needed for surface area calculations and is derived automatically by the calculator.

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Spherical Segment Calculator