Spherical segment

A spherical segment is a three-dimensional solid formed by intersecting a sphere with two parallel planes, resulting in a portion bounded by the spherical surface between the planes and the two circular bases where the planes cut the sphere.^[1] This geometric figure, also known as a spherical frustum, represents a truncated spherical cap and is distinct from a single spherical cap, which is cut by only one plane. Note that some sources, such as Harris and Stocker, use "spherical segment" to refer specifically to the spherical cap.^[1] The key dimensions of a spherical segment include its height h, the radii of the two bases a and b (where a > b), and the radius R of the original sphere.^[1] These are related through formulas such as the distance d from the sphere's center to the lower base: d = \frac{a^2 - b^2 - h^2}{2h}, and the sphere's radius: R = \sqrt{\frac{[(a - b)^2 + h^2][(a + b)^2 + h^2]}{4h^2}}.^[1] The base radii can also be expressed as a = \sqrt{R^2 - d^2} and b = \sqrt{R^2 - (d + h)^2}.^[1] The volume V of a spherical segment is given by
V = \frac{\pi h}{6} (3a^2 + 3b^2 + h^2),
which accounts for the spherical portion between the planes.^[1] For the surface area, the curved lateral surface (a spherical zone) has area S = 2\pi R h, while the total surface area includes this zone plus the areas of the two bases: $2\pi R h + \pi a^2 + \pi b^2.^[1] These formulas are fundamental in solid geometry for calculations involving spheres, such as in mensuration and engineering applications like the volume of liquid in spherical storage tanks.^[2]

Definition and Fundamentals

Definition

A spherical segment is the three-dimensional solid formed by the portion of a sphere bounded by two parallel planes that intersect the sphere. This solid is enclosed by two circular bases—the intersections of the planes with the sphere—and a curved lateral surface consisting of the spherical zone between those bases. The enclosing sphere has radius R, and the segment arises from the material between the planes, regardless of whether they are equidistant from the center.^[1] Unlike the two-dimensional circular segment, which is a plane figure bounded by a chord and the corresponding arc of a circle, the spherical segment is a volumetric solid that extends through the sphere's interior. The circular segment represents only the area in a cross-section, whereas the spherical segment captures the full depth and curvature in three dimensions.^[3]^[1] Visually, a spherical segment resembles a band or slice of the sphere obtained by making two parallel cuts, which may be asymmetric if the planes are at unequal distances from the center, producing bases of different sizes. This configuration distinguishes it as a frustum-like portion of the sphere, emphasizing its role in solid geometry rather than planar analysis.^[1]

Terminology and Notation

A spherical segment denotes the solid region of a sphere delimited by two parallel planes and the intervening portion of the sphere's surface.^[1] The curved lateral surface of this solid, excluding the planar faces, is termed the spherical zone.^[1] The two parallel circular regions formed by the intersection of the planes with the sphere are referred to as the bases of the segment.^[4] Standard notation for a spherical segment includes R for the radius of the sphere from which it is derived.^[1] The height h represents the perpendicular distance between the two parallel cutting planes.^[1] The radii of the bases are labeled a and b, with a denoting the larger base radius and b the smaller for consistency in this treatment.^[1] Some sources assign a to the lower base and b to the upper base without regard to relative size, reflecting variations in orientation.^[4] Additionally, d signifies the distance from the sphere's center to the nearer cutting plane.^[1] Note that terminology for the solid itself varies; for instance, some texts apply "spherical segment" exclusively to what is here a spherical cap (a single-base case), reserving other terms for the two-base form.^[5]

Geometric Properties

Cross-Sectional Geometry

A cross-section of a spherical segment taken perpendicular to its axis of symmetry, parallel to the bounding planes, yields a circle whose radius varies continuously along the height h of the segment. Assuming the lower base has radius a and the upper base has radius b, with the distance from the sphere's center to the lower base denoted as d, the radius r(y) at a distance y from the lower base (where $0 \leq y \leq h) is given by

r(y) = \sqrt{R^2 - (d + y)^2},

where R is the radius of the sphere.^[1] This quadratic variation reflects the spherical curvature, with r(0) = a and r(h) = b.^[1] The midplane cross-section, taken at y = h/2, forms a circle of radius \sqrt{R^2 - (d + h/2)^2}. This radius equals R if and only if d + h/2 = 0, i.e., the sphere's center lies in the midplane. If the center is inside the segment, the maximum cross-sectional radius within the segment is R, occurring at the plane through the center; otherwise, all cross-sections have radius less than R.^[1] In the plane containing the axis of symmetry (the meridional plane), the cross-section reveals the segment's profile as a region bounded by two circular arcs of radius R and two parallel straight lines corresponding to the bases of lengths $2a and $2b. These arcs trace the sphere's meridian circle between the cutting planes, forming a symmetric, lens-like shape that highlights the segment's truncation.^[1] The spherical segment is convex, inheriting this property from the enclosing sphere, which ensures all cross-sections are convex disks. The area of cross-sections parallel to the bases, A(y) = \pi r(y)^2 = \pi [R^2 - (d + y)^2], varies quadratically, reaching its peak at the height closest to the sphere's center and tapering toward the bases. This variation underscores the segment's maximal width near the equatorial plane when the center is included.^[1]

Parameter Relationships

The parameters defining a spherical segment—namely, the sphere's radius R, the segment's height h, the radii of the two parallel circular bases a and b, and the signed distance d from the sphere's center to the plane of the base of radius a—are interrelated through geometric constraints derived from the sphere's equation and the Pythagorean theorem applied in the axial cross-section.^[1] In the meridional plane, the axial cross-section appears as a circular segment bounded by two parallel chords of lengths $2a and $2b, separated by distance h, with the sphere's cross-section being a circle of radius R; these form two right triangles sharing the height h, enabling derivations of all parameters from any two or three known values. The base radii relate to the sphere radius and distances from the center via the Pythagorean theorem: a = \sqrt{R^2 - d^2} for the lower base and b = \sqrt{R^2 - (d + h)^2} for the upper base, assuming the segment lies between planes at distances d and d + h from the center along the axis perpendicular to the bases.^[1] These expressions arise directly from the right triangles in the axial plane, where the hypotenuse is R, one leg is the base radius, and the other leg is the distance from the center to the plane. Conversely, solving for the distance d from the base radii and height yields d = \frac{a^2 - b^2 - h^2}{2h}, obtained by subtracting the squared Pythagorean relations for the two bases: a^2 = R^2 - d^2 and b^2 = R^2 - (d + h)^2, then isolating d. To find the sphere radius R from the segment parameters alone, substitute the expression for d into one of the Pythagorean relations, resulting in

R = \sqrt{\frac{((a - b)^2 + h^2)((a + b)^2 + h^2)}{4 h^2}}.

This formula, derived algebraically from the differences and sums of the base radii combined with the height, confirms the enclosing sphere's size without reference to the center's position.^[1] For a spherical segment to exist, the parameters must satisfy h \leq 2R to prevent exceeding the sphere's diameter. Additionally, the base radii must not exceed the sphere radius, a \leq R and b \leq R, which follows from the non-negativity of the distances in the Pythagorean relations. The segment includes the sphere's center if the interval [d, d + h] contains zero, i.e., d < 0 < d + h, indicating the center lies between the cutting planes; otherwise, the segment is entirely on one side of the center.^[1]

Volume and Surface Area

Volume Formula

The volume V of a spherical segment, defined by the portion of a sphere of radius R between two parallel planes separated by height h, with base radii a and b, is given by the closed-form expression

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

This formula provides an exact measure of the three-dimensional space enclosed by the segment and the spherical surface, applicable for any $0 < h < 2R. To derive this formula using integral calculus, consider the sphere centered at the origin with the cutting planes perpendicular to the y-axis at positions y = d and y = d + h, where a = \sqrt{R^2 - d^2} and b = \sqrt{R^2 - (d + h)^2}. The cross-sectional area at height y is a disk of radius \sqrt{R^2 - y^2}, so the area is \pi (R^2 - y^2). The volume is then the integral of these areas:

V = \pi \int_{d}^{d+h} (R^2 - y^2) \, dy = \pi \left[ R^2 y - \frac{y^3}{3} \right]_{d}^{d+h} = \pi \left( R^2 h - \frac{(d + h)^3 - d^3}{3} \right).

Expanding the cubic difference gives (d + h)^3 - d^3 = 3d^2 h + 3 d h^2 + h^3, so

V = \pi \left( R^2 h - d^2 h - d h^2 - \frac{h^3}{3} \right) = \pi h \left( a^2 - d h - \frac{h^2}{3} \right),

since R^2 - d^2 = a^2. Substituting d = \frac{a^2 - b^2 - h^2}{2h} (obtained from the relation b^2 = a^2 - 2 d h - h^2) yields

d h = \frac{a^2 - b^2 - h^2}{2},

and further simplification results in

V = \pi \left[ \frac{h}{2} (a^2 + b^2) + \frac{h^3}{6} \right] = \frac{\pi h}{6} (3a^2 + 3b^2 + h^2).

This integration approach, based on the disk method, confirms the closed-form expression and is commonly employed in applications requiring precise volumetric calculations, such as in microgravity fluid dynamics.^[6] Alternatively, the volume of a spherical segment can be interpreted as the difference between the volumes of two spherical caps: one of height R + d and another of height R - (d + h), though the direct integration provides the primary path to the general formula. The volume scales with the cube of the linear dimensions, ensuring dimensional consistency in units like cubic meters or cubic feet. For illustration, consider a sphere of radius R = 1 unit and a segment of height h = 1 unit starting at d = 0, yielding a = 1 and b = 0 (a hemispherical cap). The volume is V = \frac{\pi \cdot 1}{6} (3 \cdot 1^2 + 3 \cdot 0^2 + 1^2) = \frac{2\pi}{3} \approx 2.0947 cubic units, exactly half the full sphere volume of \frac{4\pi}{3}. The formula is exact for h < 2R; as h \to 0, V \to 0; and as h \to 2R, V \to \frac{4}{3} \pi R^3, recovering the full sphere volume.

Surface Area Formula

The curved surface area of a spherical segment corresponds to the area of the spherical zone, the portion of the sphere's surface between the two parallel cutting planes. This area is given by the formula

A_{\text{zone}} = 2 \pi R h,

where R is the radius of the sphere and h is the height of the segment.^[1] Notably, this expression depends only on R and h, and is independent of the radii a and b of the two bases or the position of the planes along the sphere.^[7] This property, known as Archimedes's hat-box theorem, highlights the uniformity of zonal areas on a sphere for a fixed height.^[8] The formula arises from integrating the surface element over the zone. In spherical coordinates, the differential surface area on the sphere is dA = 2 \pi R \sin \theta \cdot R \, d\theta, where \theta is the polar angle. For a zone spanning angles \theta_1 to \theta_2, the integration yields

A_{\text{zone}} = 2 \pi R^2 \int_{\theta_1}^{\theta_2} \sin \theta \, d\theta = 2 \pi R^2 [-\cos \theta]_{\theta_1}^{\theta_2} = 2 \pi R^2 (\cos \theta_1 - \cos \theta_2).

The height h relates to these angles by h = R (\cos \theta_1 - \cos \theta_2), simplifying the result to $2 \pi R h.^[7] Archimedes originally proved this without calculus by comparing the zone to the lateral surface of an enclosing cylinder of the same height and radius R, demonstrating equivalence through geometric projection.^[8] This independence from base positions is particularly useful for approximating areas in thin segments, where h \ll R, treating the zone as nearly cylindrical.^[1] The total surface area of the spherical segment includes the curved zone plus the areas of the two flat circular bases:

A_{\text{total}} = A_{\text{zone}} + \pi a^2 + \pi b^2 = 2 \pi R h + \pi a^2 + \pi b^2,

where a and b are the radii of the lower and upper bases, respectively.^[1] These bases are planar disks lying in the cutting planes and do not form part of the original sphere's surface; their inclusion accounts for the complete boundary of the solid segment.^[1]

Spherical Cap

A spherical cap is a special case of a spherical segment obtained in the limit as the second base radius b approaches zero, resulting in a portion of a sphere cut off by a single plane, with height h \leq R where R is the sphere's radius, and the single base radius a = \sqrt{h(2R - h)}.^[5] This configuration forms a dome-like shape, distinct from the general segment by having only one circular base, and it represents the smaller portion when the cutting plane does not intersect the sphere's center, typically with h < R.^[5] Geometrically, the height h of the spherical cap measures the perpendicular distance from the base plane to the sphere's pole (the apex of the cap), and this h corresponds to the sagitta, or maximum depth, of the spherical surface relative to the chordal base.^[9] In architectural domes, such as the Palazzetto dello Sport in Rome (1957), the sagitta h is optimized around 30% of the base span for material efficiency in thin-shell construction, yielding a radius R via R = \frac{a^2}{2h} + \frac{h}{2}.^[10] Similarly, in optics, the sagitta quantifies the aspheric deviation or lens curvature depth, essential for fabricating spherical optical elements like domes that protect sensors while minimizing distortion.^[9] The volume of a spherical cap specializes from the general spherical segment formula by setting b = 0, yielding

V = \frac{1}{3} \pi h^2 (3R - h),

which represents the enclosed solid volume.^[5] The surface area consists of the curved portion A_{\text{curved}} = 2 \pi R h, the flat base A_{\text{base}} = \pi a^2 = \pi h (2R - h), and the total surface area A_{\text{total}} = 2 \pi R h + \pi h (2R - h).^[5]

Spherical Zone

A spherical zone is the portion of a sphere's surface enclosed between two parallel planes that intersect the sphere, forming a continuous band or strip on the surface.^[7] This geometric figure arises when the sphere is sliced by such planes, with the zone defined solely by the curved surface between them, excluding any planar bases.^[8] The distance between the planes, denoted as the height h, is measured along the axis perpendicular to the planes, and the zone's width remains constant at h irrespective of its location on the sphere.^[7] The spherical zone can be constructed as a surface of revolution by rotating a circular arc—subtended by the central angle corresponding to the zone—about an axis parallel to the bounding planes.^[7] A key property is that the zone's surface area is independent of its position on the sphere, depending only on the sphere's radius R and the height h. This surprising result, known as Archimedes' hat-box theorem, equates the zone's area to the lateral surface area of a right circular cylinder with radius R and height h:

A = 2 \pi R h

Archimedes established this formula in his treatise On the Sphere and Cylinder (c. 225 BCE), demonstrating it through geometric comparisons between spherical and cylindrical sections.^[11]^[12] In relation to a spherical segment, the zone serves as the curved lateral boundary of the solid segment formed by the same two planes, though it is analyzed independently in spherical geometry as a two-dimensional surface object.^[8] For visualization, a spherical zone resembles the band of Earth's surface between two lines of latitude, such as the region from 30° to 50° north, where the area scales directly with h but remains unaffected by the specific latitudinal position or the sphere's overall orientation.^[7]

Applications

Engineering and Design

Spherical segments play a crucial role in the design of storage tanks and pressure vessels, where their volume formulas enable precise calculations for partial liquid fills. In liquefied natural gas (LNG) storage, for instance, engineers use the height h of the liquid level to determine the volume of the spherical segment formed, optimizing tank capacity and inventory management as levels fluctuate during operations. This approach is essential for spherical tanks in LNG carriers, where multiple tanks provide total capacities up to 200,000 cubic meters of LNG, ensuring efficient space utilization in cryogenic conditions.^[13] In architectural engineering, spherical zones—annular segments between two parallel planes—approximate curved surfaces for domes and arches, facilitating material estimation through surface area computations. Stadium roofs and geodesic domes often incorporate these segments to achieve structural integrity while minimizing weight. The uniform curvature reduces bending stresses, allowing for lighter frameworks compared to flat or cylindrical alternatives. Manufacturing processes for optical components, such as lenses and mirrors, frequently involve machining spherical segments to achieve precise curvatures, with cap formulas applied to control the segment height for desired focal lengths. In precision optics production, like spherical lenses for cameras, the sagitta (depth) of the cap is calculated to ensure optical performance, enabling tolerances as fine as micrometers. This method supports high-volume fabrication techniques, including diamond turning and polishing, critical for industries like telecommunications. A practical example is the capacity calculation for a hemispherical bowl, a special cap where h = R, yielding a volume of \frac{2}{3}\pi R^3, which informs designs in food processing equipment or medical basins for accurate sizing. Similarly, frustum-like spherical segments appear in piping transitions for chemical plants, where segment geometry aids in flow optimization and weld joint planning. One key advantage of spherical segments in pressure containment is their uniform stress distribution, as the geometry minimizes hoop and longitudinal stresses compared to cylindrical vessels, enhancing safety in high-pressure applications like gas storage. This property, governed by Laplace's law for thin-walled spheres, allows designs to withstand pressures up to 100 bar with thinner materials, reducing fabrication costs.

Mathematical and Scientific Uses

Spherical segments have played a significant role in classical mensuration and early calculus, particularly in Archimedes' derivation of the sphere's volume using the method of exhaustion. In his treatise On the Sphere and Cylinder, Archimedes used inscribed and circumscribed polyhedra and applied exhaustion to bound the true volume, establishing that the sphere's volume is two-thirds that of its circumscribing cylinder. This approach prefigured integral calculus by summing infinitesimal volumes to approximate curved solids, influencing later developments in quadrature methods for volumes of revolution.^[14] In astronomy and geodesy, spherical segments and zones model layered planetary structures for computations involving light propagation and gravitational fields. Atmospheric segments, treated as spherical zones, facilitate simulations of multiple light scattering in planetary atmospheres, enabling predictions of observed brightness and polarization from cometary or gaseous envelopes.^[15] In geodesy, spherical cap harmonics expand local gravity fields over zonal segments, improving resolution for geoid modeling and anomaly detection by integrating potential over finite spherical portions rather than global harmonics.^[16] Physics applications leverage spherical segments to quantify overlaps in multi-body interactions. The intersection volume of two overlapping spheres forms symmetric lens-shaped segments, whose combined volume informs collision dynamics in particle simulations and excluded volume effects in dense systems.^[17] In molecular modeling, exact overlap volumes between atomic spheres calculate solvent-accessible surfaces and interaction energies, essential for protein-ligand binding affinity predictions. In computational geometry, spherical segments support efficient 3D rendering and optimization algorithms. Techniques for sampling projected spherical caps enable real-time importance sampling in graphics pipelines, approximating hemispherical lighting integrals for realistic shading without full sphere evaluation.^[18] These methods optimize truncation of spherical primitives in mesh generation, reducing polygon counts while preserving curvature in animations and virtual environments.^[19]