Spherical cap
A spherical cap is a portion of a sphere cut off by a plane, forming a dome-shaped segment bounded by a circular base; if the cutting plane passes through the center of the sphere, the cap is a hemisphere, but in general, it is defined by the sphere's radius r and the height h of the cap from the base to the dome's apex.[1] The volume V of a spherical cap is given by the formula V = \frac{1}{3} \pi h^2 (3r - h), which can be derived using methods of calculus such as integration via disks or the method of washers, where the cross-sectional area at a distance from the base varies as a circle of radius determined by the sphere's equation.[2] This formula originates from classical geometry, with Archimedes providing early computations in his treatise On the Sphere and Cylinder, where he used the method of exhaustion to establish relationships between spherical segments and circumscribed cylinders.[3] The surface area of the curved portion of the cap (excluding the base) is A = 2 \pi r h, which interestingly equals the lateral surface area of a cylinder of the same radius and height, a result Archimedes highlighted to show the sphere's surface area equals that of its circumscribing cylinder.[1][4] Spherical caps appear in various applications across mathematics and science; in convex geometry, they are used to study concentration of measure on spheres, where the surface measure of a cap provides bounds on probabilistic distributions. In physics, models of liquid droplets or planetary atmospheres often approximate shapes as spherical caps due to surface tension minimizing energy, leading to equilibrium forms.[5] Architecturally, spherical caps inspire dome designs, such as in geodesic structures, where the geometry ensures structural efficiency.[6] Generalizations extend to higher dimensions, known as hyperspherical caps, which play roles in optimization and packing problems on spheres.Definition and Geometry
Definition
A spherical cap is the portion of a sphere cut off by a plane.[7] When the plane intersects the interior of the sphere, it creates a circular boundary, forming a dome-shaped region bounded by the curved spherical surface and the flat circular base.[8] Geometrically, the spherical cap is characterized by the sphere's radius r and the height h, which is the perpendicular distance from the circular base to the top of the dome. The cutting plane lies at a distance a = r - h from the sphere's center along the axis of symmetry, assuming the smaller cap where h \leq r.[9] In a textual visualization, the cap resembles a rounded hill sitting atop its flat base; for h < r, the sphere's center lies outside the cap below the base, while for h = r, it forms a hemisphere with the center on the base, and for h > r, the center is inside the cap. The term "spherical cap" specifically denotes the smaller portion of the sphere on one side of the plane, often visualized as the dome. In contrast, a "spherical segment" generally refers to the solid region between two parallel planes cutting the sphere, though in some contexts with a single plane, it may synonymously describe the cap's volume; this article focuses on the solid spherical cap as the bounded region including its interior.[10]Key Parameters
A spherical cap is characterized by several key geometric parameters that define its shape and size relative to the parent sphere. The primary parameters are the radius r of the sphere and the height h of the cap, which is the perpendicular distance from the base plane to the apex of the cap. The distance a from the center of the sphere to the base plane is given by a = r - h, assuming the cap is the smaller portion where the plane intersects the sphere such that h \leq r.[1] The radius b of the circular base of the cap is derived from the Pythagorean theorem applied to the right triangle formed by the sphere's radius, the distance a, and the base radius, yielding b = \sqrt{h(2r - h)} or equivalently b = \sqrt{r^2 - a^2}. The height h is also known as the sagitta of the spherical cap, representing the maximum depth or "sag" of the curved surface from the base plane to the sphere's surface along the axis of symmetry.[1][11] Trigonometric relations further connect these parameters, where \theta is the half-angle subtended by the base circle at the sphere's center. Specifically, \cos \theta = a / r = (r - h) / r, and \sin \theta = b / r. These identities facilitate conversions between linear dimensions and angular measures in spherical geometry.[1] Boundary cases illustrate the range of the parameters. When h = 0, the cap degenerates to a single point (tangent plane). For h = r, the base plane passes through the sphere's center, resulting in a hemisphere with base radius b = r. Although h can extend up to $2r for the larger cap portion (where the plane is on the opposite side, yielding the full sphere as h \to 2r), the standard spherical cap typically considers the minor segment with h \leq r to avoid redundancy with the complementary cap.[1]Volume and Surface Area
Standard Formulas
The volume V of a spherical cap is given by V = \frac{1}{3} \pi h^2 (3r - h), where h is the height of the cap and r is the radius of the sphere.[1] The lateral surface area A_\text{lat}, which covers only the curved portion of the cap, is A_\text{lat} = 2 \pi r h. [1] The total surface area A_\text{total} includes both the curved surface and the flat base, expressed as A_\text{total} = A_\text{lat} + \pi b^2 = 2 \pi r h + \pi b^2, where b is the radius of the base circle.[1] The base area itself is \pi b^2, with the base radius b related to the other parameters by b = \sqrt{h(2r - h)}. This relation distinguishes the geometric base from the curved surface in area calculations.[1] An alternative expression for the volume uses the distance a from the sphere's center to the base plane (with h = r - a for the minor cap), yielding V = \frac{1}{3} \pi (2r + a)(r - a)^2. However, the primary formulations rely on h and r for direct computation of volumetric and areal properties.[1]Intuitive Derivation from Spherical Sector
A spherical sector is the solid generated by rotating a circular sector about its diameter, encompassing the spherical cap and the conical region from the sphere's center to the cap's base.[12] The volume of the spherical sector corresponding to a cap of height h on a sphere of radius r is given by V_\text{sector} = \frac{2}{3} \pi r^2 h.[13] To derive the lateral surface area A of the spherical cap intuitively, without calculus, draw an analogy to the volume formula for a cone, which is \frac{1}{3} times the base area times the height. For the spherical sector, the volume relates exactly to the curved zone area (the lateral surface of the cap) as V_\text{sector} = \frac{1}{3} A r, where r plays the role of the "height" from the center. Rearranging yields A = \frac{3 V_\text{sector}}{r}. Substituting the sector volume gives A = \frac{3}{r} \cdot \frac{2}{3} \pi r^2 h = 2 \pi r h. This step-by-step geometric analogy leverages the proportional structure between conical and spherical solids, treating the zone area as an effective "base" scaled by the radial distance. Note that the spherical cap volume itself arises as V_\text{cap} = V_\text{sector} - V_\text{cone}, where the cone has base radius b = \sqrt{h(2r - h)} and height h, but this subtraction pertains to volume, not the independent surface area derivation above.[14] This method provides intuition through geometric analogy, with exactness confirmed by rigorous geometric proofs like Archimedes' hat-box theorem, avoiding the need for integration.[4] Historically, such proportional derivations align with Archimedes' approaches in On the Sphere and Cylinder, where he used mechanical balances and exhaustion to relate spherical and cylindrical elements without modern limits.[15]Calculus Derivation
Consider a sphere of radius r centered at the origin in three-dimensional Cartesian coordinates. The spherical cap of height h is the portion of the sphere above the plane z = a, where a = r - h and $0 < h \leq 2r.[1] To derive the volume V of the cap using the disk method, integrate the cross-sectional areas parallel to the xy-plane from z = a to z = r. At height z, the radius of the disk is x = \sqrt{r^2 - z^2}, so the area is \pi x^2 = \pi (r^2 - z^2). Thus, V = \int_{a}^{r} \pi (r^2 - z^2) \, dz = \pi \left[ r^2 z - \frac{1}{3} z^3 \right]_{a}^{r} = \pi \left( r^2 (r) - \frac{1}{3} r^3 - r^2 (a) + \frac{1}{3} a^3 \right). Simplifying the expression yields V = \pi \left( r^3 - \frac{1}{3} r^3 - r^2 a + \frac{1}{3} a^3 \right) = \pi \left( \frac{2}{3} r^3 - r^2 a + \frac{1}{3} a^3 \right). Substituting a = r - h gives a^3 = (r - h)^3 = r^3 - 3r^2 h + 3 r h^2 - h^3, so \frac{1}{3} a^3 = \frac{1}{3} r^3 - r^2 h + r h^2 - \frac{1}{3} h^3, \quad r^2 a = r^3 - r^2 h. Then, V = \pi \left( \frac{2}{3} r^3 - (r^3 - r^2 h) + \frac{1}{3} r^3 - r^2 h + r h^2 - \frac{1}{3} h^3 \right) = \pi \left( \frac{1}{3} h^2 (3r - h) \right) = \frac{1}{3} \pi h^2 (3r - h). This derivation follows the slicing method for volumes of solids of revolution.[16] For the lateral surface area A of the cap, consider it as a surface of revolution generated by rotating the curve x = \sqrt{r^2 - z^2} about the z-axis from z = a to z = r. The surface area element is $2\pi x \, ds, where ds = \sqrt{1 + \left( \frac{dx}{dz} \right)^2} \, dz. Differentiating gives \frac{dx}{dz} = -\frac{z}{\sqrt{r^2 - z^2}}, so $1 + \left( \frac{dx}{dz} \right)^2 = 1 + \frac{z^2}{r^2 - z^2} = \frac{r^2}{r^2 - z^2}, \quad \sqrt{1 + \left( \frac{dx}{dz} \right)^2} = \frac{r}{\sqrt{r^2 - z^2}} = \frac{r}{x}. Thus, ds = \frac{r}{x} \, dz, and A = \int_{a}^{r} 2\pi x \cdot \frac{r}{x} \, dz = \int_{a}^{r} 2\pi r \, dz = 2\pi r (r - a) = 2\pi r h. This result is obtained via the formula for surfaces of revolution about the axis of integration.[17] An alternative parametric derivation confirms these formulas using spherical coordinates, where the position is (r \sin \theta \cos \phi, r \sin \theta \sin \phi, r \cos \theta) with $0 \leq \theta \leq \alpha, $0 \leq \phi < 2\pi, and \cos \alpha = a/r = 1 - h/r. For the surface area, the element is r^2 \sin \theta \, d\theta \, d\phi, so A = \int_{0}^{2\pi} \int_{0}^{\alpha} r^2 \sin \theta \, d\theta \, d\phi = 2\pi r^2 \left[ -\cos \theta \right]_{0}^{\alpha} = 2\pi r^2 (1 - \cos \alpha) = 2\pi r^2 \cdot \frac{h}{r} = 2\pi r h. The volume formula can be derived via a more involved integration in spherical coordinates, confirming consistency with the disk method.[1] The surface area formula $2\pi r h matches the result from the intuitive derivation using a spherical sector, confirming consistency between the calculus approach and geometric heuristics.Physical Properties
Center of Mass
For a spherical cap of uniform density \rho, the center of mass lies along the axis of rotational symmetry due to the object's symmetry. To find its position, consider the sphere centered at the origin with the cap extending from z = R - h to z = R, where R is the sphere's radius and h is the cap's height. The volume element is dV = \pi (R^2 - z^2) \, dz, and the z-coordinate of the center of mass relative to the sphere's center is given by \bar{z} = \frac{1}{V} \int_{R-h}^{R} z \, dV, where V = \frac{1}{3} \pi h^2 (3R - h) is the cap's volume.[1] Evaluating the integral yields \int_{R-h}^{R} z \pi (R^2 - z^2) \, dz = \frac{\pi}{4} (2Rh - h^2)^2. Thus, \bar{z} = \frac{3 (2R - h)^2}{4 (3R - h)}. This position is measured from the sphere's center toward the cap's apex.[1] To express the distance from the base plane (located at z = R - h), subtract R - h from \bar{z}, resulting in \bar{z}_\text{base} = \frac{h (4R - h)}{4 (3R - h)}. For the special case of a hemisphere where h = R, this simplifies to \bar{z}_\text{base} = \frac{3R}{8}, confirming the well-known result for a hemispherical solid.Moment of Inertia
The moment of inertia of a spherical cap with uniform density \rho about its symmetry axis—the line passing through the apex and perpendicular to the base—is computed by integrating the second moment of mass distribution perpendicular to this axis. To derive the formula, place the center of the sphere at the origin, with the spherical cap extending from z = r - h to z = r, where r is the radius of the sphere and h is the height of the cap (apex at z = r). The symmetry axis is the z-axis. The moment of inertia I about this axis is given by I = \int (x^2 + y^2) \, dm. Using the method of thin disks perpendicular to the z-axis, consider a disk at height z with thickness dz and radius b(z) = \sqrt{r^2 - z^2}. The mass of the disk is dm = \rho \pi b(z)^2 \, dz = \rho \pi (r^2 - z^2) \, dz. The moment of inertia of this uniform disk about its central axis (the z-axis) is dI = \frac{1}{2} dm \, b(z)^2 = \frac{1}{2} \rho \pi (r^2 - z^2)^2 \, dz. Thus, I = \frac{\pi \rho}{2} \int_{r-h}^{r} (r^2 - z^2)^2 \, dz. Expand the integrand: (r^2 - z^2)^2 = r^4 - 2 r^2 z^2 + z^4. The antiderivative is \int (r^4 - 2 r^2 z^2 + z^4) \, dz = r^4 z - \frac{2 r^2}{3} z^3 + \frac{1}{5} z^5. Evaluating from r - h to r, \left[ r^4 z - \frac{2 r^2}{3} z^3 + \frac{1}{5} z^5 \right]_{r-h}^{r} = \frac{8}{15} r^5 - r^4 (r - h) + \frac{2 r^2}{3} (r - h)^3 - \frac{1}{5} (r - h)^5. Simplifying the expression yields the compact form I = \frac{\pi \rho}{2} \left( \frac{4}{3} h^3 r^2 - h^4 r + \frac{1}{5} h^5 \right). The mass of the cap is M = \rho V, where the volume V = \frac{1}{3} \pi h^2 (3 r - h). Substituting \rho = M / V into the expression for I gives I in terms of M, r, and h: I = \frac{ M h \left(4 r^2 - 3 h r + \frac{3}{5} h^2 \right) }{ 2 (3 r - h ) }. For the special case of a hemisphere (h = r), this simplifies to I = \frac{2}{5} M r^2, matching the known result for a solid hemisphere about its symmetry axis. For moments about parallel axes offset from the symmetry axis, the parallel axis theorem can be applied using the center of mass location along the axis, but the computation here focuses on the symmetry axis itself.Applications
Volumes of Union and Intersection of Spheres
When two spheres of radii R_1 and R_2 have centers separated by a distance d satisfying |R_1 - R_2| < d < R_1 + R_2, their intersection forms a symmetric lens-shaped region known as a spherical lens.[18] This lens consists of two spherical caps, one from each sphere, with the dividing plane being the radical plane of the spheres.[18] The height h_1 of the cap from the first sphere is given byh_1 = R_1 - \frac{d^2 + R_1^2 - R_2^2}{2d},
and the height h_2 of the cap from the second sphere is
h_2 = R_2 - \frac{d^2 + R_2^2 - R_1^2}{2d}. [18] The volume of the intersection V_{\text{int}} is the sum of the volumes of these caps,
V_{\text{int}} = V_{\text{cap}}(R_1, h_1) + V_{\text{cap}}(R_2, h_2),
where the volume of a spherical cap of height h and sphere radius R is V_{\text{cap}}(R, h) = \frac{1}{3} \pi h^2 (3R - h).[18] A closed-form expression for the intersection volume is
V_{\text{int}} = \frac{\pi (R_1 + R_2 - d)^2 (d^2 + 2 d R_2 - 3 R_2^2 + 2 d R_1 + 6 R_2 R_1 - 3 R_1^2)}{12 d}. [18] The volume of the union of the two spheres is obtained via the inclusion-exclusion principle:
V_{\text{union}} = \frac{4}{3} \pi R_1^3 + \frac{4}{3} \pi R_2^3 - V_{\text{int}}. [18] In the special case of equal radii R_1 = R_2 = R, the intersection volume simplifies to
V_{\text{int}} = \frac{\pi (4R + d) (2R - d)^2}{12}. [18]