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Frustum

A frustum is the portion of a solid, typically a cone, pyramid, or other polyhedron, that lies between two parallel planes intersecting the solid.^[1] This geometric shape is formed by truncating the apex of the original solid with a plane parallel to its base, resulting in a three-dimensional figure with two parallel bases of unequal size and trapezoidal lateral faces.^[2] Common types include the conical frustum, derived from a right circular cone with circular bases of radii R_1 and R_2, and the pyramidal frustum, obtained from a pyramid with polygonal bases sharing the same number of sides.^[3] The height h separates the two bases, and the slant height s measures the distance along the lateral surface between them, calculated as s = \sqrt{h^2 + (R_1 - R_2)^2} for conical frustums.^[2] Key properties of a frustum include its volume and surface area formulas, which generalize across types using the areas of the bases. The volume V is given by V = \frac{1}{3} h (A_1 + A_2 + \sqrt{A_1 A_2}), where A_1 and A_2 are the areas of the larger and smaller bases, respectively.^[2] For a conical frustum, this simplifies to V = \frac{1}{3} \pi h (R_1^2 + R_1 R_2 + R_2^2).^[2] The lateral surface area (excluding bases) for a conical frustum is \pi (R_1 + R_2) s, while for a pyramidal frustum, it is \frac{1}{2} (p_1 + p_2) s, with p_1 and p_2 as the perimeters of the bases.^[3] These formulas derive from integrating cross-sectional areas or using Cavalieri's principle, highlighting the frustum's role as a prismatoid.^[2] Frustums find practical applications in engineering and design due to their efficient structural and volumetric properties. In manufacturing, they form the basis for objects like buckets, funnels, and storage hoppers, where the tapered shape facilitates material flow.^[4] Architecturally, frustums appear in truncated domes and tiered structures, such as the stepped levels of ancient Mesopotamian ziggurats, which stack pyramidal frustums to create monumental forms.^[5] In modern contexts, conical frustums model components like aircraft fuselages for streamlined aerodynamics.^[6] Additionally, the concept extends to computer graphics, where a viewing frustum defines the visible region in 3D rendering pipelines.^[7]

Definition and Basic Geometry

Definition

A frustum is a three-dimensional geometric solid formed by truncating a cone or pyramid with a plane parallel to its base, leaving the portion between the original base and the cutting plane. This results in a shape with two parallel faces: a larger base from the original solid and a smaller base from the cut, connected by a lateral surface that tapers linearly between them.^[1] The term "frustum" derives from the Latin word frustum, meaning "piece" or "bit cut off," with its earliest recorded use in English dating to 1658 in the works of Sir Thomas Browne.^[8] Although the specific terminology emerged in the modern era, the underlying concept of truncated solids traces back to ancient Greek geometry, where Euclid indirectly addressed such forms through discussions of volumes and similarity in Book XII of the Elements, employing the method of exhaustion to compare pyramidal and conical segments.^[9] To understand a frustum, it is essential to recall the parent shapes: a cone is a solid figure with a circular base that tapers smoothly to a single apex point, while a pyramid features a polygonal base connected to an apex by triangular faces. In a frustum derived from these, the bases remain circular for conical frustums or polygonal for pyramidal ones, with the lateral surface consisting of trapezoidal faces (or a smoothly curved zone in the conical case) that emphasize the tapered transition between the unequal parallel bases.^[10]

Key Geometric Elements

A frustum is characterized by two parallel bases: a larger base and a smaller base, with the perpendicular height h denoting the distance between these planes. The lateral surface connects the perimeters of the two bases, forming trapezoidal faces in the case of a pyramidal frustum or a truncated conical surface otherwise.^[1] For a conical frustum, the bases are circles with radii R for the larger base and r for the smaller base, where R > r. The slant height l measures the straight-line distance along the lateral surface from the edge of the larger base to the corresponding edge of the smaller base. The perpendicular height h is the vertical separation between the bases, distinct from the slant height, which follows the inclined generating line; in a cross-sectional view, the frustum appears as an isosceles trapezoid where h is the perpendicular height between the parallel bases and l is the length of the non-parallel sides.^[2] In a pyramidal frustum, the bases are similar polygons, with the larger base having side lengths denoted collectively as a and the smaller base as b. For a regular pyramidal frustum (where bases are regular polygons), the apothem—the perpendicular distance from the center to the midpoint of a side—applies to each base, aiding in descriptions of the lateral faces. The slant height l is the altitude of each trapezoidal lateral face, measured perpendicular to the base edges, while the perpendicular height h remains the vertical distance between bases; visually, a longitudinal section through the apex direction reveals a trapezoid with h as the height and l as the face height along the slope.^[3] These elements derive from truncating a full cone or pyramid by a plane parallel to the base, removing a smaller similar figure from the apex down to a height H from the original base, leaving the frustum portion with its defined R, r, h, and l (or polygonal equivalents) dependent on the cut position relative to the full height.^[2]^[3]

Types and Special Cases

Conical Frustum

A conical frustum, also known as a frustum of a right circular cone, is formed by slicing the top off a cone with a plane parallel to its base, resulting in two circular bases of different radii connected by a tapered lateral surface. The cutting planes are perpendicular to the cone's axis, ensuring the bases remain parallel and the figure maintains rotational symmetry about this central axis. This symmetry arises from the original cone's circular cross-sections, making the conical frustum a solid of revolution.^[2] The geometric properties of a conical frustum are characterized by its two circular bases with radii R_1 (larger) and R_2 (smaller), and the height h separating them along the axis. The lateral surface, which is smooth and curved, exhibits full rotational invariance around the axis, distinguishing it from faceted solids. When unrolled onto a plane, this lateral surface develops into a sector of an annulus, where the inner arc corresponds to the smaller base and the outer arc to the larger base, with the radial distance between arcs equal to the slant height.^[2]^[11] Special cases of the conical frustum include the standard right version, where the axis is perpendicular to both bases, and the oblique variant, in which the cutting plane remains parallel to the base but the axis is tilted, leading to non-perpendicular bases and reduced symmetry. The right conical frustum is the conventional form studied in geometry due to its simplicity and symmetry. A limiting case occurs when the smaller radius R_2 approaches zero, reducing the frustum to a full cone. Unlike the pyramidal frustum, which consists of flat trapezoidal lateral faces connecting the polygonal bases, the conical frustum has a continuously curved lateral surface generated by rotating a straight line segment around the axis.^[2]^[12]^[1]

Pyramidal Frustum

A pyramidal frustum is formed by truncating a pyramid—either regular or irregular—with two parallel planes intersecting the lateral edges, resulting in two similar polygonal bases of different sizes and a faceted lateral surface composed of trapezoidal faces. Unlike the smooth, curved lateral surface of a conical frustum, the pyramidal variant features discrete flat faces, making it a polyhedral solid suitable for applications in polyhedral geometry.^[3]^[13] The geometric properties unique to the pyramidal frustum include a number of lateral faces equal to the number of sides of the polygonal bases; for instance, a frustum derived from a square-based pyramid has four trapezoidal lateral faces connecting corresponding sides of the bases. Each lateral face is a trapezoid with two parallel sides corresponding to the edges of the upper and lower bases, and the non-parallel sides representing portions of the original pyramid's edges. This structure arises because the parallel truncation preserves the linear scaling of cross-sections similar to the base.^[3]^[13]^[14] Special cases of the pyramidal frustum include the regular variant, where the bases are regular polygons that are similar and concentrically aligned, often with isosceles trapezoidal lateral faces for symmetry. Irregular cases may arise from oblique or non-regular pyramids, resulting in misaligned or non-concentric polygons, while maintaining similarity in shape with variations in orientation or regularity. These distinctions highlight the frustum's adaptability in geometric constructions beyond the idealized conical form.^[3]^[13] In the broader context of polyhedra, the pyramidal frustum connects directly to truncated polyhedra, as the truncation of a pyramid by a plane parallel to the base yields this exact form.^[3]^[13]

Formulas and Properties

Volume

The volume of a frustum, defined as the portion of a pyramid or cone between two parallel planes intersecting the lateral faces, is calculated using the perpendicular height h between the bases and the areas A_1 and A_2 of the two bases, assuming a right frustum where the bases are perpendicular to the axis.^[3] The general formula for the volume V is

V = \frac{h}{3} \left( A_1 + A_2 + \sqrt{A_1 A_2} \right),

which arises from the similarity of cross-sections parallel to the bases; the areas scale with the square of the distance along the height, and integrating these cross-sectional areas yields the cubic term adjusted by the arithmetic and geometric means of the base areas.^[3] This formula assumes the bases are similar polygons or circles, with volume expressed in cubic units consistent with the units of h and the base areas. For a conical frustum, where the bases are circles of radii R and r (R > r), the base areas are A_1 = \pi R^2 and A_2 = \pi r^2, so the volume simplifies to

V = \frac{\pi h}{3} \left( R^2 + r^2 + R r \right).

This can be derived by considering the frustum as the difference between a large cone of height H and base radius R and a smaller similar cone of height H - h and base radius r, removed from the apex. By similarity of triangles, the ratio of similarity is k = r / R = (H - h) / H, so h = H (1 - k). The volume of the large cone is \frac{1}{3} \pi R^2 H, and the small cone is \frac{1}{3} \pi r^2 (H - h) = \frac{1}{3} \pi (k R)^2 (k H) = \frac{1}{3} \pi k^3 R^2 H. Subtracting gives

V = \frac{1}{3} \pi R^2 H - \frac{1}{3} \pi k^3 R^2 H = \frac{1}{3} \pi R^2 H (1 - k^3) = \frac{1}{3} \pi R^2 H (1 - k)(1 + k + k^2).

Substituting $1 - k = h / H and k = r / R yields \frac{1}{3} \pi h (R^2 + R r + r^2), confirming the formula.^[2] Alternatively, direct integration of the cross-sectional area \pi [r(z)]^2 from z = 0 to h, where r(z) = R + (r - R) z / h, also produces this result.^[2] For a pyramidal frustum with polygonal bases, the general formula applies directly using the base areas. For example, with square bases of side lengths a and b (a > b), A_1 = a^2 and A_2 = b^2, so \sqrt{A_1 A_2} = a b and

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

This follows the same geometric averaging as in the general case, derived analogously through similarity and integration of scaling cross-sections.^[3]

Surface Area

The surface area of a frustum is divided into the lateral surface area, which covers the side faces, and the total surface area, which includes the two bases. The lateral surface excludes the top and bottom bases, while the total encompasses all external surfaces.^[2] For a conical frustum with base radii R and r (where R > r) and slant height l, the lateral surface area is given by \pi (R + r) l. This formula arises from unrolling the lateral surface into a sector of an annulus, where the area equals the average circumference of the bases, \pi (R + r), multiplied by the slant height l. The slant height for a right conical frustum is calculated as l = \sqrt{h^2 + (R - r)^2}, with h denoting the height.^[2]^[15] The total surface area of a conical frustum is the lateral surface area plus the areas of the two circular bases: \pi (R + r) l + \pi (R^2 + r^2).^[2] For a pyramidal frustum with parallel polygonal bases of perimeters p_1 (bottom) and p_2 (top), and slant height s, the lateral surface area is \frac{1}{2} (p_1 + p_2) s. Each lateral face is a trapezoid, and the total lateral area sums these, equivalent to the average perimeter times the slant height. For a regular n-sided pyramidal frustum with base side lengths a and b, this simplifies to \frac{n}{2} (a + b) l, where l is the slant height per face, analogous to s. The slant height s (or l) is derived similarly to the conical case, using the height h and the difference in apothems of the bases. The total surface area adds the areas of the two bases: \frac{1}{2} (p_1 + p_2) s + A_1 + A_2, where A_1 and A_2 are the base areas.^[3]

Applications and Examples

Real-World Applications

In engineering, conical frustums are widely employed in the design of pipes and funnels to facilitate smooth transitions between different diameters, minimizing turbulence and optimizing fluid or material flow. For instance, custom-fabricated metal cone frustums serve as pipe increasers, reducers, and funnels in industrial applications, allowing efficient handling of gases, liquids, or powders.^[16] Similarly, conical offsets act as transitions between circular pipes of varying sizes and axes, enhancing structural integrity and flow efficiency in piping systems.^[17] In rocketry, conical nozzles, which are essentially frustums, are used to accelerate exhaust gases and improve thrust efficiency, as seen in early rocket designs and modern solid rocket boosters where a 15-degree half-angle configuration reduces divergence losses.^[18]^[19] Architectural applications of pyramidal frustums appear in both ancient and modern structures, providing stability and aesthetic progression. Ancient Mesopotamian ziggurats, such as the Ziggurat of Ur, consist of terraced levels that form stacked rectangular pyramidal frustums, creating a stepped profile that symbolizes a connection between earth and the divine while distributing structural loads effectively.^[20] In contemporary design, the Ziggurat Building in Vienna, Virginia, adopts a truncated pyramidal form—essentially a single large frustum—for enhanced security and seismic resilience, with its sloping sides reducing vulnerability to impacts.^[21] In optics and acoustics, frustum shapes contribute to precise control of light and sound propagation. Optical probes with conical-frustum tips, as used in common-path optical coherence tomography, improve reflectivity and signal collection by focusing light within the frustum's tapered geometry, achieving up to 3.7 times greater performance than traditional conical lenses.^[22] Electrowetting lenses often incorporate conical frustum cross-sections to enable variable focal lengths through liquid interface manipulation, enhancing adaptability in compact imaging systems.^[23] In acoustics, horn loudspeakers feature conical frustum flares to match impedance between the driver and air, broadening directivity and amplifying sound output; the straight-sided frustum section sets the dispersion pattern, as in constant directivity designs that maintain even coverage across frequencies.^[24] Ultrasonic machining horns also utilize cone frustum profiles to concentrate vibrational energy, improving material removal rates in precision manufacturing.^[25] Manufacturing leverages frustum geometry for efficient storage and handling of bulk materials in containers like buckets and silos, where the tapered form aids in complete discharge and structural stability. Industrial buckets are frequently shaped as conical frustums to maximize volume per material used while easing pouring and stacking, as demonstrated in experimental designs comparing frustum versus cylindrical forms for fluid dynamics.^[26] Silos and hoppers incorporate conical frustum sections at the base to promote mass flow of granules, preventing bridging and enabling controlled unloading; steel hoppers, for example, use this shape to manage flow in agricultural and industrial bulk storage.^[27] Volume calculations for these frustum-based vessels ensure accurate capacity assessment in concrete plants and grain facilities.^[28] In modern contexts, frustums play key roles in digital fabrication and visualization. In 3D printing, frustum geometries approximate tapered prototypes like nozzles or housings, with alignment to the print beam's frustum ensuring accurate rendering of sloped surfaces without artifacts.^[29] Computer graphics rendering relies on the view frustum—a pyramidal frustum defining the camera's field of vision—to perform culling and projection, optimizing scene processing by excluding objects outside the visible volume in real-time applications like games and simulations.^[30]^[31]

Mathematical Examples

Consider a conical frustum with larger base radius R = 5 cm, smaller base radius r = 2 cm, and height h = 10 cm. The volume is calculated using the formula V = \frac{1}{3} \pi h (R^2 + R r + r^2). Substituting the values gives V = \frac{1}{3} \pi (10) (25 + 10 + 4) = \frac{10}{3} \pi (39) = 130 \pi cm³, approximately 408.41 cm³.^[2] The total surface area includes the areas of the two bases and the lateral surface. The areas of the bases are \pi R^2 = 25\pi cm² and \pi r^2 = 4\pi cm². The slant height l is \sqrt{h^2 + (R - r)^2} = \sqrt{100 + 9} = \sqrt{109} cm. The lateral surface area is \pi (R + r) l = \pi (7) \sqrt{109} cm². Thus, the total surface area is $29\pi + 7\pi \sqrt{109} cm², approximately 91.11 cm² for the bases plus 229.63 cm² for the lateral area, totaling about 320.74 cm².^[2] For a pyramidal frustum with square bases of side lengths a = 4 m (larger) and b = 2 m (smaller), and height h = 3 m, the volume is given by V = \frac{h}{3} (A_1 + A_2 + \sqrt{A_1 A_2}), where A_1 = a^2 = 16 m² and A_2 = b^2 = 4 m². This yields V = \frac{3}{3} (16 + 4 + \sqrt{64}) = 20 + 8 = 28 m³.^[3] The lateral surface area consists of four identical trapezoidal faces. The perimeters are p_1 = 4a = 16 m and p_2 = 4b = 8 m. The slant height for each face is the altitude of the trapezoid, calculated as l = \sqrt{h^2 + \left( \frac{a - b}{2} \right)^2} = \sqrt{9 + 1} = \sqrt{10} m. The lateral area is then \frac{1}{2} (p_1 + p_2) l = \frac{1}{2} (24) \sqrt{10} = 12 \sqrt{10} m², approximately 37.95 m².^[3] In the limiting case where the smaller radius r approaches the larger radius R in a conical frustum, the volume approaches that of a cylinder: V \to \pi R^2 h. Similarly, for a pyramidal frustum with square bases, as b \to a, the volume approaches the prism volume a^2 h. This illustrates the frustum as a generalization bridging cones/pyramids and prisms/cylinders.^[2]^[3] The volume of a frustum differs from that of a full cone or pyramid by subtracting the volume of a smaller similar cone or pyramid from the apex. For the conical example above, if the full cone has height H such that the frustum height is h = H - h' with h'/H = r/R = 2/5, then h' = (2/5) H and h = (3/5) H = 10 cm implies H = 50/3 cm and h' = 20/3 cm; the full cone volume is \frac{1}{3} \pi R^2 H = \frac{1}{3} \pi (25) (50/3) = \frac{1250}{9} \pi cm³, and subtracting the small cone volume \frac{1}{3} \pi r^2 h' = \frac{1}{3} \pi (4) (20/3) = \frac{80}{9} \pi cm³ yields the frustum volume of \frac{1170}{9} \pi = 130 \pi cm³, confirming the formula. A similar subtraction holds for pyramidal frustums using similarity ratios.^[2]^[3]

References

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A frustum (unfortunately commonly misspelled "frustrum") is that portion of a solid which lies between two parallel planes cutting the solid.Missing: definition | Show results with:definition
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A conical frustum is a frustum created by slicing the top off a cone (with the cut made parallel to the base).Missing: definition | Show results with:definition
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### Summary of Pyramidal Frustum from Wolfram MathWorld
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