Solid of revolution¶

In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary).

Version: v1 · 2026-09-28 · History
Domain-specific #: 12159
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Solid Geometry, Integral Calculus → Mathematics

Core Idea¶

Solid of revolution is treated here as the recurring solid geometry identity summarized by this source-grounded definition: In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary).

In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary). The surface created by this revolution and which bounds the solid is the surface of revolution. Assuming that the curve does not cross the axis, the solid's volume is equal to the length of the circle described by the figure's centroid multiplied by the figure's area (Pappus's second centroid theorem).

A representative disc is a three-dimensional volume element of a solid of revolution. The element is created by rotating a line segment (of length ) around some axis (located units away), so that a cylindrical volume of units is enclosed. When a curve is defined by its parametric form in some interval , the volumes of the solids generated by revolving the curve around the -axis or the -axis are given by.

For Solid of revolution, the abstraction is narrower than the article's general subject matter: a positive case must preserve In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary). Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in solid geometry, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — The volume of the solid formed by rotating the area between the curves of and and the lines and about the -axis is given by.
Constitutive relation — A more rigorous justification can be given by attempting to evaluate a triple integral in cylindrical coordinates with two different orders of integration.
Operating condition — The method can be visualized by considering a thin horizontal rectangle at between on top and on the bottom, and revolving it about the -axis; it forms a ring (or disc in the case that ), with outer radius and inner radius.
Recognition evidence — Assuming the applicability of Fubini's theorem and the multivariate change of variables formula, the disk method may be derived in a straightforward manner by (denoting the solid as D).
Admissible variation — The method can be visualized by considering a thin vertical rectangle at with height , and revolving it about the -axis; it forms a cylindrical shell.
Characteristic consequence — study of a vase as a solid of revolution by Paolo Uccello.
Failure boundary — When a curve is defined by its parametric form in some interval , the volumes of the solids generated by revolving the curve around the -axis or the -axis are given by.

What It Is Not¶

Not the whole field of solid geometry. The node requires the specific identity stated by In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary).
Not an over-broad reading. In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary).
Not an over-broad reading. A more rigorous justification can be given by attempting to evaluate a triple integral in cylindrical coordinates with two different orders of integration.
Not an over-broad reading. This method may be derived with the same triple integral, this time with a different order of integration.
Not automatically Cross Section (Geometry). Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Solid of revolution applies literally inside solid geometry wherever the source-defined carrier and relation can be established. Its documented habitats include:

Disc method. The disc method is used when the slice that was drawn is perpendicular to the axis of revolution; i.e. when integrating parallel to the axis of revolution.
Shell Method of Integration. The shell method (sometimes referred to as the "cylinder method") is used when the slice that was drawn is parallel to the axis of revolution; i.e. when integrating perpendicular to the axis of revolution.
Finding the volume. Two common methods for finding the volume of a solid of revolution are the disc method and the shell method of integration.
Disc method. The method can be visualized by considering a thin horizontal rectangle at between on top and on the bottom, and revolving it about the -axis; it forms a ring (or disc in the case that ), with outer radius and inner radius.
Disc method. Assuming the applicability of Fubini's theorem and the multivariate change of variables formula, the disk method may be derived in a straightforward manner by (denoting the solid as D).
Shell Method of Integration. The method can be visualized by considering a thin vertical rectangle at with height , and revolving it about the -axis; it forms a cylindrical shell.

Outside solid geometry, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of Solid of revolution names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary). The strongest recognition evidence in the frozen account is: Assuming the applicability of Fubini's theorem and the multivariate change of variables formula, the disk method may be derived in a straightforward manner by (denoting the solid as D). A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary). so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Solid of revolution compresses multiple solid geometry details into a stable diagnostic relation. The source shows both the central mechanism—a more rigorous justification can be given by attempting to evaluate a triple integral in cylindrical coordinates with two different orders of integration.—and the practical consequence—study of a vase as a solid of revolution by Paolo Uccello. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the solid geometry entities to which the claim applies.
State the relation. Use the source-grounded identity: In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary).
Check operation and conditions. The method can be visualized by considering a thin horizontal rectangle at between on top and on the bottom, and revolving it about the -axis; it forms a ring (or disc in the case that ), with outer radius and inner radius.
Demand recognition evidence. Assuming the applicability of Fubini's theorem and the multivariate change of variables formula, the disk method may be derived in a straightforward manner by (denoting the solid as D).
Test variation. Change an implementation or setting while preserving the method can be visualized by considering a thin vertical rectangle at with height , and revolving it about the -axis; it forms a cylindrical shell.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about Solid of revolution transfers literally when a new case preserves the same carrier type, relation, and recognition test. The disc method is used when the slice that was drawn is perpendicular to the axis of revolution; i.e. when integrating parallel to the axis of revolution. The shell method (sometimes referred to as the "cylinder method") is used when the slice that was drawn is parallel to the axis of revolution; i.e. when integrating perpendicular to the axis of revolution.

Beyond the home domain. No canonical parent is asserted for Solid of revolution. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

If (e.g. revolving an area between the curve and the -axis), this reduces to. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary); recognition evidence → Assuming the applicability of Fubini's theorem and the multivariate change of variables formula, the disk method may be derived in a straightforward manner by (denoting the solid as D)

Applied / In Practice¶

The method can be visualized by considering a thin horizontal rectangle at between on top and on the bottom, and revolving it about the -axis; it forms a ring (or disc in the case that ), with outer radius and inner radius. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Disc method; invariant → In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary); boundary → the case exits the class when in geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary)

Structural Tensions¶

T1 — Stable identity versus admissible variation. In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. A more rigorous justification can be given by attempting to evaluate a triple integral in cylindrical coordinates with two different orders of integration. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. This method may be derived with the same triple integral, this time with a different order of integration. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. Assuming that the curve does not cross the axis, the solid's volume is equal to the length of the circle described by the figure's centroid multiplied by the figure's area (Pappus's second centroid theorem). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. The volume of the solid formed by rotating the area between the curves of and and the lines and about the -axis is given by. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Solid of revolution literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. A more rigorous justification can be given by attempting to evaluate a triple integral in cylindrical coordinates with two different orders of integration. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Solid of revolution distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Solid of revolution is mixed or framed-leaning. Its structural side is the repeatable organization summarized by In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary). Its framed side is the solid geometry vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: The method can be visualized by considering a thin horizontal rectangle at between on top and on the bottom, and revolving it about the -axis; it forms a ring (or disc in the case that ), with outer radius and inner radius. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary). The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: The volume of the solid formed by rotating the area between the curves of and and the lines and about the -axis is given by. A more rigorous justification can be given by attempting to evaluate a triple integral in cylindrical coordinates with two different orders of integration. It further constrains recognition and variation through: The method can be visualized by considering a thin horizontal rectangle at between on top and on the bottom, and revolving it about the -axis; it forms a ring (or disc in the case that ), with outer radius and inner radius. Assuming the applicability of Fubini's theorem and the multivariate change of variables formula, the disk method may be derived in a straightforward manner by (denoting the solid as D).

What is domain-bound. solid geometry supplies the operative entities, technical vocabulary, warrants, and exceptions that make Solid of revolution literal. Its documented scope includes the condition that The disc method is used when the slice that was drawn is perpendicular to the axis of revolution; i.e. when integrating parallel to the axis of revolution. Another bounded application condition is that The shell method (sometimes referred to as the "cylinder method") is used when the slice that was drawn is parallel to the axis of revolution; i.e. when integrating perpendicular to the axis of revolution. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—The method can be visualized by considering a thin vertical rectangle at with height , and revolving it about the -axis; it forms a cylindrical shell.—and future graph densification may discover a defensible relation only if it preserves that boundary.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Solid of revolution. The reviewed identity is: In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary). The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Neighborhood in Abstraction Space¶

Solid of revolution sits in a moderately populated region (51^st percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Continuum Mechanics & Field Models (42 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary)?
Cross Section (Geometry). Expose the geometry of a body by intersecting it with a plane or, in higher dimension, a hyperplane, retaining the induced lower-dimensional figure together with the cutter's position and orientation. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Pi. The dimensionless mathematical constant equal to a Euclidean circle's circumference divided by its diameter, equivalently characterized through analysis, with an irrational transcendental value near 3.14159. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Color Solid. Represent a declared set of surface or reproducible colors as a bounded three-dimensional region in specified color coordinates, exposing its gamut boundary, neutral structure, slices, and coverage without treating the region's shape as coordinate- or condition-invariant. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Solid of revolution remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside solid geometry lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Solid_of_revolution (revision 1328243575).
Preserved source candidate: https://books.google.com/books?id=V_WxjYMKuUAC&pg=PA168
Preserved source candidate: https://books.google.com/books?id=oQ1y1HCpeowC&pg=SA6-PA90
Preserved source candidate: http://www.cliffsnotes.com/study_guide/topicArticleId-39909,articleId-39907.html
Preserved source candidate: https://web.archive.org/web/20120319195953/http://www.cliffsnotes.com/study_guide/topicArticleId-39909,articleId-39907.html

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.