A ring torus is sometimes colloquially referred to as a donut or doughnut. The main types of toruses include ring toruses, horn toruses, and spindle toruses. In geometry, a torus (plural tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. A ring torus with aspect ratio 3, the ratio between the diameters of the larger (magenta) circle and the smaller (red) circle. For other uses, see Torus (disambiguation).Ī ring torus with a selection of circles on its surface As the distance from the axis of revolution decreases, the ring torus becomes a horn torus, then a spindle torus, and finally degenerates into a double-covered sphere. In the UK, the three-bar equal sign ≡ (U+2261) is sometimes used.This article is about the mathematical surface. For example, if two triangles have been shown to be congruent by the SSS criteria and a statement that corresponding angles are congruent is needed in a proof, then CPCTC may be used as a justification of this statement.Ī related theorem is CPCFC, in which "triangles" is replaced with "figures" so that the theorem applies to any pair of polygons or polyhedrons that are congruent.Ī symbol commonly used for congruence is an equals symbol with a tilde above it, ≅, corresponding to the Unicode character 'approximately equal to' (U+2245). The statement is often used as a justification in elementary geometry proofs when a conclusion of the congruence of parts of two triangles is needed after the congruence of the triangles has been established. If triangle ABC is congruent to triangle DEF, the relationship can be written mathematically as: Two triangles are congruent if their corresponding sides are equal in length, and their corresponding angles are equal in measure. Specifying two sides and an adjacent angle (SSA), however, can yield two distinct possible triangles. The shape of a triangle is determined up to congruence by specifying two sides and the angle between them (SAS), two angles and the side between them (ASA) or two angles and a corresponding adjacent side (AAS). (Most definitions consider congruence to be a form of similarity, although a minority require that the objects have different sizes in order to qualify as similar.) The related concept of similarity applies if the objects have the same shape but do not necessarily have the same size. In this sense, two plane figures are congruent implies that their corresponding characteristics are "congruent" or "equal" including not just their corresponding sides and angles, but also their corresponding diagonals, perimeters, and areas.
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