Area is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve or the volume of a solid . Two different regions may have the same area ; by synecdoche, "area" sometimes is used to refer to the region, as in a "polygonal area".
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
The steradian or square radian is the unit of solid angle in the International System of Units (SI). It is used in three dimensional geometry, and is analogous to the radian, which quantifies planar angles. A solid angle in the form of a right circular cone can be projected onto a sphere, defining a spherical cap where the cone intersects the sphere. The magnitude of the solid angle expressed in steradians is defined as the quotient of the surface area of the spherical cap and the square of the sphere's radius. This is analogous to the way a plane angle projected onto a circle defines a circular arc on the circumference, whose length is proportional to the angle. Steradians can be used to measure a solid angle of any shape. The solid angle subtended is the same as that of a cone with the same projected area.
In geometry, a torus 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. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut.
In geometry, a solid angle is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle at that point.
The Method of Mechanical Theorems, also referred to as The Method, is one of the major surviving works of the ancient Greek polymath Archimedes. The Method takes the form of a letter from Archimedes to Eratosthenes, the chief librarian at the Library of Alexandria, and contains the first attested explicit use of indivisibles. The work was originally thought to be lost, but in 1906 was rediscovered in the celebrated Archimedes Palimpsest. The palimpsest includes Archimedes' account of the "mechanical method", so called because it relies on the center of weights of figures (centroid) and the law of the lever, which were demonstrated by Archimedes in On the Equilibrium of Planes.
A Gabriel's horn is a type of geometric figure that has infinite surface area but finite volume. The name refers to the Christian tradition where the archangel Gabriel blows the horn to announce Judgment Day. The properties of this figure were first studied by Italian physicist and mathematician Evangelista Torricelli in the 17th century.
In mathematics, an annulus is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word anulus or annulus meaning 'little ring'. The adjectival form is annular.
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.
A cylinder has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. The boundary of a cross-section in three-dimensional space that is parallel to two of the axes, that is, parallel to the plane determined by these axes, is sometimes referred to as a contour line; for example, if a plane cuts through mountains of a raised-relief map parallel to the ground, the result is a contour line in two-dimensional space showing points on the surface of the mountains of equal elevation.
A Taylor cone refers to the cone observed in electrospinning, electrospraying and hydrodynamic spray processes from which a jet of charged particles emanates above a threshold voltage. Aside from electrospray ionization in mass spectrometry, the Taylor cone is important in field-emission electric propulsion (FEEP) and colloid thrusters used in fine control and high efficiency thrust of spacecraft.
A circular arc is the arc of a circle between a pair of distinct points. If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the center of the circle that is less than π radians ; and the other arc, the major arc, subtends an angle greater than π radians. The arc of a circle is defined as the part or segment of the circumference of a circle. A straight line that connects the two ends of the arc is known as a chord of a circle. If the length of an arc is exactly half of the circle, it is known as a semicircular arc.
In geometry, a hypercone is the figure in the 4-dimensional Euclidean space represented by the equation
An oloid is a three-dimensional curved geometric object that was discovered by Paul Schatz in 1929. It is the convex hull of a skeletal frame made by placing two linked congruent circles in perpendicular planes, so that the center of each circle lies on the edge of the other circle. The distance between the circle centers equals the radius of the circles. One third of each circle's perimeter lies inside the convex hull, so the same shape may be also formed as the convex hull of the two remaining circular arcs each spanning an angle of 4π/3.
In geometry, a bicone or dicone is the three-dimensional surface of revolution of a rhombus around one of its axes of symmetry. Equivalently, a bicone is the surface created by joining two congruent, right, circular cones at their bases.
In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows:
In geometry, a developable roller is a convex solid whose surface consists of a single continuous, developable face. While rolling on a plane, most developable rollers develop their entire surface so that all the points on the surface touch the rolling plane. All developable rollers have ruled surfaces. Four families of developable rollers have been described to date: the prime polysphericons, the convex hulls of the two disc rollers, the polycons and the Platonicons.
In geometry, a polycon is a kind of a developable roller. It is made of identical pieces of a cone whose apex angle equals the angle of an even sided regular polygon. In principle, there are infinitely many polycons, as many as there are even sided regular polygons. Most members of the family have elongated spindle like shapes. The polycon family generalizes the sphericon. It was discovered by the Israeli inventor David Hirsch in 2017.
