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Proposition 2: The area of circles is proportional to the square of their diameters. [3] Proposition 5: The volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases. [4] Proposition 10: The volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height. [5]
The lateral surface area of a right circular cone is = where is the radius of the circle at the bottom of the cone and is the slant height of the cone. [4] The surface area of the bottom circle of a cone is the same as for any circle, . Thus, the total surface area of a right circular cone can be expressed as each of the following: Radius and ...
Utilizing the pyramid (or cone) volume formula of = ′, where is the infinitesimal area of each pyramidal base (located on the surface of the sphere) and ′ is the height of each pyramid from its base to its apex (at the center of the sphere).
As can be seen, the area of the circle defined by the intersection with the sphere of a horizontal plane located at any height equals the area of the intersection of that plane with the part of the cylinder that is "outside" of the cone; thus, applying Cavalieri's principle, it could be said that the volume of the half sphere equals the volume ...
The volume of a conical or pyramidal frustum is the volume of the solid before slicing its "apex" off, minus the volume of this "apex": =, where B 1 and B 2 are the base and top areas, and h 1 and h 2 are the perpendicular heights from the apex to the base and top planes. Considering that
Archimedes could also find the volume of the cone using the mechanical method, since, in modern terms, the integral involved is exactly the same as the one for area of the parabola. The volume of the cone is 1/3 its base area times the height. The base of the cone is a circle of radius 2, with area , while the height is 2, so the area is ...
In geometry, a spherical sector, [1] also known as a spherical cone, [2] is a portion of a sphere or of a ball defined by a conical boundary with apex at the center of the sphere. It can be described as the union of a spherical cap and the cone formed by the center of the sphere and the base of the cap.
In speaking about these processes, the measure (length or area) of a figure's base is often referred to as its "base." By this usage, the area of a parallelogram or the volume of a prism or cylinder can be calculated by multiplying its "base" by its height; likewise, the areas of triangles and the volumes of cones and pyramids are fractions of ...