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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:
The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is =. It is also A = Ω r 2 {\displaystyle A=\Omega r^{2}} where Ω is the solid angle of the spherical sector in steradians , the SI unit of solid angle.
The external surface area A of the cap equals r2 only if solid angle of the cone is exactly 1 steradian. Hence, in this figure θ = A/2 and r = 1. The solid angle of a cone with its apex at the apex of the solid angle, and with apex angle 2 θ, is the area of a spherical cap on a unit sphere
A sphere of radius r has surface area 4πr 2.. The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies. [1] The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with ...
The curved surface area of the spherical segment bounded by two parallel disks is the difference of surface areas of their respective spherical caps. For a sphere of radius r {\displaystyle r} , and caps with heights h 1 {\displaystyle h_{1}} and h 2 {\displaystyle h_{2}} , the area is
Such polygons may have any number of sides greater than 1. Two-sided spherical polygons—lunes, also called digons or bi-angles—are bounded by two great-circle arcs: a familiar example is the curved outward-facing surface of a segment of an orange. Three arcs serve to define a spherical triangle, the principal subject of this article.
Given a closed curve in E 3, find a surface having the curve as boundary with minimal area. Such a surface is called a minimal surface. In 1776 Jean Baptiste Meusnier showed that the differential equation derived by Lagrange was equivalent to the vanishing of the mean curvature of the surface:
The intersection of a normal plane and the surface will form a curve called a normal section and the curvature of this curve is the normal curvature. For most points on most “smooth” surfaces, different normal sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures , call these κ ...