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A spherical wedge with radius r and angle of the wedge α. In geometry, a spherical wedge or ungula is a portion of a ball bounded by two plane semidisks and a spherical lune (termed the wedge's base). The angle between the radii lying within the bounding semidisks is the dihedral α.
For example, assuming the Earth is a sphere of radius 6371 km, the surface area of the arctic (north of the Arctic Circle, at latitude 66.56° as of August 2016 [7]) is 2π ⋅ 6371 2 | sin 90° − sin 66.56° | = 21.04 million km 2 (8.12 million sq mi), or 0.5 ⋅ | sin 90° − sin 66.56° | = 4.125% of the total surface area of the Earth.
A full circle lune, {2} 2π The surface area of a spherical lune is 2θ R 2, where R is the radius of the sphere and θ is the dihedral angle in radians between the two half great circles.
The surface area, or properly the -dimensional volume, of the -sphere at the boundary of the (+) -ball of radius is related to the volume of the ball by the differential equation
If the radius of the sphere is denoted by r and the height of the cap by h, the volume of the spherical sector is =.. This may also be written as = (), where φ is half the cone aperture angle, i.e., φ is the angle between the rim of the cap and the axis direction to the middle of the cap as seen from the sphere center.
Surface area. Conical frustum 3D model of a conical frustum. For a right circular conical frustum [5] [6] the slant height is = +, the lateral surface area is ...
The wedge prism is used to take measurements in both land management and in timber procurement. Other tools often used to accompany the wedge prism in taking forest inventory are clinometers, Biltmore sticks, relascopes, and diameter tapes. A wedge prism can also be used with a target placed at plot center, to establish fixed radius plots.
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