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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.
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.
The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area. [11] The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the surface tension locally minimizes surface area.
The surface area of a sphere of radius R is . Given the surface area of a non-spherical object A , one can calculate its surface area-equivalent radius by setting 4 π R eq 2 = A {\displaystyle 4\pi R_{\text{eq}}^{2}=A}
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle ) is called the reference plane (sometimes fundamental plane ).
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
The sphere's radius is taken as unity. For specific practical problems on a sphere of radius R the measured lengths of the sides must be divided by R before using the identities given below. Likewise, after a calculation on the unit sphere the sides a, b, and c must be multiplied by R.
The surface of the spherical segment (excluding the bases) is called spherical zone. Geometric parameters for spherical segment. If the radius of the sphere is called R , the radii of the spherical segment bases are a and b , and the height of the segment (the distance from one parallel plane to the other) called h , then the volume of the ...