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The ratio of the volume of a sphere to the volume of its circumscribed cylinder is 2:3, as was determined by Archimedes. The principal formulae derived in On the Sphere and Cylinder are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder.
where V is the volume of a sphere and r is the radius. S A = 4 π r 2 {\displaystyle SA=4\pi r^{2}} where SA is the surface area of a sphere and r is the radius.
There is an isoperimetric inequality for the sphere with its usual metric and spherical measure (see Ledoux & Talagrand, chapter 1): . If A ⊆ S n−1 is any Borel set and B⊆ S n−1 is a ρ n-ball with the same σ n-measure as A, then, for any r > 0,
The benefit of the expansion in terms of the real ... where ω n−1 is the volume of the (n−1)-sphere. ... determined by the selection rules for the ...
The volume of a sphere with radius r is 4 / 3 πr 3. The surface area of a sphere with radius r is 4πr 2. Some of the formulae above are special cases of the volume of the n-dimensional ball and the surface area of its boundary, the (n−1)-dimensional sphere, given below. Apart from circles, there are other curves of constant width.
Subtracting the volume of the cone from the volume of the cylinder gives the volume of the sphere: V S = 4 π − 8 3 π = 4 3 π . {\displaystyle V_{S}=4\pi -{8 \over 3}\pi ={4 \over 3}\pi .} The dependence of the volume of the sphere on the radius is obvious from scaling, although that also was not trivial to make rigorous back then.
The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. A sphere with a spherical triangle on it.
The cosine rule may be used to give the angles A, B, and C but, to avoid ambiguities, the half angle formulae are preferred. Case 2: two sides and an included angle given (SAS). The cosine rule gives a and then we are back to Case 1. Case 3: two sides and an opposite angle given (SSA). The sine rule gives C and then we have Case 7. There are ...