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This term has other meanings, see Shar (meanings). The surface of the ball is a sphere

r is the radius of the ball

A ball is a geometric body; the set of all points of space that are at a distance from the center is not more than the specified one. This distance is called the radius of the ball. The ball is formed by rotating a semicircle near its fixed diameter. This diameter is called the axis of the ball, and both ends of the specified diameter - the poles of the ball. The surface of the ball is called a sphere: a closed ball includes this sphere, the open ball excludes. Content

If the cutting plane passes through the center of the ball, then the section of the ball is called a large circle. Other flat sections of the ball are called small circles. The area of these sections is calculated by the formula πR².

Surface area S {\ displaystyle S} and volume V {\ displaystyle V} < img src = "https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" alt = "V"> ball of radius r {\ displaystyle r} (and diameter d = 2 r {\ displaystyle d = 2r} ) are defined by the formulas: S = 4 π r 2 {\ displaystyle S = \ 4 \ pi r ^ {2}} S = π d 2 {\ displaystyle S = \ \ pi d ^ {2}} V = 4 3 π r 3 {\ displaystyle V = {\ frac {4} {3} } \ pi r ^ {3}} V = π d 3 6 {\ displaystyle V = {\ frac {\ pi d ^ {3}} {6}}

The concept of a ball in a metric space naturally generalizes the concept of a ball in Euclidean geometry.

Let the given space be (X, ρ) {\ displaystyle (X, \ rho)} . Then A ball (or an open ball) with a center at x 0 ∈ X {\ displaystyle x_ {0} \ in X} and radius r> 0 {\ displaystyle r> 0} is the set B r (x 0) = {x ∈ X ∣ ρ (x, x 0)

Sphere radius r {\ displaystyle r} with center x 0 {{displaystyle x 0}} is also called r {\ displaystyle r} - by the neighborhood of the point x 0 {\ displaystyle x_ {0}} . The ball is an open set in the topology, generated by the metric ρ {\ displaystyle \ rho} . A closed ball is closed by a set in the topology generated by the metric ρ {\ displaystyle \ rho} are its base. Obviously, B r (x 0) ⊂ D r (x 0) {\ displaystyle B_ {r} (x_ {0}) \ subset D_ {r} (x_ {0})} . However, generally speaking, the closure of an open ball may not coincide with a closed ball: B r (x 0) ¯ D r (x 0). {\ displaystyle {\ overline {B_ {r} (x_ {0})}} \ neq D_ {r} (x_ {0}).} For example: let (X, ρ) {\ displaystyle (X, \ rho)} discrete metric space, and X {\ displaystyle X} consists of more than two points . Then for any x ∈ X {\ displaystyle x \ in X} we have: B 1 (x) = {x}, B 1 (x) ¯ = {x}, D 1 (x) = X. {\ displaystyle B_ {1} (x) = \ {x \}, \; {\ overline {B_ {1} (x)}} = \ {x \}, \; D_ {1} (x) = X .} Let R d {\ displaystyle \ mathbb {R} ^ {d}} is a Euclidean space with regular Euclidean distance. Then if d = 1 {\ displaystyle d = 1} (space - line), then B r (x 0) = {x ∈ R ∣ | x - x 0 |

Source - https://ru.wikipedia.org/w/index.php?title=Shar&oldid=94085009

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