Definition.
Sphere (ball surface) is the collection of all points in three-dimensional space that are the same distance from a single point, called the center of the sphere(O).A sphere can be described as a three-dimensional figure that is formed by rotating a circle around its diameter by 180° or a semicircle around its diameter by 360°.
Definition.
Ball is the collection of all points in three-dimensional space, the distance from which does not exceed a certain distance to a point called ball center(O) (set of all points three-dimensional space scoped).A ball can be described as a three-dimensional figure, which is formed by rotating a circle around its diameter by 180 ° or a semicircle around its diameter by 360 °.
Definition. Sphere (ball) radius(R) is the distance from the center of the sphere (ball) O to any point of the sphere (surface of the ball).
Definition. Sphere (ball) diameter(D) is a segment connecting two points of the sphere (the surface of the ball) and passing through its center.
Formula. Ball volume:
| V = | 4 | π R 3 = | 1 | π D 3 |
| 3 | 6 |
Formula. Surface area of a sphere through radius or diameter:
S = 4π R 2 = π D 2
Sphere Equation
1. Equation of a sphere with radius R and center at the origin of the Cartesian coordinate system:
x 2 + y 2 + z 2 = R 2
2. Equation of a sphere with radius R and center at a point with coordinates (x 0 , y 0 , z 0) in the Cartesian coordinate system:
(x - x 0) 2 + (y - y 0) 2 + (z - z 0) 2 = R 2
Definition. diametrically opposed points are any two points on the surface of a ball (sphere) that are connected by a diameter.
Basic properties of a sphere and a ball
1. All points of the sphere are equally distant from the center.
2. Any section of a sphere by a plane is a circle.
3. Any section of a sphere by a plane is a circle.
4. The sphere has largest volume among all spatial figures with the same surface area.
5. Through any two diametrically opposite points, you can draw many large circles for a sphere or circles for a ball.
6. Through any two points, except for diametrically opposite points, it is possible to draw only one large circle for a sphere or a large circle for a ball.
7. Any two great circles of one ball intersect along a straight line passing through the center of the ball, and the circles intersect at two diametrically opposite points.
8. If the distance between the centers of any two balls is less than the sum of their radii and greater than the modulus of the difference between their radii, then such balls intersect, and a circle is formed in the plane of intersection.
The secant, chord, secant plane of the sphere and their properties
Definition. The secant of the spheres is a straight line that intersects the sphere at two points. The points of intersection are called puncture points surface or entry and exit points on the surface.
Definition. Chord of a sphere (ball) is a segment connecting two points of a sphere (the surface of a ball).
Definition. cutting plane is the plane that intersects the sphere.
Definition. Diametral plane- this is a secant plane passing through the center of a sphere or ball, the section forms, respectively great circle and big circle. The great circle and the great circle have a center that coincides with the center of the sphere (ball).
Any chord passing through the center of a sphere (ball) is a diameter.
A chord is a segment of a secant line.
The distance d from the center of the sphere to the secant is always less than the radius of the sphere:
d< R
The distance m between the cutting plane and the center of the sphere is always less than the radius R:
m< R
The section of the cutting plane on the sphere will always be minor circle, and on the ball the section will be small circle. A small circle and a small circle have their centers that do not coincide with the center of the sphere (ball). The radius r of such a circle can be found by the formula:
r \u003d √ R 2 - m2,
Where R is the radius of the sphere (ball), m is the distance from the center of the ball to the cutting plane.
Definition. Hemisphere (hemisphere)- this is half of the sphere (ball), which is formed when it is cut by a diametrical plane.
Tangent, tangent plane to the sphere and their properties
Definition. Tangent to sphere is a straight line that touches the sphere at only one point.
Definition. Tangent plane to sphere is a plane that touches the sphere at only one point.
The tangent line (plane) is always perpendicular to the radius of the sphere drawn to the point of contact
The distance from the center of the sphere to the tangent line (plane) is equal to the radius of the sphere.
Definition. ball segment- this is the part of the ball that is cut off from the ball by a cutting plane. The backbone of the segment call the circle that formed at the site of the section. segment height h is the length of the perpendicular drawn from the middle of the base of the segment to the surface of the segment.
Formula. Outer surface area of a sphere segment with height h in terms of sphere radius R:
S = 2π Rh
In Chapter 2, we will continue “building geometry” and talk about the structure and properties of the most important spatial figures - a ball and a sphere, cylinders and cones, prisms and pyramids. Most of the objects created by human hands are buildings, cars, furniture, utensils, etc. ., etc., consists of parts having the shape of these figures.
§ 4. SPHERE AND BALL
After straight lines and planes, the sphere and ball are the simplest, but very important and rich in various properties. spatial figures. Entire books have been written about the geometric properties of a ball and its surface - a sphere. Some of these properties were already known to the ancient Greek geometers, and some were found quite recently, in last years. These properties (together with the laws of natural science) explain why, for example, celestial bodies and fish eggs have the shape of a ball, why bathyscaphes and soccer balls are made in the shape of a ball, why ball bearings are so common in technology, etc. We can only prove the simplest properties of a ball. The proof of other properties, although very important, often requires the use of methods that are not at all elementary, although the formulations of such properties can be very simple: for example, among all bodies having a given surface area, the sphere has the largest volume.
4.1. Definitions of sphere and sphere.
A sphere and a ball are defined in space in exactly the same way as a circle and a circle on a plane. A sphere is a figure consisting of all points in space remote from a given
point to the same (positive) distance.
This point is called the center of the sphere, and the distance is called its radius (Fig. 4.1).
So, a sphere with center O and radius R is a figure formed by all points X of the space for which
A ball is a figure formed by all points in space that are at a distance not greater than a given (positive) distance from a given point. This point is called the center of the ball, and this distance is called its radius.
So, a ball with center O and radius R is a figure formed by all points X of the space for which
Those points X of the ball with center O and radius R for which they form a sphere. It is said that this sphere bounds the given sphere, or that it is its surface.
The symbol of the globe is the globality of the globe of the Earth. A symbol of the future, it differs from the cross in that the latter personifies suffering and human death. AT Ancient Egypt first came to the conclusion that the earth is spherical. This assumption served as the basis for numerous reflections on the immortality of the earth and the possibility of immortality of the living organisms inhabiting it.
This point(O) is called the center of the sphere. Any segment connecting the center and some point of the sphere is called the radius of the sphere (R-radius of the sphere). The segment connecting two points of the sphere and passing through its center is called the diameter of the sphere. Obviously, the diameter of the sphere is 2R.
Definition of a ball A ball is a body that consists of all points in space that are at a distance not greater than a given distance from a given point (or a figure bounded by a sphere). A body bounded by a sphere is called a sphere. The center, radius, and diameter of a sphere are also called the center, radius, and diameter of a sphere. Ball
The plane passing through the center of the ball is called the diametral plane. The plane passing through the center of the ball is called the diametral plane. The section of the ball by the diametrical plane is called the great circle, and the section of the sphere is called the great circle. The section of the ball by the diametrical plane is called the great circle, and the section of the sphere is called the great circle.
X²+y²=R²-d² If d>R, then the sphere and the plane have no common points. R, then the sphere and plane have no common points."> R, then the sphere and plane have no common points."> R, then the sphere and plane have no common points." title="(!LANG:x²+y²=R² -d² If d>R, then the sphere and the plane do not have common points."> title="x²+y²=R²-d² If d>R, then the sphere and the plane have no common points."> !}
A tangent plane to a sphere by a tangent plane to a sphere A plane that has only one common point with the sphere is called the tangent plane to the sphere, the tangent point A of the plane and the sphere. And their common point is called the tangent point A of the plane and the sphere.
Theorem: The radius of a sphere drawn at the point of contact between the sphere and the plane is perpendicular to the tangent plane. Proof: Consider a plane α tangent to a sphere with center O at point A. Prove that OA is perpendicular to α. Let's assume it's not. Then the radius OA is inclined to the plane α, and hence the distance from the center of the sphere to the plane is less than the radius of the sphere. Therefore, the sphere and the plane intersect in a circle. This is contrary to what is tangent, i.e. a sphere and a plane have only one common point. The resulting contradiction proves that OA is perpendicular to α.
Sphere and ball
The word "sphere" comes from the Greek word "sfire", which is translated into Russian as "ball".
SHAR is a symbol of the future.
The symbol of the globe is the globality of the globe of the Earth. A symbol of the future, it differs from the cross in that the latter personifies suffering and human death. In ancient Egypt, they first came to the conclusion that the earth is spherical. This assumption served as the basis for numerous reflections on the immortality of the earth and the possibility of immortality of the living organisms inhabiting it.
A man holding a ball in his hands symbolizes a subject bearing the burdens of the world. It is not by chance that some stations in Western Europe, for example, in Helsinki, are decorated with similar sculptures: here the burdens that fall on the shoulders of the traveler are depicted.
Thus, the ball and the globe are signs of providence, conduct, eternity, power and power of crowned persons.
The stone hemisphere of the sphere is embodied in religious temples - domes Orthodox churches in Russia; stupas associated with the residence of bodhisattvas in India. In Indonesia, stupas have acquired the shape of a bell with a stone spire at the top and are called dagobas.
In Greco-Roman mythology, the ball symbolized good luck, fate, being associated with Tihe (Fortune) standing on the ball. The famous painting by Picasso "Girl on a Ball" is a dancing Fortune.
Ball Shape in Nature Many berries are ball shaped.
The planets are spherical.
Some trees are spherical.
Definition of a sphere A sphere is a surface consisting of all points in space located at a given distance from a given point.
A sphere is a surface obtained by rotating a semicircle around a diameter
This point (O) is called the center of the sphere. Any segment connecting the center and some point of the sphere is called the radius of the sphere (R-radius of the sphere). The segment connecting two points of the sphere and passing through its center is called the diameter of the sphere. Obviously, the diameter of the sphere is 2R.
Definition of a ball A ball is a body that consists of all points in space that are at a distance not greater than a given distance from a given point (or a figure bounded by a sphere). A body bounded by a sphere is called a sphere. The center, radius, and diameter of a sphere are also called the center, radius, and diameter of a sphere.
Spherical segment A spherical segment is a part of a ball cut off from it by some plane.
Spherical layer A spherical layer is a part of a sphere enclosed between two parallel cutting planes.
Spherical sector A spherical sector is a body obtained by rotating a circular sector with an angle less than 900 around a straight line containing one of the radii bounding the circular sector.
The plane passing through the center of the ball is called the diametrical plane. The cross section of a ball with a diametral plane is called a great circle, and the cross section of a sphere is called a great circle. Ball section
We fix Solve the problem No. 573, No. 574 (a)
Equation of a sphere in a rectangular coordinate system M(x;y;z) is an arbitrary point belonging to the sphere. /MC/= v(x-x0)2+(y-y0)2+(z-z0)2 MC=R, then (x-x0)2+(y-y0)2+(z-z0)2=R2
Task 1. Find the coordinates of the center and radius of the sphere given by the equation: x?+y?+z?=49 (X-3)?+(y+2)?+z?=2 2. Write the equation for a sphere of radius R with center And if A(2;-4;7) R=3 A(0;0;0) R=v2 A(2;0;0) R=4 3. Solve problem No. 577(a)
Mutual arrangement of the sphere and the plane Let us denote the radius of the sphere by the letter R, and the distance from its center to the plane? by the letter d. Let us introduce a coordinate system so that the plane Oxy coincides with the plane?, and the center C of the sphere lies on the positive semiaxis Oz.
In this coordinate system, the point C (o; o; d), so the sphere has the equation x2+y2+(z-d)2=R? The plane coincides with the coordinate plane Oxy, and therefore its equation is z=0
Thus, the question of the mutual arrangement of the sphere and the plane is reduced to the study of a system of equations. Substituting z=0 into the second equation, we get x?+y?=R?-d? 3 cases are possible:
x?+y?=R?-d? If d>R, then the sphere and the plane have no common points.
x?+y?=R?-d? If d=R, then the sphere and the plane name only one common point. In this case? called the tangent plane to the sphere
x?+y?=R?-d? If d We fix Solve the problem No. 580, No. 581 Tangent plane to the sphere A plane that has only one common point with the sphere is called the tangent plane to the sphere, and their common point is called the tangent point A of the plane and the sphere. Theorem: The radius of a sphere drawn at the point of contact between the sphere and the plane is perpendicular to the tangent plane. Proof: Consider a plane? touching a sphere with center O at point A. Prove that OA is perpendicular to?. Let's assume it's not. Then the radius OA is inclined to the plane?, and, consequently, the distance from the center of the sphere to the plane is less than the radius of the sphere. Therefore, the sphere and the plane intersect in a circle. This is contrary to what is tangent, i.e. a sphere and a plane have only one common point. The resulting contradiction proves that OA is perpendicular?. Converse theorem: If the radius of a sphere is perpendicular to a plane passing through its end lying on the sphere, then this plane is tangent to the sphere. We fix Solve problem No. 592 Area of a Sphere A sphere cannot be flattened! A polyhedron described near a sphere is a polyhedron all of whose faces are touched by the sphere. A sphere is said to be inscribed in a polyhedron Task: The cross-sectional area of a sphere passing through its center is 9m2. Find the area of the sphere. Solution: The section passing through the center of the sphere is a circle. Ssec =?r2, 9=?R2, R=v9/? . Sspheres=4 ?r2 , Sspheres=4? · nine/? =36m2
