Definition of a ball. Sphere, ball, segment and sector. Formulas and properties of the sphere. Tangent, tangent plane to a sphere and their properties
Ball (sphere)
Spherical surface. Ball (sphere). Ball sections: circles.
Archimedes' theorem. Parts of the ball: spherical segment,
spherical layer, spherical belt, spherical sector.
Spherical surface - This locus of points(those. manynumber of all points)in space, equidistant from one point O , which is called the center of the spherical surface (Fig.90). Radius AOi diameter AB are determined in the same way as in a circle.
Ball (sphere) - This a body bounded by a spherical surface. Can get the ball by rotating the semicircle ( or circle ) around the diameter. All plane sections of the ball are circles ( Fig.90 ). The largest circle lies in a section passing through the center of the ball and is called big circle. Its radius is equal to the radius of the ball. Any two large circles intersect along the diameter of the ball ( AB, Fig.91 ).This diameter is also the diameter of intersecting great circles. Through two points of a spherical surface located at the ends of the same diameter(A and B, Fig.91 ), you can draw countless large circles. For example, an infinite number of meridians can be drawn through the Earth's poles.
The volume of the sphere is one and a half times less than the volume of the cylinder circumscribed around it. (Fig.92 ), A the surface of the ball is one and a half times less than the total surface of the same cylinder ( Archimedes' theorem):
Here S ball And V ball - the surface and volume of the ball, respectively;
S cyl And V cyl - the total surface and volume of the circumscribed cylinder.
Parts of the ball. Part of a ball (sphere) ), cut off from it by some plane ( ABC, Fig.93), called ball(spherical ) segment. Circle ABC called basis ball segment. Line segment MN perpendicular drawn from the center N circle ABC until it intersects with a spherical surface, is called height ball segment. Dot M called top ball segment.
Part of a sphere enclosed between two parallel planes ABC and DEF intersecting a spherical surface (Fig. 93), called spherical layer; the curved surface of a spherical layer is called ball belt(zone). Circles ABC and DEF – grounds ball belt. Distance N.K. between the bases of the spherical belt - its height. The part of the ball bounded by the curved surface of a spherical segment ( AMCB, Fig.93) and conical surface OABC , the base of which is the base of the segment ( ABC ), and the vertex is the center of the ball O , called spherical sector.
When people are asked the difference between a sphere and a ball, many simply shrug their shoulders, thinking that in fact they are the same thing (the analogy with a circle and a circle). Indeed, do all of us know geometry well from the school curriculum and can immediately answer this question? A sphere has some differences from a ball, which not only schoolchildren need to know in order to get a good grade for their demonstrated knowledge, but also many other people, for example, whose work is directly related to drawings.
Definition
Ball– the set of all points in space. All these points are located from the center of the geometric body at a distance that is no more than a given one. This distance itself is called the radius. A ball, as a geometric body, is formed as follows: a semicircle rotates near its diameter. As for the sphere, this is the surface of the ball (for example, a closed ball includes it, an open one does not). Calculating the area or volume of a ball involves entire geometric formulas that are very complex, despite the apparent simplicity of the geometric figure itself.
Sphere, as noted above, is the surface of the ball, its shell. All points in space are equidistant from the center of the sphere. As for the radius of a geometric body, it is called any segment, one point of which is directly the center of the sphere, and the other can be located at any point on the surface. We can say that a sphere is the shell of a ball without any content (more specific examples will be given below). Just like a ball, a sphere is a body of revolution. By the way, many also wonder what is the difference between a circle and a circle from a sphere and a ball. Everything is simple here: in the first case these are figures on a plane, in the second - in space.
Comparison
It has already been said that a sphere is the surface of a ball, which already makes it possible to talk about one significant sign of difference. The difference between the two geometric bodies is observed in some other aspects:
- All points of the ball are at the same distance from the center, while the body is limited by the surface (a sphere that is empty inside). In other words, the sphere is hollow. Usually, for ease of understanding, a simple example is given with a balloon and a billiard ball. Both of these objects are called balls, but in the first case we are dealing with a sphere, and in the second with a full-fledged ball with its own contents inside.
- A sphere has its own area, but it has no volume. A sphere is the opposite: its volume can be calculated, while it has no area. Some may say that this is the main sign of difference, but it only appears if it is necessary to make some calculations (complex geometric formulas). Therefore, the main difference is that the sphere is hollow, and the ball is a body with contents inside.
- Another difference lies in the radius. For example, the radius of a sphere is not only the distance of points to the center. A radius can be any segment connecting a point on a sphere to its center. All these segments are equal to each other. As for the ball, the points lying inside it are removed from the center by less than a radius (precisely because of the sphere bounding it).
Conclusions website
- A sphere is hollow, while a ball is a body filled inside. For example, a hot air balloon is a sphere, a billiard ball is a full-fledged ball.
- A sphere has area and no volume, but a sphere does the opposite.
- The third difference is the measurement of the radius of two geometric bodies.
A ball and a sphere are, first of all, geometric figures, and if a ball is a geometric body, then a sphere is the surface of a ball. These figures were of interest many thousands of years ago BC.
Subsequently, when it was discovered that the Earth is a ball and the sky is a celestial sphere, a new fascinating direction in geometry was developed - geometry on a sphere or spherical geometry. In order to talk about the size and volume of a ball, you must first define it.
Ball
A ball of radius R with a center at point O in geometry is a body that is created by all points in space that have a common property. These points are located at a distance not exceeding the radius of the ball, that is, they fill the entire space less than the radius of the ball in all directions from its center. If we consider only those points that are equidistant from the center of the ball, we will consider its surface or the shell of the ball.
How can I get the ball? We can cut a circle out of paper and start rotating it around its own diameter. That is, the diameter of the circle will be the axis of rotation. The formed figure will be a ball. Therefore, the ball is also called a body of rotation. Because it can be formed by rotating a flat figure - a circle.
Let's take some plane and cut our ball with it. Just like we cut an orange with a knife. The piece that we cut off from the ball is called a spherical segment.
In Ancient Greece, they knew how to not only work with a ball and sphere as geometric figures, for example, use them in construction, but also knew how to calculate the surface area of a ball and the volume of a ball.
A sphere is another name for the surface of a ball. A sphere is not a body - it is the surface of a body of rotation. However, since both the Earth and many bodies have a spherical shape, for example a drop of water, the study of geometric relationships inside the sphere has become widespread.
For example, if we connect two points of a sphere with each other by a straight line, then this straight line is called a chord, and if this chord passes through the center of the sphere, which coincides with the center of the ball, then the chord is called the diameter of the sphere.
If we draw a straight line that touches the sphere at just one point, then this line will be called a tangent. In addition, this tangent to the sphere at this point will be perpendicular to the radius of the sphere drawn to the point of contact.
If we extend the chord to a straight line in one direction or the other from the sphere, then this chord will be called a secant. Or we can say it differently - the secant to the sphere contains its chord.
Ball volume
The formula for calculating the volume of a ball is:
where R is the radius of the ball.
If you need to find the volume of a spherical segment, use the formula:
V seg =πh 2 (R-h/3), h is the height of the spherical segment.
Surface area of a ball or sphere
To calculate the area of a sphere or the surface area of a ball (they're the same thing):
where R is the radius of the sphere.
Archimedes was very fond of the ball and sphere, he even asked to leave a drawing on his tomb in which a ball was inscribed in a cylinder. Archimedes believed that the volume of a ball and its surface are equal to two-thirds of the volume and surface of the cylinder in which the ball is inscribed.”
In Chapter 2 we will continue “structural geometry” and talk about the structure and properties of the most important spatial figures - ball and sphere, cylinders and cones, prisms and pyramids. Most objects created by human hands - buildings, cars, furniture, dishes, etc. ., etc., consists of parts shaped like these figures.
§ 4. SPHERE AND BALL
After straight lines and planes, the sphere and ball are the simplest, but very important spatial figures rich in various properties. Whole books have been written about the geometric properties of a ball and its surface - a sphere. Some of these properties were known to ancient Greek geometers, and some were discovered more recently, in recent years. These properties (together with the laws of natural science) explain why, for example, celestial bodies and fish eggs are spherical in shape, why bathyscaphes and soccer balls are made in the shape of a ball, why ball bearings are so common in technology, etc. We can prove only the simplest properties of the ball. Proofs of other, albeit very important properties, often require the use of completely non-elementary methods, although the formulation of such properties can be very simple: for example, among all bodies having a given surface area, the ball has the largest volume.
4.1. Definitions of sphere and ball.
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 one.
different points to the same (positive) distance.
This point is called the center of the sphere, and the distance is its radius (Fig. 4.1).
So, a sphere with center O and radius R is a figure formed by all points X of space for which
A ball is a figure formed by all points in space located at a distance no greater than a given (positive) distance from a given point. This point is called the center of the ball, and this distance is its radius.
So, a ball with center O and radius R is a figure formed by all points X of space for which
Those points X of a ball with center O and radius R for which they form a sphere. They say that this sphere encloses a given ball or that it is its surface.
About the same points X of the ball for which they say that they lie inside the ball.
The radius of a sphere (and ball) is called not only the distance, but also any segment connecting the center with a point on the sphere.
