Hello! In this article, we will consider problems with balls. Rather, there will be a combination of bodies here: a ball or, in other words, a cylinder described near the ball (which is the same thing) and a cube inscribed in the ball.

The blog has already considered a group of tasks with balls, . In the presented tasks, we will talk about finding the volume and surface area of ​​​​the indicated bodies.you need to know!

Sphere volume formula:

The formula for the surface area of ​​a sphere is:

The formula for the volume of a cylinder is:

The formula for the surface area of ​​a cylinder is:

More about cylinder side surface area:

It is a rectangle "twisted" into a cylinder, one side of which is equal to the circumference of the base - this is 2ПiR, the other side is equal to the height of the cylinder - this is N.

What should be noted regarding the presented tasks?

1. If a ball is inscribed in a cylinder, then they have a common radius.

2. The height of a cylinder circumscribed about a sphere is equal to two of its radii (or diameter).

3. If a cube is inscribed in a sphere, then the diagonal of this cube is equal to the diameter of the sphere.

245348. The cylinder is described near the ball. The volume of the cylinder is 33. Find the volume of the sphere.

Sphere volume formula:

We need to find the radius of the sphere.

A sphere and a cylinder have a common radius. The base of the cylinder is a circle with radius R, the height of the cylinder is equal to two radii. So the volume of the cylinder is calculated by the formula:

Substitute the volume given in the condition into the formula and express the radius:

Let's leave the expression in this form, it is not necessary to express the radius (extract the root of the third degree), since we need exactly R 3 .

Thus, the volume of the sphere will be equal to:

Answer: 22

245349. The cylinder is described near the ball. The volume of the sphere is 24. Find the volume of the cylinder.

This task is the reverse of the previous one.

Sphere volume formula:

The volume of a cylinder is calculated by the formula:

Since the volume of the sphere is known, we can express the radius and then find the volume of the cylinder:

Thus:

Answer: 36

316557. The ball is inscribed in the cylinder. The surface area of ​​the sphere is 111. Find the total surface area of ​​the cylinder.

Sphere surface formula:

Cylinder surface formula:

Let's simplify:

Since the surface area of ​​the ball is given to us, we can express the radius:

Answer: 166.5

A BALL ABOUT A CYLINDER AND A CONE is called (a) if the top of the cone lies on the surface of the ball, and the base of the cone is the section of the ball. A ball can always be circumscribed near a right circular cone. The center of a ball circumscribed near a cone lies at the height of the cone. The center of the ball described near the cone can be both inside and outside the cone, and also coincide with the center of the base.

is called) if the bases of the cylinder are sections of a sphere. (a A right circular cylinder can be circumscribed. The center of a sphere circumscribed about a cylinder lies at the height of the cylinder.

The center of the circumcircle of a triangle is the point of intersection of the perpendicular bisectors of the sides of the triangle. The center of the circumcircle of a triangle can be outside the triangle. For a right triangle: R= The center of the circumcircle of a right triangle is the midpoint of the hypotenuse. For a regular quadrilateral: R= a side; R is the radius of the inscribed circle

No. 645. A cylinder is inscribed in a sphere. Find the ratio of the area of ​​the total surface of the cylinder to the area of ​​the sphere if the height of the cylinder is equal to the diameter of the base. R R Given: a sphere with center O, a cylinder is inscribed, h=2 R Find: R Analysis of conditions: O R

A ball can be circumscribed near a pyramid if and only if a circle can be circumscribed near its base.

To build the center O of this ball, you need:

1. Find the center O, the circle circumscribed near the base.

2. Through point O, draw a straight line perpendicular to the plane of the base.

3. Through the middle of any side edge of the pyramid, draw a plane perpendicular to this edge.

4. Find the point O of the intersection of the constructed line and plane.

Special case: the side edges of the pyramid are equal. Then:

the ball can be described;

the center O of the ball lies at the height of the pyramid;

Where is the radius of the circumscribed sphere; - side rib; H is the height of the pyramid.

5.2. ball and prism

A sphere can be circumscribed near a prism if and only if the prism is straight and a circle can be circumscribed near its base.

The center of the ball is the middle of the segment connecting the centers of the circles described near the bases.

where is the radius of the circumscribed sphere; is the radius of the circumscribed circle near the base; H is the height of the prism.

5.3. ball and cylinder

A sphere can always be described near a cylinder. The center of the sphere is the center of symmetry of the axial section of the cylinder.

5.4. ball and cone

A sphere can always be described near a cone. the center of the ball; serves as the center of a circle circumscribed about the axial section of the cone.

The world around us, despite the variety of objects and phenomena occurring with them, is full of harmony due to the clear action of the laws of nature. Behind the apparent freedom with which nature draws the outlines and creates the forms of things, there are clear rules and laws that involuntarily suggest the presence of some higher power in the process of creation. On the verge of pragmatic science, which gives a description of occurring phenomena from the position of mathematical formulas and theosophical worldviews, there is a world that gives us a whole bunch of emotions and impressions from the things that fill it and the events that occur with them.

A ball as is the most common form found in nature for physical bodies. Most of the bodies of the macrocosm and microcosm have the shape of a ball or tend to approach one. In fact, the ball is an example of an ideal shape. The generally accepted definition for a ball is considered to be the following: it is a geometric body, a set (set) of all points in space that are located at a distance from the center not exceeding a given one. In geometry, this distance is called the radius, and in relation to a given figure it is called the radius of the ball. In other words, the volume of the sphere contains all points located at a distance from the center not exceeding the length of the radius.

The ball is also considered as the result of the rotation of a semicircle around its diameter, which at the same time remains motionless. At the same time, to such elements and characteristics as the radius and volume of the ball, the axis of the ball (fixed diameter) is added, and its ends are called the poles of the ball. The surface of a sphere is called a sphere. If we are dealing with a closed ball, then it includes this sphere, if with an open one, then it excludes it.

Considering additional definitions related to the ball, it should be said about cutting planes. The secant plane passing through the center of the ball is called the great circle. For other flat sections of the ball, it is customary to use the name "small circles". When calculating the areas of these sections, the formula πR² is used.

Calculating the volume of a ball, mathematicians encountered some rather fascinating patterns and peculiarities. It turned out that this value either completely repeats, or is very close in terms of the method of determination, to the volume of a pyramid or a cylinder described around a ball. It turns out that the volume of the ball is equal if its base has the same area as the surface of the ball, and the height is equal to the radius of the ball. If we consider a cylinder described around the ball, then we can calculate the pattern according to which the volume of the ball is one and a half times less than the volume of this cylinder.

Attractive and original is the method of withdrawing the ball using the Cavalieri principle. It consists in finding the volume of any figure by adding the areas obtained by its section by an infinite number. For the conclusion, let's take a hemisphere of radius R and a cylinder having a height R with a base-circle of radius R (the bases of the hemisphere and the cylinder are located in the same plane). In this cylinder we enter a cone with a vertex in the center of its lower base. Having proved that the volume of the hemisphere and the parts of the cylinder that are outside the cone are equal, we can easily calculate the volume of the ball. Its formula takes the following form: four thirds of the product of the cube of radius and π (V= 4/3R^3×π). This is easy to prove by drawing a common cutting plane through a hemisphere and a cylinder. The areas of a small circle and a ring bounded from the outside by the sides of a cylinder and a cone are equal. And, using the Cavalieri principle, it is easy to come to the proof of the main formula, with the help of which we determine the volume of the ball.

But not only the problem of studying natural bodies is connected with finding ways to determine their various characteristics and properties. Such a figure of stereometry as a ball is very widely used in practical human activities. The mass of technical devices has in their designs details not only of a spherical shape, but also made up of elements of a ball. It is the copying of ideal natural solutions in the process of human activity that gives the highest quality results.

When a pyramid inscribed in a ball is given in the problem, the following theoretical information will be useful in solving it.

If the pyramid is inscribed in a ball, then all its vertices lie on the surface of this ball (on the sphere), respectively, the distances from the center of the ball to the vertices are equal to the radius of the ball.

Each face of a pyramid inscribed in a ball is a polygon inscribed in some circle. The bases of the perpendiculars dropped from the center of the ball on the plane of the faces are the centers of these circumscribed circles. Thus, the center of the sphere described near the pyramid is the point of intersection of the perpendiculars to the faces of the pyramid, drawn through the centers of the circles described near the faces.

More often, the center of the ball described near the pyramid is considered as the point of intersection of the perpendicular drawn to the base through the center of the circle circumscribed near the base, and the perpendicular bisector to the lateral edge (the perpendicular bisector lies in the plane passing through this lateral edge and the first perpendicular (drawn to the base). If a circle cannot be inscribed near the base of a pyramid, then this pyramid cannot be inscribed in a ball.It follows that a ball can always be inscribed near a triangular pyramid, and a quadrangular pyramid inscribed in a ball with a parallelogram at the base can have a rectangle or square base.

The center of the sphere described near the pyramid can lie inside the pyramid, on the surface of the pyramid (on the side face, on the base), and outside the pyramid. If the condition of the problem does not say exactly where the center of the described ball lies, it is advisable to consider how various options for its location can affect the solution.

Near any regular pyramid, a sphere can be described. Its center is the point of intersection of the line containing the height of the pyramid and the perpendicular bisector to the side edge.

When solving problems on a pyramid inscribed in a ball, some triangles are most often considered.

Let's start with triangle SO1C. It is isosceles, since its two sides are equal as the radii of the ball: SO1=O1С=R. Therefore, O1F is its height, median and bisector.

Right triangles SOC and SFO1 are similar in acute angle S. Hence

SO=H is the height of the pyramid, SC=b is the length of the side edge, SF=b/2, SO1=R, OC=r is the radius of the circle circumscribed near the base of the pyramid.

In a right triangle OO1C, the hypotenuse is O1C=R, the legs are OC=r, OO1=H-R. According to the Pythagorean theorem:

If we continue the height SO, we get the diameter SM. Triangle SCM is right-angled (because the inscribed angle SCM rests on the diameter). In it, OC is the height drawn to the hypotenuse, SO and OM are the projections of the legs SC and CM onto the hypotenuse. According to the properties of a right triangle,