Definition.
Sphere (ball surface) is the collection of all points in three-dimensional space that are at the same distance from one point, called center of the sphere(ABOUT).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 a collection of all points in three-dimensional space, the distance from which does not exceed a certain distance to a point called center of the ball(O) (the set of all points of three-dimensional space limited by a sphere).A ball 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. Radius of the sphere (ball)(R) is the distance from the center of the sphere (ball) O to any point on the sphere (surface of the ball).
Definition. Sphere (ball) diameter(D) is a segment connecting two points of a sphere (the surface of a ball) and passing through its center.
Formula. Sphere 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 opposite 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 ball by a plane is a circle.
4. The sphere has the largest volume among all spatial figures with the same surface area.
5. Through any two diametrically opposite points you can draw many great circles for a sphere or circles for a ball.
6. Through any two points, except diametrically opposite points, you can 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 of their radii, then such balls intersect, and a circle is formed in the intersection plane.
Secant, chord, secant plane of a sphere and their properties
Definition. Sphere secant is a straight line that intersects the sphere at two points. The intersection points are called piercing points surfaces or entry and exit points on the surface.
Definition. Chord of a sphere (ball)- this is a segment connecting two points on 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 accordingly large circle And big circle. The great circle and 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 location of the section of the cutting plane on the sphere will always be small circle, and on the ball the section will be small circle. The small circle and small circle have their own centers that do not coincide with the center of the sphere (ball). The radius r of such a circle can be found using the formula:
r = √R 2 - m 2,
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 a sphere (ball), which is formed when it is cut by a diametrical plane.
Tangent, tangent plane to a sphere and their properties
Definition. Tangent to a sphere is a straight line that touches the sphere at only one point.
Definition. Tangent plane to a 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. Basis of the segment called 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 through the radius of the sphere R:
S = 2πRh
We give here a very simple, although not entirely rigorous, derivation of the formula for the area of a spherical surface; in its idea it is very close to the methods of integral calculus. So, let us be given a certain ball of radius R. Let us select some small area on its surface (Fig. 412) and consider a pyramid or cone with its vertex in the center of the ball O, having this area as its base; strictly speaking, we are only conditionally speaking about a cone or a pyramid, since the base is not flat, but spherical. But if the size of the base is small compared to the radius of the ball, it will differ very little from a flat one (for example, when measuring a not very large plot of land, they neglect the fact that it lies not on a plane, but on a sphere).
Then, denoting the base of the “pyramid” through the area of this section, we find its volume as the product of one third of the height by the area of the base (the height is the radius of the ball):
If we now decompose the entire surface of the ball into a very large number N of such small areas, thereby the volume of the ball into N volumes of “pyramids” having these areas as their bases, then the entire volume will be represented by the sum
where the last sum is equal to the total surface of the ball:
So, the volume of a sphere is equal to one third of the product of its radius and surface area. Hence, for the surface area we have the formula
The last result is formulated as follows:
The surface area of a sphere is equal to four times the area of its great circle.
The above conclusion is also suitable for the surface area of a sphere sector (we mean only the base, i.e., the spherical surface, or “cap”; see Fig. 409). And in this case, the volume of the sector is equal to one third of the product of the radius of the ball and the area of its spherical base:
where we find the formula for the area of the cap
The spherical surface of the spherical layer is called a spherical belt (see Fig. 408). To calculate the surface area of the spherical belt, we find the difference between the surfaces of two spherical caps:
where is the height of the layer. So, the surface area of the spherical belt for a given ball depends only on the height of the corresponding layer, but not on its position on the ball.
Task. The lateral surface of a cone circumscribed around a ball has an area equal to one and a half times the surface area of the ball. Find the height of the cone if the radius of the ball is .
Solution. For convenience, let us introduce the angle a between the height and the generatrix of the cone (Fig. 413). Let us find the expressions for the height, base radius and generatrix of the cone
Having just one formula and knowing initially what the diameter or radius is, you can easily calculate the surface area of the ball. The formula will look like S =4πR2, where pi is multiplied by 4, then by the radius of the ball to the square power. But before direct calculations, you should immediately understand the terms.
This you should know:
- Ball- a geometric object resulting from rotational semicircular movements around the center. Any point on the surface of the ball is at the same distance from the center.
- Sphere- not the same as a ball. If it is a volumetric object and includes internal space, then a sphere is only the surface of this object and has only its own area. In other words, it cannot be said that a sphere has such and such a volume, unlike a ball.
- Pi" is a constant number equal to the ratio of the circumference of a circle to its diameter. In abbreviated form, it is usually denoted by a number equal to 3.14. But in fact, after the three there are more than a thousand numbers!
- The radius of the ball is equal to ½ its diameter. The exact diameter can be calculated using several flat and even objects. You just need to clamp the ball between these objects that clamp the ball and are located perpendicular to each other, and then measure the resulting diameter.
- Square degree denoted as a two and means that this number must be multiplied by itself once. If the power of a number were in the form of three, then you would need to multiply by itself twice. By writing down the expression on paper, you can understand why two and three are used, and not one and two.
- Volume– a quantity indicating the size in space occupied by an object. The volume of the ball depends on the diameter. The formula will be equal to four thirds multiplied by pi and again multiplied by its radius cubed.
- Square– a quantity indicating the size of the surface of an object, but not the internal space.
Interesting facts
This is interesting:
- The number "pi" has its own fan clubs all over the world. Members of society try to remember as many signs from this number as possible, and also try to unravel the universal secrets hidden in the number.
- The Earth's land area is only 29.2% of its total surface. The exact number of the area is difficult to give due to the uneven topography of the Earth, such as depressions and mountains.
- Knowledge about the formula for the area of a sphere can be used in everyday life. Also, with this knowledge you can suppress your opponent in a dispute.
By demonstrating the extent of your knowledge in the field of geometry, you can initially gain respect, and you can make it clear to repairmen and sellers that you simply cannot be fooled.
Application of the formula
Let's look at an example, how to calculate the area of a round ball, the diameter of which is 50 cm. Following the formula, you need to divide 50 by two (to get the radius), square the resulting number and multiply the whole thing first by 4, then by 3.14. As a result, we get a number of 7,850 square centimeters.
Formula for calculating area It is used not only among teachers at school and researchers in the laboratory. This formula may be useful for the average painter. After all, if the ball is large and there is not enough paint, then the question arises: will this mixture be enough to paint the entire object? And this is far from the only everyday case where the formula can be useful.
Formula for calculating volume It may also be useful for the construction team making repairs. And it doesn’t matter what kind of object it is - an industrial building, a small house or an ordinary apartment. This is what distinguishes professionals - they know how to apply their knowledge in practice.
But what to do if it is not possible to measure the object? This question may arise in the case of the enormous size of the object or its inaccessibility. In this case, electronic technologies can help, the operation of which is based on scanning space with certain frequencies and lasers. With modern technology, it is not necessary to know all the formulas by heart. It is enough to have an Internet connection and go to any online calculator.
It is generally accepted that the first person to find and derive the formula for the volume and area of a sphere , was Archimedes. This is the greatest ancient Greek scientist who lived 300 years BC. He was not only a mathematician, but also a physicist and engineer. He is one of the first people who tried to “digitize” the world around us. His theorems and works are still used today.
It was Archimedes who determined the boundaries of the number "pi" and identified them without having any modern gadgets. Archimedes himself was very proud of the formula he found, with the help of which the volume of a sphere is calculated. In honor of this, his descendants depicted a cylinder and a ball on his gravestone.
If by some miracle he were reborn in our time, he would immediately be able to transform this world and take it to a new level.
Video
Using this video as an example, it will be easy for you to understand how to find the surface area of a ball.
If the length of the radius (r) is known, then square surfaces spheres(S) will be the quadruple product of the squared radius and the number Pi (π): S=4∗π∗r². For example, with a radius length spheres three meters away square will be 4∗3.14∗3²=113.04 square meters.
If you know (V) of the space bounded by the sphere, then you can first find its diameter (d), and then use the formula given in the first step. Since the volume of one sixth of Pi per cubed length of diameter spheres(V=π∗d³/6), then the diameter can be calculated as the cube root of six volumes divided by Pi: d=³√(6∗V/π). Substituting this value into the formula from the first step, we get: S=π∗(³√ (6∗V/π))². For example, with a space limited by a sphere equal to 500 cubic meters, the calculation of its area will look like this: 3.14∗(³√(6∗500/3.14))² = 3.14∗(³√955.41)² = 3, 14∗9.85² = 3.14∗97.02 = 304.64 square meters.
It is quite difficult to make all these calculations in your head, so you will have to use one of the calculators. For example, this could be a calculator built into the Google or Nigma search engines. Google differs for the better in that it can independently determine the order of operations, while Nigma will require you to carefully all the parentheses. To calculate area spheres According to the data, for example, from the second step, the search query that must be entered into Google will look like this: “4*pi*3^2”. And for the most complex case with calculating the cube root and squaring from the third step, the request will be: “pi*(6*500/pi)^(2/3)”.
All planets in the solar system have the shape ball. In addition, many objects created by man, including parts of technical devices, have a spherical or close to such shape. A ball, like any body of rotation, has an axis that coincides with its diameter. However, this is not the only important property ball. Below we discuss the main properties of this geometric figure and the method for finding its area.
Instructions
If you take a circle and rotate it around its axis, you get a body called a ball. In other words, a ball is a body bounded by a sphere. The sphere is a shell ball, and its circumference. From ball it differs in that it is hollow. Axle like ball, so for a sphere it coincides with the diameter and passes through the center. Radius ball called a segment drawn from its center to any external point. In contrast to the sphere, section ball are circles. Most celestial bodies have a shape close to spherical. At different points ball there are identical in shape, but unequal in size, so-called sections - circles of different areas.
A ball and a sphere are interchangeable bodies, unlike a cone, despite the fact that it is also a body of revolution. Spherical surfaces always form a circle in their cross-section, regardless of whether it is horizontal or vertical. A conical surface is obtained only by rotating the triangle along its axis perpendicular to the base. Therefore, a cone, unlike ball, and is not considered an interchangeable body of revolution.
The largest possible circle is obtained by cutting ball passing through the center O. All circles that pass through the center O intersect each other in the same diameter. The radius is always equal to half the diameter. Through two points A and B, located anywhere on the surface ball, can go through an infinite number of circles or circles. It is for this reason that an unlimited number of meridians can be drawn through the Earth.
When finding the area ball is considered, first of all, square spherical surface.Area ball, or rather, the sphere forming its surface, can be calculated on a base with the same radius R. Since square circle is the product of the semicircle and the radius, it can be calculated as follows:S = ?R^2 Since through the center ball pass four main large circles, then, accordingly square ball(sphere) is equal to:S = 4 ?R^2
This can be useful if either the diameter or radius is known ball or spheres. However, these parameters are not given as conditions in all geometric problems. There are also problems in which a ball is inscribed in a cylinder. In this case, you should use Archimedes’ theorem, the essence of which is that square surfaces ball one and a half times less than the total surface of the cylinder: S = 2/3 S cylinder, where S cylinder. - square full surface of the cylinder.
Video on the topic
Knowing only the length diameter circles, you can calculate not only square circle, but also the area of some other geometric figures. This follows from the fact that the diameters of circles inscribed or circumscribed around such figures coincide with the lengths of their sides or diagonals.
Instructions
If you need to find square(S) according to its known length diameter(D), multiply pi (π) by its length diameter, and divide the result by four: S=π ²*D²/4. For example,
A ball is a set of all points in space extending from the center point at a distance of a certain radius R. The radius, in turn, is the segment connecting the center ball with every point on its surface.
You will need
- – formula for the surface area of a ball;
- – formula for the volume of a ball;
- – arithmetic skills.
Instructions
1. In everyday life there is often a need to calculate square spherical surface or part of it in order to calculate, say, material consumption. Having calculated the volume ball, you can use specific gravity to calculate the mass of the substance that makes up the contents of the sphere. In order to discover square and volume ball, it is enough to know its radius or diameter. Using the formulas that today's schoolchildren derive in the 11th grade of a secondary school, you can easily calculate these parameters.
2. Let's say, the diameter of a soccer ball, according to each FIFA requirement, should be in the range of 21.8-22.2 cm. Average for ease of calculation to 22 cm. Consequently, the radius (R) will be equal to (22:2) - 11 cm. Tea interesting to know what square surface of a soccer ball?
3. Take the surface area formula ball:S ball= 4tmR2 Substitute the radius of the soccer ball into the above formula - 11 cm. S = 4 x 3.14 x 11x11.
4. After carrying out simple mathematical operations, you get the result: 1519.76. Thus, square The surface area of a soccer ball is 1,519.76 square centimeters.
5. Now calculate the volume of the ball. Take the formula for calculating volume ball: V = 4/3tmR3 Substitute again the value of the radius of the soccer ball - 11 cm. V = 4/3 x 3.14 x 11 x 11 x 11.
6. After calculations, say, on a calculator you get: 5576.89. It turns out that the volume of air in a soccer ball is 5,576.89 cubic centimeters.
A ball is the simplest three-dimensional geometric figure, to indicate the size of which each one parameter is sufficient. The boundaries of this figure are usually called a sphere. The volume of space limited by the sphere can be calculated both with the support of appropriate trigonometric formulas and with available means.
Instructions
1. Use the classic formula for the volume (V) of a sphere, if its radius (r) is known from the conditions - raise the radius to the third power, multiply by the number Pi, and increase the total by another third. This formula can be written as follows: V=4*?*r?/3.
2. If it is possible to measure the diameter (d) of the sphere, then divide it in half and use it as the radius in the formula from the previous step. Or find one-sixth of the cubed diameter multiplied by Pi: V=?*d?/6.
3. If we know the volume (v) of the cylinder, in which the sphere is inscribed, then to find its volume, determine what two-thirds of the known volume of the cylinder is equal to: V=?*v.
4. If you know the average density (p) of the material that makes up the sphere and its mass (m), then this is also enough to determine the volume - divide the second by the first: V=m/p.
5. Use some measuring containers as a handy means to measure the volume of a spherical vessel. Let's say, fill it with water, using a measuring container to measure the amount of liquid being poured. Convert the resulting value in liters to cubic meters - this unit is adopted in the international SI system for measuring volume. As an indicator for converting from liters to cubic meters, use the number 1000, because one liter is equal to one cubic decimeter, and each cubic meter contains exactly a thousand of them.
6. Use the opposite measurement rule to that described in the previous step if a spherical body cannot be filled with liquid, but can be immersed in it. Fill the measuring vessel with water, sweep the tier, immerse the spherical body being measured in the liquid and, based on the difference in tiers, determine the amount of water displaced. After this, convert the resulting total from liters to cubic meters in the same way as described in the previous step.
Video on the topic
Repairs, moving, painting an object - all this will require calculating the area. It is not a crime to remember the school curriculum.
Instructions
1. Let us remember what area is. Area is the measure of a plane figure in relation to a standard figure. Or a correct value, the numerical value of which has the following properties: If a figure can be divided into parts that will be primitive figures, then the area of such a figure will be equal to the sum of the areas of its parts The area of a square with a side that is equal to the unit of measurement is equal to one Equal figures have equal areas From these rules it follows that the area is not a certain quantity, that is, the area gives only a conditional collation to some figure. When you need to find the area of an arbitrary figure, you need to calculate how many squares with a side (which is equal to one) this figure can accommodate.
2. Example: Let's take a figure - a rectangle, one in which a square centimeter fits six times. Then the area of such a rectangle will be equal to 6 cm2. If we take a more difficult figure, say, a trapezoid, then it turns out that: If the trapezoid is such a size that a square centimeter fits into it only twice, and the third part does not fit entirely and a small triangle remains. In order to measure the area of this remaining triangle, you need to apply fractions of a square centimeter to it; you can take a millimeter. True, this method is not very comfortable for difficult figures. Consequently, there are different formulas for calculating the area of various figures. If you need to calculate the area of a certain figure, then you can take a geometry textbook and remember the material that you once studied at school. So, the formula for the area of a cube: the area of the cube is equal to the number of faces multiplied by the area of the face, i.e. 6*a2
Video on the topic
All planets of the clear system have the shape ball. In addition, many objects made by man, including parts of technical devices, are spherical or close to such a shape. A ball, like any body of revolution, has an axis that coincides with its diameter. However, this is not an exceptional main quality ball. Below we discuss the main properties of this geometric figure and the method for finding its area.
Instructions
1. If you take a semicircle or circle and rotate it around its axis, you get a body called a ball. In other words, a ball is a body bounded by a sphere. The sphere is a shell ball, and its cross section is a circle. From ball it differs in that it is hollow. Axle like ball, so for a sphere it coincides with the diameter and passes through the center. Radius ball called a segment drawn from its center to any external point. In contrast to the sphere, section ball are circles. Many planets and celestial bodies have a shape close to spherical. At various points ball there are identical in shape, but unequal in size, so-called sections - circles of different areas.
2. A ball and a sphere are interchangeable bodies, unlike a cone, despite the fact that the cone is also a body of revolution. Spherical surfaces invariably form a circle in their cross-section, regardless of how exactly it rotates - horizontally or vertically. A conical surface is obtained only by rotating the triangle along its axis perpendicular to the base. Consequently, the cone, unlike ball, and is not considered an interchangeable body of revolution.
3. The largest possible circle is obtained by cutting ball plane passing through the center O. All circles that pass through the center O intersect each other in the same diameter. The radius is always equal to half the diameter. Through two points A and B, located anywhere on the surface ball, can go through an unlimited number of circles or circles. It is for this reason that an unlimited number of meridians can be drawn through the Earth's poles.
4. When finding the area ball considered before anyone else square spherical surface.Area ball, or rather, the sphere forming its surface, can be calculated based on the area of a circle with the same radius R. From the fact that square of a circle is the product of a semicircle and a radius, it can be calculated in the following way: S = ?R^2 Since through the center ball pass four main huge circles, then, accordingly square ball(sphere) is equal to:S = 4 ?R^2
5. This formula can be suitable if we know either the diameter or the radius ball or spheres. However, these parameters are not given as conditions in all geometric problems. There are also problems in which a ball is inscribed in a cylinder. In this case, you should use Archimedes’ theorem, the essence of which is that square surfaces ball one and a half times less than the total surface of the cylinder: S = 2/3 S cylinder, where S cylinder. – square full surface of the cylinder.
Video on the topic
A ball is the simplest three-dimensional figure of a geometrically positive shape, all points of space within the boundaries of which are removed from its center at a distance not exceeding the radius. The surface formed by the majority of points furthest from the center is called a sphere. To quantitatively express the measure of space contained within a sphere, a parameter is provided, which is called the volume of the ball.
Instructions
1. If you want to measure the volume of a ball not theoretically, but only with improvised means, then this can be done, say, by determining the volume of water displaced by it. This method is applicable in the case where there is a possibility of placing the ball in some container commensurate with it - a beaker, glass, jar, bucket, barrel, pool, etc. In this case, before placing the ball, sweep the layer of water, do this again after it is completely immersed, and then find the difference between the marks. Traditionally, factory-produced measuring containers have divisions showing the volume in liters and units derived from it - milliliters, decalitres, etc. If the obtained value needs to be converted into cubic meters and multiple units of volume, then proceed from the fact that one liter corresponds to one cubic decimeter or one thousandth of a cubic meter.
2. If the material from which the ball is made is known, and the density of this material can be found out, say, from a reference book, then the volume can be determined by weighing the given object. Simply divide the weighing result by the reference density of the manufacturing substance: V=m/p.
3. If the radius of the ball is determined from the conditions of the problem or it can be measured, then the corresponding mathematical formula can be used to calculate the volume. Multiply the quadruple number Pi by the third power of the radius, and divide the resulting total by three: V=4*?*r?/3. Let's say, with a radius of 40 cm, the volume of the ball will be 4 * 3.14 * 40?/3 = 267946.67 cm? ? 0.268m?.
4. Measuring the diameter is often easier than measuring the radius. In this case, there is no need to divide it in half to use with the formula from the previous step - it is better to simplify the formula itself. In accordance with the converted formula, multiply the number Pi by the diameter to the third power, and divide the total by six: V=?*d?/6. Let's say, a ball with a diameter of 50 cm should have a volume of 3.14 * 50?/6 = 65416.67 cm? ? 0.654m?.
Problems involving calculating the area of a circle are often found in school geometry courses. In order to discover square circle, you need to know the length diameter or the radius of the circle in which it is enclosed.
You will need
- – length of the diameter of the circle.
Instructions
1. A circle is a figure on a plane consisting of many points located at the same distance from another point, called the center. A circle is a flat geometric figure that consists of a lot of points enclosed in a circle, which is the boundary of the circle. A diameter is a line segment connecting two points on a circle and passing through its center. A radius is a segment connecting a point on a circle and its center. ? - number “pi”, mathematical constant, continuous value. It shows the ratio of the circumference of a circle to its length diameter. Calculate the exact value of a number? impossible. In geometry, the approximate value of this number is used: ? ? 3.14
2. The area of a circle is equal to the product of the square of the radius and the number and is calculated by the formula: S=?R^2, where S - square circle, R is the length of the radius of the circle.
3. From the definition of radius it follows that it is equal to half diameter. Consequently, the formula takes the form: S=?(D/2)^2, where D is the length diameter circles. Substitute the value into the formula diameter, calculate square circle.
4. The area of a circle is measured in units of area - mm2, cm2, m2, etc. In what units is the information you receive expressed? square circle depends on the units in which the diameter of the circle was given.
5. If you need to calculate square ring, use the formula: S=?(R-r)^2, where R, r are the radii of the outer and inner circles of the ring, respectively.
Helpful advice
There is International Pi Day, which is celebrated on March 14th. The exact time of the triumphal date is 1 hour 59 minutes 26 seconds, according to the numbers of the date - 3.1415926...
Video on the topic
Note!
Interesting: the volume of a ball with a diameter three times greater than the diameter of another ball is 9 times larger than the total volume of 3 such balls.
Helpful advice
In order to develop children's passion for mathematical calculations, offer surrounding objects as examples for calculation: a ball, a watermelon, a ball of grandma's yarn. It is visual and therefore fascinating.