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 sphere and the sky is celestial sphere, a new exciting direction in geometry has 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 having general 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. Educated figure- there 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 a sphere as with 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 revolution. However, since both the Earth and many bodies have a spherical shape, for example a drop of water, the study of geometric relationships within 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 will be called a chord, and if this the chord will pass 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.”
where V is the required volume of the ball, π – 3.14, R – radius.
Thus, with a radius of 10 centimeters volume of the ball equal to:
| V | 3.14 × 10 3 | = 4186,7 | cubic centimeters. |
In geometry ball is defined as a certain body, which is a collection of all points in space that are located from the center at a distance no more than a given one, called the radius of the ball. The surface of the ball is called a sphere, and the ball itself is formed by rotating a semicircle around its diameter, remaining motionless.
This geometric body is often encountered by design engineers and architects, who often have to calculate the volume of a sphere. For example, in the design of the front suspension of the vast majority of modern cars, so-called ball joints are used, in which, as you can easily guess from the name itself, balls are one of the main elements. With their help, the hubs of the steered wheels and levers are connected. On how correct it will be calculated their volume largely depends not only on the durability of these units and the correctness of their operation, but also on traffic safety.
In technology widest distribution received such parts as ball bearings, with the help of which the axes are fastened in the fixed parts of various components and assemblies and their rotation is ensured. It should be noted that when calculating them, designers need find the volume of the sphere(or rather, balls placed in a cage) with high degree accuracy. As for the manufacture of metal bearing balls, they are produced from metal wire using a complex process that includes the stages of forming, hardening, rough grinding, finishing and cleaning. By the way, those balls that are included in the design of all ballpoint pens, are manufactured using exactly the same technology.
Quite often, balls are used in architecture, where they are most often decorative elements of buildings and other structures. In most cases they are made of granite, which often requires high costs manual labor. Of course, comply with this high accuracy the manufacture of these balls, like those used in various units and mechanisms, is not required.
Such an interesting and popular game as billiards is unthinkable without balls. For their production they are used various materials(bone, stone, metal, plastics) and various technological processes are used. One of the main requirements for billiard balls is their high strength and ability to withstand high mechanical loads (primarily shock). In addition, their surface must be an exact sphere in order to ensure smooth and even rolling on the surface of the pool tables.
Finally, without such geometric bodies, like balls, not a single New Year or Christmas tree is complete. These decorations are made in most cases from glass using the blowing method, and during their production greatest attention The focus is not on dimensional accuracy, but on the aesthetics of the products. Technological process At the same time, the Christmas balls are almost completely automated and the Christmas balls are only packed manually.
The radius of a ball (denoted as r or R) is the segment that connects the center of the ball with any point on its surface. As with a circle, the radius of a ball is an important quantity needed to find the ball's diameter, circumference, surface area, and/or volume. But the radius of the ball can also be found by given value diameter, circumference and other quantities. Use a formula into which you can substitute these values.
Steps
Formulas for calculating radius
- For example, given a ball with a diameter of 16 cm. The radius of this ball: r = 16/2 = 8 cm. If the diameter is 42 cm, then the radius is 21 cm (42/2=21).
-
Calculate the radius from the circumference. Use the formula: r = C/2π. Since the circumference of a circle is C = πD = 2πr, then divide the formula for calculating the circumference by 2π and get the formula for finding the radius.
- For example, given a ball with a circumference of 20 cm. The radius of this ball is: r = 20/2π = 3.183 cm.
- The same formula is used to calculate the radius and circumference of a circle.
-
Calculate the radius from the volume of the sphere. Use the formula: r = ((V/π)(3/4)) 1/3. The volume of the ball is calculated by the formula V = (4/3)πr 3. Isolating r on one side of the equation, you get the formula ((V/π)(3/4)) 3 = r, that is, to calculate the radius, divide the volume of the ball by π, multiply the result by 3/4, and raise the resulting result to a power 1/3 (or take the cube root).
- For example, given a ball with a volume of 100 cm 3 . The radius of this ball is calculated as follows:
- ((V/π)(3/4)) 1/3 = r
- ((100/π)(3/4)) 1/3 = r
- ((31.83)(3/4)) 1/3 = r
- (23.87) 1/3 = r
- 2.88 cm= r
- For example, given a ball with a volume of 100 cm 3 . The radius of this ball is calculated as follows:
-
Calculate the radius from the surface area. Use the formula: g = √(A/(4 π)). The surface area of the ball is calculated by the formula A = 4πr 2. Isolating r on one side of the equation, you get the formula √(A/(4π)) = r, that is, to calculate the radius, you need to extract Square root from the surface area divided by 4π. Instead of taking the root, the expression (A/(4π)) can be raised to the power of 1/2.
- For example, given a sphere with a surface area of 1200 cm 3 . The radius of this ball is calculated as follows:
- √(A/(4π)) = r
- √(1200/(4π)) = r
- √(300/(π)) = r
- √(95.49) = r
- 9.77 cm= r
Determination of basic quantities
-
Remember the basic quantities that are relevant to calculating the radius of a ball. The radius of a ball is the segment that connects the center of the ball to any point on its surface. The radius of a ball can be calculated from given values of diameter, circumference, volume, or surface area.
Use the values of these quantities to find the radius. Radius can be calculated from given values of diameter, circumference, volume, and surface area. Moreover, the indicated values can be found from a given radius value. To calculate the radius, simply convert the formulas to find the values shown. Below are the formulas (which include radius) for calculating diameter, circumference, volume, and surface area.
Finding the radius from the distance between two points
-
Find the coordinates (x,y,z) of the center of the ball. Ball radius equal to the distance between its center and any point lying on the surface of the ball. If the coordinates of the center of the ball and any point lying on its surface are known, you can find the radius of the ball using a special formula by calculating the distance between two points. First find the coordinates of the center of the ball. Keep in mind that since the ball is three-dimensional figure, then the point will have three coordinates (x,y,z), and not two (x,y).
- Let's look at an example. Given a ball with center coordinates (4,-1,12) . Use these coordinates to find the radius of the ball.
-
Find the coordinates of a point lying on the surface of the ball. Now we need to find the coordinates (x,y,z) any point lying on the surface of the ball. Since all points lying on the surface of the ball are located at the same distance from the center of the ball, you can choose any point to calculate the radius of the ball.
- In our example, let us assume that some point lying on the surface of the ball has coordinates (3,3,0) . By calculating the distance between this point and the center of the ball, you will find the radius.
-
Calculate the radius using the formula d = √((x 2 - x 1) 2 + (y 2 - y 1) 2 + (z 2 - z 1) 2). Having found out the coordinates of the center of the ball and a point lying on its surface, you can find the distance between them, which is equal to the radius of the ball. The distance between two points is calculated by the formula d = √((x 2 - x 1) 2 + (y 2 - y 1) 2 + (z 2 - z 1) 2), where d is the distance between the points, (x 1, y 1 ,z 1) – coordinates of the center of the ball, (x 2 , y 2 , z 2) – coordinates of a point lying on the surface of the ball.
- In the example under consideration, instead of (x 1 ,y 1 ,z 1) substitute (4,-1,12), and instead of (x 2 ,y 2 ,z 2) substitute (3,3,0):
- d = √((x 2 - x 1) 2 + (y 2 - y 1) 2 + (z 2 - z 1) 2)
- d = √((3 - 4) 2 + (3 - -1) 2 + (0 - 12) 2)
- d = √((-1) 2 + (4) 2 + (-12) 2)
- d = √(1 + 16 + 144)
- d = √(161)
- d = 12.69. This is the desired radius of the ball.
- In the example under consideration, instead of (x 1 ,y 1 ,z 1) substitute (4,-1,12), and instead of (x 2 ,y 2 ,z 2) substitute (3,3,0):
-
Keep in mind that in general cases r = √((x 2 - x 1) 2 + (y 2 - y 1) 2 + (z 2 - z 1) 2). All points lying on the surface of the ball are located at the same distance from the center of the ball. If in the formula for finding the distance between two points “d” is replaced by “r”, you get a formula for calculating the radius of the ball from the known coordinates (x 1,y 1,z 1) of the ball’s center and the coordinates (x 2,y 2,z 2 ) any point lying on the surface of the ball.
- Square both sides of this equation and you get r 2 = (x 2 - x 1) 2 + (y 2 - y 1) 2 + (z 2 - z 1) 2. Note that this equation corresponds to the equation of a sphere r 2 = x 2 + y 2 + z 2 with its center at coordinates (0,0,0).
- Don't forget about the order of execution mathematical operations. If you don't remember this order, and your calculator can work with parentheses, use them.
- This article talks about calculating the radius of a ball. But if you're having trouble learning geometry, it's best to start by calculating the quantities associated with a ball using known value radius.
- π (Pi) is a letter of the Greek alphabet that denotes a constant, equal to the ratio diameter of a circle to its circumference. Pi is irrational number, which is not written as a relation real numbers. There are many approximations, for example, the ratio 333/106 will allow you to find the number Pi with an accuracy of four digits after decimal point. As a rule, they use approximate value Pi, which is 3.14.
- For example, given a sphere with a surface area of 1200 cm 3 . The radius of this ball is calculated as follows:
Calculate the radius from the diameter. Radius equal to half diameter, so use the formula g = D/2. This is the same formula that is used to calculate the radius and diameter of a circle.
Many bodies that we meet in life or that we have heard about are spherical in shape, such as a soccer ball, a falling drop of water during rain, or our planet. In this regard, it is relevant to consider the question of how to find the volume of a sphere.
Ball figure in geometry
Before answering the question about the ball, let’s take a closer look at this body. Some people confuse it with a sphere. Outwardly, they are really similar, but a ball is an object filled inside, while a sphere is only the outer shell of a ball of infinitesimal thickness.
From the point of view of geometry, a ball can be represented by a collection of points, and those of them that lie on its surface (they form a sphere) are at the same distance from the center of the figure. This distance is called the radius. In fact, radius is the only parameter that can be used to describe any properties of a ball, such as its surface area or volume.
The picture below shows an example of a ball.
If you look closely at this perfect round object, you can guess how to get it from an ordinary circle. To do this, just rotate this flat figure around an axis coinciding with its diameter.
One of the famous ancient literary sources, which discusses in sufficient detail the properties of this volumetric figure, is the work of the Greek philosopher Euclid - “Elements”.
Surface area and volume
When considering the question of how to find the volume of a ball, in addition to this value, a formula for its area should be given, since both expressions can be related to each other, as will be shown below.
So, to calculate the volume of a ball, you should apply one of the following two formulas:
- V = 4/3 *pi * R3;
- V = 67/16 * R3.
Here R is the radius of the figure. The first formula given is accurate, however, to take advantage of this, you must use corresponding number decimal places for pi. The second expression gives completely good result, differing from the first by only 0.03%. For a row practical problems this accuracy is more than enough.
Equal to this value for a sphere, that is, expressed by the formula S = 4 * pi * R2. If we express the radius from here and then substitute it into the first formula for volume, then we get: R = √ (S / (4 * pi)) = > V = S / 3 * √ (S / (4 * pi)).
Thus, we examined the questions of how to find the volume of a ball through the radius and through its surface area. These expressions can be successfully applied in practice. Later in the article we will give an example of their use.
Raindrop problem
Water, when in weightlessness, takes the form of a spherical drop. This is due to the presence of strength surface tension, which tend to minimize the surface area. The ball, in turn, has the lowest value among all geometric shapes with the same mass.
During rain, a falling drop of water is in weightlessness, so its shape is a sphere (here we neglect the force of air resistance). It is necessary to determine the volume, surface area and radius of this drop if it is known that its mass is 0.05 grams.
The volume is easy to determine; to do this, divide the known mass by the density of H 2 O (ρ = 1 g/cm 3). Then V = 0.05 / 1 = 0.05 cm 3.
Knowing how to find the volume of a ball, we should express the radius from the formula and substitute the resulting value, we have: R = ∛ (3 * V / (4 * pi)) = ∛ (3 * 0.05 / (4 * 3.1416)) = 0.2285 cm.
Now we substitute the radius value into the expression for the surface area of the figure, we get: S = 4 * 3.1416 * 0.22852 = 0.6561 cm 2.
Thus, knowing how to find the volume of a ball, we received answers to all the questions of the problem: R = 2.285 mm, S = 0.6561 cm 2 and V = 0.05 cm 3.
Before you begin to study the concept of a ball, what the volume of a ball is, and consider formulas for calculating its parameters, you need to remember the concept of a circle, studied earlier in the geometry course. After all, most actions in three-dimensional space are similar to or follow from two-dimensional geometry, adjusted for the appearance of the third coordinate and third degree.
What is a circle?
A circle is a figure on a Cartesian plane (shown in Figure 1); the most common definition is “ locus all points on the plane, the distance from which to given point(center) does not exceed a certain non-negative number, called the radius."
As we can see from the figure, point O is the center of the figure, and the set of absolutely all points that fill the circle, for example, A, B, C, K, E, are no further given radius(do not go beyond the circle shown in Fig. 2).
If radius equal to zero, then the circle turns into a point.
Problems with understanding
Students often confuse these concepts. It's easy to remember with an analogy. The hoop that children spin in class physical culture, - circle. By understanding this or remembering that the first letters of both words are “O,” children will mnemonically understand the difference.
Introduction of the concept of "ball"
A ball is a body (Fig. 3) bounded by a certain spherical surface. What kind of “spherical surface” it is will become clear from its definition: this is the geometric locus of all points on the surface, the distance from which to a given point (center) does not exceed a certain non-negative number called the radius. As you can see, the concepts of a circle and a spherical surface are similar, only the spaces in which they are located differ. If we depict a ball in two-dimensional space, we get a circle whose boundary is a circle (the boundary of a ball is a spherical surface). In the figure we see a spherical surface with radii OA = OB.
Ball closed and open
In vector and metric spaces, two concepts related to the spherical surface are also considered. If the ball includes this sphere, then it is called closed, but if not, then the ball is open. These are more “advanced” concepts; they are studied in institutes when introducing analysis. For a simple one, even household use Those formulas that are studied in the stereometry course for grades 10-11 will be sufficient. These are the ones that are accessible to almost every average person. educated person concepts will be discussed further.
Concepts you need to know for the following calculations
Radius and diameter.
The radius of a ball and its diameter are determined in the same way as for a circle.
Radius is a segment connecting any point on the boundary of the ball and the point that is the center of the ball.
Diameter is a segment connecting two points on the boundary of a ball and passing through its center. Figure 5a clearly demonstrates which segments are the radii of the ball, and Figure 5b shows the diameters of the sphere (segments passing through point O).
Sections in a sphere (ball)
Any section of a sphere is a circle. If it passes through the center of the ball, it is called a large circle (circle with diameter AB), the remaining sections are called small circles (circle with diameter DC).
The area of these circles is calculated using the following formulas:
Here S is the designation for area, R for radius, D for diameter. There is also a constant equal to 3.14. But do not confuse that to calculate area great circle they use the radius or diameter of the ball (sphere) itself, and to determine the area, the dimensions of the radius of the small circle are required.
An infinite number of such sections that pass through two points of the same diameter lying on the boundary of the ball can be drawn. As an example, our planet: two points on the North and South Poles, which are the ends earth's axis, and in geometric sense- the ends of the diameter, and the meridians that pass through these two points (Figure 7). That is, the number large circles the number of a sphere tends to infinity.
Ball parts
If you cut off a “piece” from the sphere using a certain plane (Figure 8), then it will be called a spherical or spherical segment. It will have a height - perpendicular from the center of the cutting plane to the spherical surface O 1 K. Point K on the spherical surface at which the height comes is called the vertex spherical segment. A small circle with radius O 1 T (in in this case, according to the figure, the plane did not pass through the center of the sphere, but if the section passes through the center, then the circle of section will be large), formed when cutting off the spherical segment, will be called the base of our piece of the ball - a spherical segment.
If we connect each base point of a spherical segment to the center of the sphere, we get a figure called a “spherical sector”.
If two planes pass through a sphere and are parallel to each other, then that part of the sphere that is enclosed between them is called a spherical layer (Figure 9, which shows a sphere with two planes and a separate spherical layer).
The surface (highlighted part in Figure 9 on the right) of this part of the sphere is called a belt (again, for better understanding, an analogy can be drawn with the globe, namely with his climatic zones- arctic, tropical, temperate, etc.), and the cross-sectional circles will be the bases of the spherical layer. The height of the layer is part of the diameter drawn perpendicular to the cutting planes from the centers of the bases. There is also the concept of a spherical sphere. It is formed when planes that are parallel to each other do not intersect the sphere, but touch it at one point each.
Formulas for calculating the volume of a ball and its surface area
The ball is formed by rotating around the fixed diameter of a semicircle or circle. For calculations different parameters this object will not need much data.
The volume of a sphere, the formula for calculating which is given above, is derived through integration. Let's figure it out point by point.
We consider a circle in a two-dimensional plane, because, as mentioned above, it is the circle that underlies the construction of the ball. We use only its fourth part (Figure 10).
We take a circle with unit radius and center at the origin. The equation of such a circle looks like in the following way: X 2 + Y 2 = R 2. We express Y from here: Y 2 = R 2 - X 2.
Be sure to note that the resulting function is non-negative, continuous and decreasing on the segment X (0; R), because the value of X in the case when we consider a quarter of a circle lies from zero to the value of the radius, that is, to unity.
The next thing we do is rotate our quarter circle around the x-axis. As a result, we get a hemisphere. To determine its volume, we will resort to integration methods.
Since this is the volume of only a hemisphere, we double the result, from which we find that the volume of the ball is equal to:
Small nuances
If you need to calculate the volume of a ball through its diameter, remember that the radius is half the diameter, and substitute this value into the above formula.
You can also reach the formula for the volume of a ball through the area of its bordering surface - the sphere. Let us recall that the area of a sphere is calculated by the formula S = 4πr 2, integrating which we also arrive at the above formula for the volume of a sphere. From the same formulas you can express the radius if the problem statement contains a volume value.