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 revolution. 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, then the study geometric relationships has become widespread within the sphere.
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.”
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 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
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); most often the definition sounds like “the geometric location of all points on the plane, the distance from which to a 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 located no further than a given radius (do not go beyond the circle shown in Fig. .2).
If the radius is 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 a “spherical surface” 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 as part of their introduction to analysis. For a simple one, even household use Those formulas that are studied in the stereometry course for grades 10-11 will be sufficient. It is these concepts that are accessible to almost every average educated person that 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 be confused that to calculate the area of a large circle, the radius or diameter of the ball (sphere) itself is used, 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 of the earth's axis, and in a geometric sense - the ends of the diameter, and the meridians that pass through these two points (Figure 7). That is, the number of large circles on 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 - a 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 of the 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 a better understanding, an analogy can be drawn with the globe, namely with its 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. To calculate various parameters of a given object, not much data is needed.
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 is as follows: 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.
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, it is enough to rotate this flat figure around an axis that coincides with its diameter.
One of the famous ancient literary sources, which discusses the properties of this three-dimensional figure in sufficient detail, 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, but to take advantage of this, you must use the appropriate number of decimal places for pi. The second expression gives completely good result, differing from the first by only 0.03%. For a number of practical tasks, 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 surface tension forces, which tend to minimize the surface area. The ball, in turn, has the lowest value among all geometric figures 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.
Instructions
note
^ - sign indicating exponentiation;
^1/2 is essentially taking the square root;
^1/3 - cube root extraction.
Sources:
- diameter is
A circle is a geometric figure on a plane that consists of all points of this plane that are at the same distance from a given point. The given point is called the center circle, and the distance at which the points circle are from its center - radius circle. The area of the plane bounded by a circle is called a circle. There are several calculation methods diameter circle, the choice of a specific one depends on the available initial data.
Instructions
Video on the topic
When constructing various geometric shapes, it is sometimes necessary to determine their characteristics: length, width, height, and so on. If we're talking about about a circle or circle, you often have to determine its diameter. A diameter is a straight line segment that connects the two points furthest from each other located on a circle.
You will need
- - yardstick;
- - compass;
- - calculator.
Instructions
In the simplest case, determine the diameter using the formula D = 2R, where R is the radius of the circle with the center at point O. This is convenient if you are drawing a circle with a predetermined . For example, if, when constructing a figure, you set the opening of the compass legs to 50 mm, then the diameter of the resulting circle will be equal to twice the radius, that is, 100 mm.
If you know the circumference that makes up the outer boundary of the circle, then use the formula to determine the diameter:
D = L/p, where
L – circumference;
p is the number “pi”, equal to approximately 3.14.
For example, if the length is 180 mm, then the diameter will be approximately: D = 180 / 3.14 = 57.3 mm.
If you have a pre-drawn circle with radius, diameter and circumference, then use a measuring ruler to estimate the diameter. The difficulty is to find
