How to calculate the volume of a sphere. How to find the volume of a ball: basic formulas and an example of their use

Definition of a ball

Ball is a body all points of which are located from a given point at a distance not exceeding R.

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The given point referred to in the definition of a ball is called center this ball. And the mentioned distance is radius of this ball.

A ball, by analogy with a circle, also has a diameter D D D, which is twice the radius in length:

D = 2 ⋅ R D=2\cdot R D=2 ⋅ R

Formula for the volume of a ball in terms of its radius

The volume of the ball is calculated using the following formula:

Formula for the volume of a ball in terms of radius

V = 4 3 ⋅ π ⋅ R 3 V=\frac(4)(3)\cdot\pi\cdot R^3V=3 4 ​ ⋅ π ⋅ R 3

R R R- the radius of this ball.

Let's look at a few examples.

Problem 1

A ball is inscribed in a cube, diagonal d d d which is equal to 500 cm. \sqrt(500)\text( cm.)5 0 0 ​ cm . Find the volume of the ball.

Solution

D = 500 d=\sqrt(500) d =5 0 0 ​

First you need to determine the length of the side of the cube. We will assume that it is equal a a a. Therefore, the diagonal of the cube is equal (based on the Pythagorean theorem):

D = a 2 + a 2 + a 2 d=\sqrt(a^2+a^2+a^2)d =a 2 + a 2 + a 2 ​

D = 3 ⋅ a 2 d=\sqrt(3\cdot a^2)d =3 ⋅ a 2 ​

D = 3 ⋅ a d=\sqrt(3)\cdot ad =3 ​ ⋅ a

500 = 3 ⋅ a \sqrt(500)=\sqrt(3)\cdot a5 0 0 ​ = 3 ​ ⋅ a

A = 500 3 a=\sqrt(\frac(500)(3))a =3 5 0 0 ​ ​

A ≈ 12.9 a\approx12.9 a ≈1 2 . 9

If a ball is inscribed in a cube, then its radius is equal to half the length of the side of this cube. As a result we have:

R = 1 2 ⋅ a R=\frac(1)(2)\cdot aR=2 1 ​ ⋅ a

R = 1 2 ⋅ 12.9 ≈ 6.4 R=\frac(1)(2)\cdot 12.9\approx6.4R=2 1 ​ ⋅ 1 2 . 9 ≈ 6 . 4

The final stage is finding the volume of the ball using the formula:

V = 4 3 ⋅ π ⋅ R 3 ≈ 4 3 ⋅ π ⋅ (6.4) 3 ≈ 1097, 5 cm 3 V=\frac(4)(3)\cdot\pi\cdot R^3\approx\frac(4 )(3)\cdot\pi\cdot (6.4)^3\approx1097.5\text( cm)^3V=3 4 ​ ⋅ π ⋅ R 3 ≈ 3 4 ​ ⋅ π ⋅ (6 . 4 ) 3 ≈ 1 0 9 7 , 5 cm3

Answer

1097.5 cm3. 1097.5\text( cm)^3.1 0 9 7 , 5 cm3 .

Formula for the volume of a ball in terms of its diameter

The volume of a ball can also be found through its diameter. To do this, we use the relationship between the radius and diameter of the ball:

D = 2 ⋅ R D=2\cdot R D=2 ⋅ R

R = D 2 R=\frac(D)(2) R=2 D​

Let's substitute this expression into the formula for the volume of the ball:

V = 4 3 ⋅ π ⋅ R 3 = 4 3 ⋅ π ⋅ (D 2) 3 = π 6 ⋅ D 3 V=\frac(4)(3)\cdot\pi\cdot R^3=\frac(4 )(3)\cdot\pi\cdot\Big(\frac(D)(2)\Big)^3=\frac(\pi)(6)\cdot D^3V=3 4 ​ ⋅ π ⋅ R 3 = 3 4 ​ ⋅ π ⋅ ( 2 D​ ) 3 = 6 π ​ ⋅ D 3

Volume of a ball through diameter

V = π 6 ⋅ D 3 V=\frac(\pi)(6)\cdot D^3V=6 π ​ ⋅ D 3

D D D- the diameter of this ball.

Problem 2

The diameter of the ball is 15 cm. 15\text( cm.) 1 5 cm . Find its volume.

Solution

D=15 D=15 D=1 5

Immediately substitute the diameter value into the formula:

V = π 6 ⋅ D 3 = π 6 ⋅ 1 5 3 ≈ 1766.25 cm 3 V=\frac(\pi)(6)\cdot D^3=\frac(\pi)(6)\cdot 15^3\ approx1766.25\text( cm)^3V=6 π ​ ⋅ D 3 = 6 π ​ ⋅ 1 5 3 ≈ 1 7 6 6 . 2 5 cm3

Answer

$1766.25\text{cm3 .}$

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 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 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 we see, the concepts of circle and spherical surface They 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 associated with a spherical surface are also discussed. 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 simple, even everyday use, the 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 of area, R is radius, D is 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 at the North and South Poles, which are the ends of the 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 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. And a small circle with a radius of O 1 T (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 cross-section circle will be large), formed by cutting off the spherical segment, will be called the base of our piece ball - 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 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.

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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 from a given value of diameter, circumference and other quantity. Use a formula into which you can substitute these values.

Steps

Formulas for calculating radius

Calculate the radius from the diameter. The radius is equal to half the 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.

• 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).

1. 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.

2. 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

3. 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

   1. 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

   1. Find the coordinates (x,y,z) of the center of the ball. The radius of a ball is 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 a ball is a three-dimensional figure, the point will have three coordinates (x, y, z), rather than 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.

   2. 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.

   3. 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.

   4. 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 center of the ball 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 performing 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 a known radius value.
   • π (Pi) is a letter of the Greek alphabet that denotes a constant equal to the ratio of the diameter of a circle to the length of its circumference. Pi is an irrational number that is not written as a ratio of real numbers. There are many approximations, for example, the ratio 333/106 will allow you to find Pi to within four decimal places. As a rule, they use the approximate value of Pi, which is 3.14.

A ball is a geometric body of revolution formed by rotating a circle or semicircle around its diameter. Also, a ball is a space bounded by a spherical surface. There are many real spherical objects and related problems that require determining the volume of a sphere.

Ball and sphere

Circle - the oldest geometric figure, and ancient scientists attached sacred meaning to it. The circle is a symbol of endless time and space, a symbol of the Universe and existence. According to Pythagoras, the circle is the most beautiful of figures. In three-dimensional space, a circle turns into a sphere, as ideal, cosmic and beautiful as a circle.

Sphere means "ball" in ancient Greek. A sphere is a surface formed by an infinite number of points equidistant from the center of the figure. The space bounded by a sphere is a ball. A ball is an ideal geometric figure, the shape of which many real objects take. For example, in real life, cannonballs, bearings or balls have the shape of a ball, in nature - drops of water, tree crowns or berries, in space - stars, meteors or planets.

Ball volume

Determining the volume of a spherical figure is a difficult task, because such a geometric body cannot be divided into cubes or triangular prisms, the volume formulas of which are already known. Modern science allows you to calculate the volume of a ball using a definite integral, but how was the volume formula derived in Ancient Greece when no one had ever heard of integrals? Archimedes calculated the volume of a sphere using a cone and a cylinder, since the formulas for the volumes of these figures had already been determined by the ancient Greek philosopher and mathematician Democritus.

Archimedes represented half a sphere using identical cones and cylinders, with the radius of each figure being equal to its height R = h. The ancient scientist imagined the cone and cylinder divided into an infinite number of small cylinders. Archimedes realized that if he subtracts the volume of the cone Vk from the volume of the cylinder Vc, he obtains the volume of one hemisphere Vsh:

0.5 Vsh = Vc − Vk

The volume of a cone is calculated using a simple formula:

Vk = 1/3 × So × h,

but knowing that So in this case is the area of ​​the circle, and h = R, then the formula is transformed into:

Vk = 1/3 × pi × R × R 2 = 1/3 pi × R 3

The volume of the cylinder is calculated by the formula:

Vc = pi × R 2 × h,

but assuming that the height of the cylinder is equal to its radius, we get:

Vc = pi × R 3 .

Using these formulas, Archimedes obtained:

0.5 Vsh = pi × R 3 - 1/3 pi × R 3 or Vsh = 4/3 pi × R 3

The modern definition of the formula for the volume of a ball is derived from the integral of the area of ​​the spherical surface, but the result remains the same

Vsh = 4/3 pi × R 3

Calculating the volume of a ball may be needed both in real life and when solving abstract problems. To calculate the volume of a sphere using an online calculator, you will need to know only one parameter to choose from: the diameter or radius of the sphere. Let's look at a couple of examples.

Examples from life

Cannonballs

Let's say you want to know how much cast iron is needed to cast a six-foot caliber cannonball. You know that the diameter of such a core is 9.6 centimeters. Enter this number into the “Diameter” cell of the calculator and you will receive the answer as

Thus, to smelt a cannonball of a given caliber you will need 463 cubic centimeters or 0.463 liters of cast iron.

Balloons

Let you be curious about how much air is needed to pump hot air balloon ideal spherical shape. You know that the radius of the selected ball is 10 cm. Enter this value into the “Radius” calculator cell and you will get the result

This means that to inflate one such balloon you will need 4188 cubic centimeters or 4.18 liters of air.

Conclusion

The need to determine the volume of a ball may arise in the most different situations: from abstract school problems to scientific research and production issues. To solve questions of any complexity, use our online calculator, which will instantly present you with the exact result and the necessary mathematical calculations.