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How to find the volume of a sphere if the diameter is known. Ball and sphere, volume of a ball, area of a sphere, formulas

The radius of a ball (denoted as r or R) is the line segment that connects the center of the ball to any point on its surface. As with a circle, the radius of a ball is an important quantity that is 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 the diameter, circumference, and other quantities. Use a formula in which you can substitute these values.

Steps

Formulas for calculating the radius

Calculate the radius from the diameter. The radius is half the diameter, so use the formula d = D/2. This is the same formula 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).

Calculate the radius from the circumference of the circle. Use the formula: r = C/2π. Since the circumference 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 . Separating r on one side of the equation, you get the formula ((V / π) (3/4)) 3 \u003d r, that is, to calculate the radius, divide the volume of the ball by π, multiply the result by 3/4, and raise the result to the power 1/3 (or take the cube root).
- For example, given a ball with a volume of 100 cm 3. The radius of this sphere 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
Calculate the radius from the surface area. Use the formula: r = √(A/(4 π)). The surface area of the ball is calculated by the formula A \u003d 4πr 2. By isolating r on one side of the equation, you get the formula √(A/(4π)) = r, that is, to calculate the radius, you need to take the square root of 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 sphere is calculated as follows:
  - √(A/(4π)) = r
  - √(1200/(4π)) = r
  - √(300/(π)) = r
  - √(95.49) = r
  - 9.77 cm= r
Definition of basic quantities
1. Remember the basic quantities that are relevant to calculating the radius of the ball. The radius of a ball is a segment that connects the center of the ball to any point on its surface. The radius of a sphere can be calculated from given values of diameter, circumference, volume, or surface area.
  
  Use the values of these quantities to find the radius. The radius can be calculated from given values of diameter, circumference, volume, and surface area. Moreover, these values can be found from a given value of the radius. To calculate the radius, simply convert the formulas to find the given values. Below are the formulas (in which there is a radius) to calculate the 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 sphere is equal to the distance between its center and any point lying on the surface of the sphere. 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 a three-dimensional figure, the point will have three coordinates (x, y, z), and not two (x, y).
  - Consider an example. Given a ball centered with coordinates (4,-1,12) . Use these coordinates to find the radius of the ball.
2. Find the coordinates of a point on the surface of the sphere. Now you need to find the coordinates (x, y, z) any point on the surface of the sphere. Since all points lying on the surface of the ball are located at the same distance from the center of the ball, any point can be chosen to calculate the radius of the ball.
  - In our example, let's 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 \u003d √ ((x 2 - x 1) 2 + (y 2 - y 1) 2 + (z 2 - z 1) 2). Having learned the coordinates of the center of the ball and the 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 \u003d √ ((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) are the coordinates of the center of the ball, (x 2 ,y 2 ,z 2) are the coordinates of a point lying on the surface of the ball.
  - In this example, 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 \u003d √ ((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 sphere.
  - 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 centered at (0,0,0).
- Don't forget about the order in which the math operations are performed. If you do not remember this order, and your calculator knows how to 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 values associated with a ball in terms of a known radius value.
- π (Pi) is the letter of the Greek alphabet, which means 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 the number Pi with an accuracy of up to four digits after the decimal point. As a rule, they use the approximate value of pi, which is 3.14.

Many bodies that we see in life or that we have heard of 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 ball.

Figure ball in geometry

Before answering the ball's question, let's take a closer look at this body. Some people confuse it with a sphere. Outwardly, they are really similar, but the ball is an object filled inside, while the sphere is only the outer shell of a ball of infinitely small thickness.

From the point of view of geometry, a ball can be represented by a set 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, the radius is the only parameter with which you can describe any properties of a ball, such as its surface area or volume.

The figure below shows an example of a ball.

If you look closely at this ideal 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 coinciding with its diameter.

One of the well-known ancient literary sources, in which the properties of this three-dimensional figure are considered in sufficient detail, is the work of the Greek philosopher Euclid - "Elements".

Surface area and volume

Considering the question of how to find the volume of a ball, in addition to this quantity, 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, one of the following two formulas should be applied:

V = 4/3 *pi * R3;
V = 67/16 * R3.

Here R is the radius of the figure. The first of the above formulas is exact, however, to take advantage of this, you must use the appropriate number of decimal places for the number pi. The second expression gives quite a good result, differing from the first by only 0.03%. For a number of practical problems, this accuracy is more than enough.

It is equal to this value for a sphere, that is, it 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 considered the questions of how to find the volume of a ball through the radius and through the area of its surface. These expressions can be successfully applied in practice. Below in the article we will give an example of their use.

Rain drop problem

Water, when in zero gravity, 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 smallest value among all geometric shapes with the same mass.

During rain, a falling drop of water is in zero gravity, so its shape is a ball (we neglect the force of air resistance here). It is necessary to determine the volume, surface area and radius of this drop if its mass is known to be 0.05 grams.

The volume is easy to determine, for this you should divide the known mass by the density of H 2 O (ρ \u003d 1 g / cm 3). Then V \u003d 0.05 / 1 \u003d 0.05 cm 3.

Knowing how to find the volume of the ball, you 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 got answers to all questions of the problem: R = 2.285 mm, S = 0.6561 cm 2 and V = 0.05 cm 3.

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 the volume of a sphere to be determined.

Ball and sphere

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

Sphere in ancient Greek means "ball". 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 sphere. A ball is an ideal geometric figure, the shape of which is taken by many real objects. 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 certain integral, but how was the volume formula derived in Ancient Greece, when no one had heard of integrals yet? 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 of the ball using the same cone and cylinder, while the radius of each figure was equal to its height R = h. The ancient scientist presented a cone and a cylinder broken into an infinite number of small cylinders. Archimedes realized that if the volume of the cylinder Vc is subtracted from the volume of the cone Vk, he will get 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 a circle, and h \u003d R, then the formula is transformed into:

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

The volume of a 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 sphere is derived from the integral of the area of a spherical surface, but the result remains the same

Vsh = 4/3 pi × R 3

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

Real life examples

Cannonballs

Let's say you want to know how much iron is needed to cast a six-foot caliber cannonball. You know that the diameter of such a nucleus is 9.6 centimeters. Enter this number in the cell of the calculator "Diameter", and you will receive an answer in the form

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 inflate a perfect spherical balloon. You know that the radius of the selected ball is 10 cm. Type this value into the "Radius" calculator cell and you will get the result

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

Conclusion

The need to determine the volume of a ball can arise in a variety of 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.

Definition.

Sphere (ball surface) is the collection of all points in three-dimensional space that are the same distance from a single point, called the center of the sphere(O).

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 the collection of all points in three-dimensional space, the distance from which does not exceed a certain distance to a point called ball center(O) (set of all points of three-dimensional space bounded by a sphere).

A ball can be described as a three-dimensional figure, which is formed by rotating a circle around its diameter by 180 ° or a semicircle around its diameter by 360 °.

Definition. Sphere (ball) radius(R) is the distance from the center of the sphere (ball) O to any point of the sphere (surface of the ball).

Definition. Sphere (ball) diameter(D) is a segment connecting two points of the sphere (the surface of the ball) and passing through its center.

Formula. Ball 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 opposed 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 sphere 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 large circles for a sphere or circles for a ball.

6. Through any two points, except for diametrically opposite points, it is possible to 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 between their radii, then such balls intersect, and a circle is formed in the plane of intersection.

The secant, chord, secant plane of the sphere and their properties

Definition. The secant of the spheres is a straight line that intersects the sphere at two points. The points of intersection are called puncture points surface or entry and exit points on the surface.

Definition. Chord of a sphere (ball) is a segment connecting two points of 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, respectively great circle and big circle. The great circle and the 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 section of the cutting plane on the sphere will always be minor circle, and on the ball the section will be small circle. A small circle and a small circle have their centers that do not coincide with the center of the sphere (ball). The radius r of such a circle can be found by the formula:

r \u003d √ R 2 - m2,

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 the sphere (ball), which is formed when it is cut by a diametrical plane.

Tangent, tangent plane to the sphere and their properties

Definition. Tangent to sphere is a straight line that touches the sphere at only one point.

Definition. Tangent plane to 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. The backbone of the segment call 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 in terms of sphere radius R:

S = 2π Rh

Instruction

note

^ - sign denoting exponentiation;
^1/2 - in fact, the extraction of the square root;
^1/3 - extracting the cube root.

Sources:

diameter is

A circle is a geometric figure on a plane, which consists of all points of this plane that are at the same distance from a given point. The given point is called the center. circles, and the distance at which the points circles are from its center - radius circles. The area of the plane bounded by a circle is called a circle. There are several calculation methods diameter circles, the choice of a specific envy from the available initial data.

Instruction

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How to find the volume of a sphere if the diameter is known. Ball and sphere, volume of a ball, area of ​​a sphere, formulas