What is the surface area of \u200b\u200bthe sphere. Sphere, ball, segment and sector. Formulas and properties of the sphere. The main properties of the sphere and ball
The ball and sphere are primarily geometric shapes, and if the ball is a geometric body, then the sphere is the surface of the ball. These figures were interested in many thousands of years ago BC.
Subsequently, when it was discovered that the Earth is a ball, and the sky is a heavenly sphere, a new fascinating direction in geometry - geometry on the sphere or spherical geometry has developed. In order to argue about the size and volume of the ball, you must first give it a definition.
Ball
Radius RF with center at point o geometry call the body that is created by all dots space having general property. These points are at a distance not exceeding the radius of the ball, that is, fill the entire space less than the ball radius 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 a bowl.
How can I get a ball? We can cut a circle from paper and start moving it around its 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 the body of rotation. Because it can be formed by rotating a flat shape - a circle.
Take some plane and cut our ball. Just as we cut the orange knife. A piece that we cut from the ball is called a ball segment.
IN Ancient Greece They could not only work with a ball and a sphere, as with geometric shapes, for example, to use them during construction, and also knew how to calculate the surface area of \u200b\u200bthe ball and the volume of the ball.
The sphere is different called the surface of the ball. The sphere is not a body - this is the surface of the body of rotation. However, since the Earth and many bodies have a spherical shape, such as a drop of water, then the study of geometric ratios within the sphere was greatly distributed.
For example, if we connect two points of the sphere between themselves straight line, then this straight line will call chord, and if this chord is held through the center of the sphere, which coincides with the center of the ball, then the chord will be called the diameter of the sphere.
If we feed a straight line that will affect the sphere just at one point, this line will be called tangent. In addition, this tangent to the sphere at this point will be perpendicular to the sphere radius carried out to the point of touch.
If we continue chord to a straight line in the other side of the sphere, then this chord will be called the Sale. Or can be said otherwise - the sequential to the sphere contains her chord in itself.
Bowl
The formula for calculating the volume of the ball has the form:
where R is a ball radius.
If you need to find the volume of the ball segment - use the formula:
V Seg \u003d πh 2 (R-H / 3), H is the height of the ball segment.
Surface surface of the ball or sphere
To calculate the spheres area or the surface area of \u200b\u200bthe ball (this is the same):
where R is the radius of the sphere.
Archimedy loved the ball and the sphere, he even asked to leave the drawing on his tomb on which a ball entered the cylinder. Archimeda believed that the volume of the ball and its surface is equal to the two thirds of the volume and surface of the cylinder, in which the ball is inscribed.
Many of us love to play football or at least almost every one of us heard about this famous sports game. Everyone knows that football play the ball.
If you ask the passerby, the form of which geometric Figure He has a ball, some people will say that the shape of the ball, and part that the forms of the sphere. So who is right? And what is the difference between the sphere and a ball?
Important!
Ball - This is a spatial body. Inside the ball something is filled. Therefore, the ball can find the volume.
Examples of a bowl in life: watermelon and steel ball.
Ball and sphere, like a circle and circle, have a center, radius and diameter.
Important!
Sphere - The surface of the ball. The sphere can find the surface area.
Examples of the sphere in life: a volleyball ball and a ball tennis ball.
How to find the area of \u200b\u200bthe sphere
Remember!
Formula of the area of \u200b\u200bthe sphere: S \u003d 4. π R 2.
In order to find the area of \u200b\u200bthe sphere, it is necessary to remember what the degree of date. Knowing the degree of degree, it can be written by the Formula of the Square of the Sphere as follows.
S \u003d 4. π r 2 \u003d 4π r · r;
Secure the knowledge gained and we will solve the task of the area of \u200b\u200bthe sphere.
Zubareva Grade 6. Number 692 (a)
The task:
- Calculate the area of \u200b\u200bthe sphere if its radius is equal
1 \u003d 3 · \u003d \u003d / (4 · 3) \u003d) \u003d \u003d) \u003d
= = =
= 188 88 - R 3 \u003d 1
- R \u003d 1 m
Important!
Dear Parents!
With the final calculation of the radius, it is not necessary to force a child to count the cubic root. Students of the 6th grade have not yet passed and do not know the definition of roots in mathematics.
In grade 6, when solving such a task, use the generation method.
Ask the student, what a number, if you multiply 3 times on myself will give an unit.
Definition.
Sphere (surface of the ball) - This is a combination of all points in three-dimensional space, which are at the same distance from one point, called center of sphere (ABOUT).The sphere can be described as a volume figure, which is formed by the rotation of the circle around its diameter by 180 ° or a semicircle around its diameter by 360 °.
Definition.
Ball - This is a combination of all points in three-dimensional space, the distance from which does not exceed a certain distance to the point called center of Shara. (O) (the totality of all points of the three-dimensional space of limited sphere).The ball can be described as a volumetric figure, which is formed by the rotation of the 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 of the sphere (surface of the ball).
Definition. The diameter of the sphere (ball) (D) is a segment connecting two points of the sphere (ball surface) and passing through its center.
Formula. Bowl:
| V \u003d. | 4 | π R 3 \u003d | 1 | π D 3. |
| 3 | 6 |
Formula. Surface area of \u200b\u200bthe sphere Through the radius or diameter:
S \u003d 4π R 2 \u003d π d 2
Equation sphere
1. The equation of sphere with radius R and the center at the beginning of the Cartesian coordinate system:
x 2 + y 2 + z 2 \u003d R 2
2. Equation of the 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 \u003d R 2
Definition. Diametrically opposite dots They are called any two points on the surface of the ball (sphere), which are connected by a diameter.
The main properties of the sphere and ball
1. All points of the sphere are equally removed from the center.
2. Any cross section of the spheres is a circle.
3. Any section of the ball with a plane is circle.
4. Sphere has the greatest volume Among all spatial figures with the same surface area.
5. Through any two diametrically opposite points, a variety of large circles for the sphere or circles for the ball can be held.
6. Through any two points, in addition to diametrically opposite points, you can spend only one large circle for a sphere or a large circle for a ball.
7. Any two large circles of one ball intersect in a straight line passing through the center of the ball, and the circles intersect in two diametrically opposite points.
8. If the distance between the centers of any two balls is less than the sum of their radii and more difference module of their radii, then such balls crossand in the intersection plane a circle is formed.
Sequer, chord, sequential plane of sphere and their properties
Definition. Sequer sphere - This is a straight line that crosses the sphere in two points. Point of intersection are called points of singers Surfaces or points of input and output on the surface.
Definition. Chord spheres (ball) - This is a segment connecting two points of the sphere (surface of the ball).
Definition. Sequer plane - This is a plane that crosses the sphere.
Definition. Diameter plane - This is a securing plane passing through the center of the sphere or a ball, Sechenm forms, respectively large circle and big circle. A large circle and a large circle have a center that coincide with the center of the sphere (ball).
Any chord passing through the center of the sphere (ball) is a diameter.
Chord is a segment of a secure straight.
The distance d from the center of the sphere to the section is always less than the radius of the sphere:
d.< R
The distance M between the secant plane and the center of the sphere is always less than the R radius:
m.< R
The location of the section of the sequential plane on the sphere will always be small circle, and on the ball, the section will be small circle. Small circle and small circle have their centers that do not match the center of the sphere (ball). The radius of R such a circle can be found by the formula:
r \u003d √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 secular plane.
Definition. Hemisphere (Halfs) - It is half the sphere (ball), which is formed when it is crossped with a diametral plane.
Tangential, tangent plane to the sphere and their properties
Definition. Tangent to the sphere - This is a straight line that concerns the sphere only at one point.
Definition. Tangential - This is a plane that comes into contact with the sphere only at one point.
Tanner direct (plane) is always perpendicular to the sphere radius spent to the point of contact
The distance from the center of the sphere to the tangent direct (plane) is equal to the radius of the sphere.
Definition. Shara segment - This is part of a ball that cuts off from the ball by the secant plane. The basis of the segment Call a circle that was formed at the cross section. Segment height H is called the length of the perpendicular spent from the middle of the base of the segment to the surface of the segment.
Formula. Square of the outer surface of the segment of the sphere With a height of H through the radius of the sphere R:
S \u003d 2π RH
A task
In the sphere, a cone forming which is equal to L, and an angle at the top of the axial cross section is 60 degrees. Find the area of \u200b\u200bthe sphere. Decision.
The area of \u200b\u200bthe sphere will find by the formula:
Since the cone entered the sphere, we will carry out a section through the top of the cone that will be an equally traded triangle. Since the angle at the top of the axial cross section is 60 degrees, the triangle is equilateral (the sum of the corners of the triangle is 180 degrees, which means the remaining angles (180-60) / 2 \u003d 60, that is, all the angles are equal).
From where the radius of the sphere is equal to the radius of the circle described around the equilateral triangle. The side of the triangle by condition is equal to l. I.e
Thus, the area of \u200b\u200bthe sphere
S \u003d 4π (√3 / 3 L) 2
S \u003d 4 / 3πL 2
Answer: The area of \u200b\u200bthe sphere is equal to 4 / 3πl 2.
A task
The container has the shape of the hemisphere (hemisphere). The base circumference length is 46 cm. 1 square meter is consumed 300 grams of paint. How much paint is needed to paint the container?
Decision.
The surface area of \u200b\u200bthe figure will be equal to half the area of \u200b\u200bthe sphere and the area of \u200b\u200bthe cross section of the sphere.
Since we know the length of the base circumference, we will find its radius:
L \u003d 2πr.
From
R \u003d L / 2π
R \u003d 46 / 2π
R \u003d 23 / π
From where the base area is equal
S \u003d πR 2
S \u003d π (23 / π) 2
S \u003d 529 / π
The area of \u200b\u200bthe sphere will find by the formula:
S \u003d 4πr 2
Accordingly, the area of \u200b\u200bthe hemisphere
S \u003d 4πr 2/2
S \u003d 2π (23 / π) 2
S \u003d 1058 / π
The total surface area of \u200b\u200bthe figure is equal to:
529 / π + 1058 / π \u003d 1587 / π
Now we calculate the paint consumption (we take into account that the consumption is given per square meter, and the calculated value in square centimeters, that is, in one meter 10,000 square centimeters)
1587 / π * 300/10 000 \u003d 47.61 / π grams ≈ 15.15 g
A task
Decision. Ryshmy.
To explain the decision, we will comment on each of the above formulas
| For the explanation of the river, the promenonuto-skin of the formulas
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 | 8. We divide the volume of the first and second ball on each other. 9. We will reduce the resulting fraction. Note that the ratio of the volume of two balls is equal to the ratio of cubes of their radii. We take into account the expression obtained by us earlier in Formula 4 and substitute it. Since square root is a number of 1/2 degree, we transform expression 10. Recall brackets and write the resulting ratio in the form of proportion. The answer is received. | 8. Ridgelimo about "єima і і і ї kuli one on one 9. Rootimo DRIB, Scho Viyshov. Vіdmіtimo, Scho Spіvіtimnya about "Loof Dor Dorіvniuє Spevvimynesyannyh Kubіv ї Radіusіv. Vosheєmo Virazhenna, Revisani by us in the formula 4 І pіdstavimo yogo 10. Roskrymo's handkeeper І VisDo Recisean Spevvimdenshene at the Visigandi proportion. Vіdpovіda otrimana. |
The area of \u200b\u200bthe curved surface, which can not be deployed to the plane, calculate so. They split the surface on such pieces, which are already different enough from flat. Then they find the area of \u200b\u200bthese pieces, as if they were flat (for example, replacing them with projections on the plane, from which the surface deviates little). The sum of their squares and will give an approximate surface area. This is done in practice: the surface area of \u200b\u200bthe dome is obtained as the sum of the areas covering it with lumps of sheet metal (Fig. 17.5). Yet
it is better to be visible on the example of the earth's surface. It is twisted - approximately spherical. But sites, small in comparison with the size of the entire Earth, are measured as flat.
By calculating the plane of the sphere, describe the multifaceted surface close to it. Her faces will approximately represent pieces of sphere, and its area gives approximately the area of \u200b\u200bthe sphere itself. Its further calculation is based on the following lemma.
Lemma. The volume of the polyhedron P described around the sphere of radius R, and its surface area is associated with the relation
Note: The area of \u200b\u200bthe polygon q, described around the radius circle and its perimeter (Fig. 17.6) are associated with a similar ratio.
We describe around the sphere.
Each such face lies in the tangent plane of the sphere and, it means perpendicular to the sphere radius at the touch point. It means that this radius is the height of the pyramid so its volume will be:
where - the area of \u200b\u200bthe face The sum of these areas gives the surface area of \u200b\u200bthe polyhedron P, and the amount of the pyramids - its volume is therefore
Theorem (about the area of \u200b\u200bthe sphere). The area of \u200b\u200bthe radius R is expressed by the formula:
Let it be given the sphere of radius R. Take on it on the points that do not lie in one hemisphere, and will carry out the tangent plane to the sphere. These planes will limit the polyhedron described around the sphere. Let be the volume of the polyhedron - its surface area, V is the volume of the ball limited to the sphere under consideration, and S is its area.
