08.08.2020
The area of the sphere. The volume of the ball. Sphere, ball, segment and sector. Formulas and properties of a sphere From which follows the formula for the area of a sphere
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(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 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
A sphere and a ball are an analogue of a circle and a circle in three-dimensional space. It is worth talking about each of these figures, highlighting the similarities and differences, as well as the formulas inherent in these figures.
Most of the geometric constructions are made in a plane, but in high school they begin to study three-dimensional figures. Two-dimensional space has only two characteristics: length and width. Height is added in 3D regions. In grade 6 mathematics, individual 3D figures are studied.
On the plane, the figure was characterized by area and perimeter. In three-dimensional objects, volume is added to them.
Rice. 1. Three-dimensional space.
In addition, there are a number of specific properties of 3D shapes. They can be crossed by a straight line and a plane, there may be secant planes that take the form of other figures.
The use of 3D shapes for composing tasks greatly complicates them, but at the same time makes them much more interesting. We give the definitions of a ball and a sphere, after which we will try to highlight the differences between these figures.
Ball
A sphere and a sphere are an analogue of a circle and a circle in a plane. A ball is a figure obtained by rotating a semicircle around one point.
The ball has a surface area: $S=4pir^2$
A radius is a line segment that connects the center of the ball and any of the points on its surface.
Volume formula for a sphere$V=(4pir^3\over3)$
Volume shows how much space a figure occupies. To understand what volume is, you need to imagine a hollow figure. Then the volume is the amount of water that can be poured into this figure
A ball, like any other three-dimensional figure, can be cut by a plane. The secant plane of the ball is a circle, the center of which can be found by dropping a perpendicular from the center of the ball onto the circle.
Rice. 2. Section of the ball.
A sphere is a figure that is a set of points in space equidistant from the center of the sphere. Sphere:
- It has the same volume and surface area formulas as a sphere.
- The cutting plane of a sphere is a circle
- The center of the secant circle is found in the same way as in the case of a ball
Rice. 3. Sphere.
What is the difference
Then the question arises, what is the difference between a ball and a sphere, except for the definition? The fact is that the differences between a ball and a sphere are much more blurred than the differences between a circle and a circle. A sphere also has volume and surface area.
Perhaps, apart from the definition, the difference lies in the fact that the volume of the sphere is never found in problems. As a rule, they are looking for the volume of the ball. This does not mean that the sphere has no volume. This is a three-dimensional figure, so it has volume.
An analogy is simply drawn with a circle that has no area. This is not a rule, but rather a tradition that needs to be remembered: in geometry, the formulation of the volume of a sphere is not welcome.
Another difference that can be considered more or less significant: the cutting plane of a sphere: a circle that has no internal space, but has a length. Sectional Plane of a Sphere: A circle that has an area and no circumference. Therefore, it is worth being careful in the wording of the problem so that there are no errors due to such trifles.
What have we learned?
We learned what a sphere and a ball are. We talked about their similarities and differences. We learned that there are almost no differences between these figures. We decided that it is not necessary to give such a formulation as the volume of a sphere.
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Many of us love to play football, or at least almost all of us have heard of this famous sports game. Everyone knows that football is played with a ball.
If you ask a passer-by what geometric shape the ball has, then some people will say that the shape of a ball, and some that the shape of a sphere. So which one is right? And what is the difference between a sphere and a sphere?
Important!
Ball is a space body. Inside the ball is filled with something. Therefore, the sphere can find the volume.
Examples of a ball in life: a watermelon and a steel ball.
A ball and sphere, like a circle and a circle, have a center, a radius, and a diameter.
Important!
Sphere is the surface of the sphere. You can find the surface area of a sphere.
Examples of a sphere in life: a volleyball and a table tennis ball.
How to find the area of a sphere
Remember!
Sphere area formula: S=4 π R 2
In order to find the area of a sphere, you need to remember what a power of a number is. Knowing the definition of the degree, we can write the formula for the area of a sphere as follows.
S=4 π R 2 \u003d 4π R R;
Consolidate the acquired knowledge and solve the problem for the area of a sphere.
Zubareva 6th grade. Number 692(a)
The task:
- Calculate the area of a sphere if its radius is
1 = 3 = = / (4 3) = ) = = ) =
= = =
= 188 88 - R3 = 1
- R = 1 m
Important!
Dear parents!
In the final calculation of the radius, it is not necessary to force the child to calculate the cube root. 6th grade students have not yet passed and do not know the definition of roots in mathematics.
In the 6th grade, when solving such a problem, use the enumeration method.
Ask the student what number, if multiplied 3 times by itself, will give one.
