Math How are the formulas for the surface area and volume of a sphere derived?

Derive the volume and surface area of the sphere.

The process of calculating the formula goes like this:

Assuming that the radius of the sphere and the radius of the bottom surface of the cylinder are equal, both are r, then the height of the cylinder is 2r, or d, and then the volume and surface area of the cylinder are expressed by letters and symbols, and then multiplied separately.

v cylinder = r2 2r

r2×（r+r）

R3 2v ball = R3 2

R3S cylinder = R2 2+ D D

dr+πdd

r+d)d3r×2πr

6 r2s ball = 6 r2

Volume derivation: The surface of the sphere is divided into an infinite number of facets, and then the center of the sphere is used as the vertex to connect these facets to form an infinite number of approximate cones.

The sum of the base areas of these cones is the surface area of the ball, which is approximately the radius of the ball.

So the sum of volumes is: (4 r)*r 3=4 rrr 3:

It is calculated using points. Before the invention of the integral operation, the volume of the ball was made using the principle of Zu Xuan. It is as if the surface area is made only when there is a limit operation, which is similar to the principle of integration.

Microelement method. Use knowledge to the limit.

There are deductions in high school math books.

Divide the upper hemisphere of a ball with radius r into n parts with the square of each part of the same height, and treat each part as a cylinder, where the radius is equal to the radius of the circle on its bottom surface, then the side area of the kth cylinder from bottom to top s(k) = 2 r(k) * h, where h = r n r (k) = root number [r -(kh) ]s (k) = root number [r - (kr n) ]2 r n = 2 r * root number [1 n -(k n ) then s(1)+s(2)+....s(n) When n takes the limit (infinity), it is the surface area of the hemisphere 2 r multiplied by 2 and it is the surface area of the whole sphere 4 r

The sphere area formula is derived as follows:

Square is denoted by .

Cut the upper hemisphere of a ball with radius r into n equal heights.

And think of each part as a cylinder, where the radius is equal to the radius of its base circle.

Then the side area of the k-th cylinder from bottom to top is s(k) = 2 r(k)*h.

where h=r n r(k) = root number [r -(kh)].

s(k) = root number [r - (kr n)]2 r n.

2 r * root number [1 n - (k n)

then s(1)+s(2)+....s(n) is the hemispheric surface area when n takes the limit (infinity)2 r

Multiply by 2 to get the surface area of the entire ball 4 r

Sphere Area Formula:

The formula for calculating the area of the sphere is: s=4*r 2*, if it is a hemisphere, only half of the area of the sphere and the area of the bottom circle need to be calculated, and the result is s=1 2s.

Ball + S bottom = 2 r 2 + r 2 = 3 r 2.

The formula for the surface area of the ball.

Let the radius of the ball be $r$, and the surface area of the ball is uniquely determined by the radius $r$, so its surface area $s$ is a function of $r$ as an independent variable, i.e., $s ball = 4 r 2$.

1. Definition: The surface area of the sphere refers to the surface area of the geometry enclosed by the sphere, which includes the sphere and the space enclosed by the sphere.

Dig out a contour of the same height in the center of a cylinder with a base radius r and a height of r. The remaining part is equal to the area of a hemisphere when it is cut in a flat plane. Wait for the conclusion that they are equal in volume.

And the volume of the excavated body is easy to find. It's the volume of the hemisphere. vii2 3tra3.

Hence the volume of a whole sphere is 4 3 tr 3 and the ball is formed by circular rotation. The area of the circle is s=tr 2, then the sphere is its integral, and according to the integral formula, the corresponding volume formula of the sphere is v=4 3tr a3

1 Derivation of the volume formula of the ball Basic idea method: First, the ball is cut by the plane of the center of the sphere, and the ball is divided into two hemispheres of equal size by the cross-section, and the cross-section is called the bottom surface of the obtained hemisphere (l) The first step: Divide the hemisphere into layers with a set of planes parallel to the bottom surface (2) The second step:

Finding the approximate sum Each layer is a "small disc" that approximates the shape of a cylinder, and we replace the volume of the "small disc" with the volume approximation of the small cylindrical shape, and their sum is the approximate value of the volume of the hemisphere (3) Step 3: Transform from approximate sum to exact sum When the infinite increases, the approximate volume of the hemisphere tends to the exact volume (see the textbook for the specific process) 2 Theorem: The volume of a sphere with radius of is formulated as:

3 Application of the volume formula There is only one condition to find the volume of the ball, that is, the radius of the ball The cube of the radius ratio of the two balls is equal to the volume ratio of the two balls The ball is cut into the cube, and the diameter of the ball is equal to the edge length of the cube; The cube is connected to the ball, and the radius of the ball is equal to the length of the edge of the cube (i.e., half of the diagonal of the sphere); The radius of the inscribed sphere of a regular tetrahedron with an edge length of is , and the radius of the outer sphere is can also be found by calculus, but it is difficult to write.

It is derived through calculus in higher mathematics.

There is a circle x 2+y 2=r 2, and a sphere is obtained by rotating the circle around the x-axis in the xoy axis.

The microelement of the sphere volume is dv= [r 2-x 2)] 2dx dv= r 2-x 2)] 2dx integral interval is [-r,r], and the result is .

4/3πr^3

The diameter of the bar is the shaft; The center of the sphere is placed at the coordinate origin; The selected diameter coincides with the z-axis. Then the radius of the cross-sectional circle perpendicular to the axis at the center of the sphere z on the shaft is r= (r 2-z 2).Its area is ·r 2= ·

<>1. The volume formula of the ball: v=(4 3) The "Zu Huang Principle" independently developed by Zu Chongzhi's father and son is richer than Archimedes' research and involves more complex problems. The sum of Zu ChongHis son Zu Xuan together, with ingenious methodsSolved the problem of calculating the volume of the sphere.

3. In the "Nine Chapters of Arithmetic", it is believed that the ratio of the volume of the inscribed cylinder to the sphere of the sphere is equal to the ratio of the area of the square to its inscribed circle, and Liu Hui pointed out in his annotation for the "Nine Chapters of Arithmetic" that the statement of the original book is incorrect, and only the ratio of the "Mu Hefang Gai" (the volume of the common part of two cylinders that intersect vertically) to the volume of the sphere is exactly equal to the ratio of the area of the square to its inscribed circle. However, Liu Hui did not find the volume formula of the perpendicular intersection of the two cylinders, so he could not get the volume formula of the sphere. Zu Chong's stuffy pin Lu father and son application "equal heights."Two three-dimensional cubes with equal cross-sectional areas must also have equal volumes", and the volume of the "Mou Closed Square Cover" was obtained.

And the sphere volume is equal to 4 times the volume of the "Mou Square Cover", thusFinally, the volume of the sphere is calculated, and this formula is the famous "Zu Huang Axiom". 4. It can be seen that: (1 2) v ball bucket burning = (2 3) r3, the final available, v ball = (4 3) r3.

The formula for the volume of the sphere is derived from this.

Cut the ball into an infinite number of small rings, the width of the ring is rd (arc micro macro element), the length is the circumference of the circle 2 rsin area element: ds=2 rsin (rd)=2 (r 2)sin d integral:s table = [0, ]2 (r 2)sin d =2 (r 2) [0, ]sin d =-2 (r 2)cos |[0,π]r=4πr^2

The formula regarding the surface area of the ball is derived as follows:

The surface area of a ball refers to the area of space occupied by the surface of the ball. The surface area of the ball can be expressed by the formula s=4 r2, where r is the radius of the ball. First, the sphere is projected onto the xyz coordinate system, and the surface area of the ball can be regarded as composed of circular surfaces on the xyz coordinate system.

Suppose the radius of the sphere is r, then the radius of the circular surface is also r, and the radii are all equal. Next, let's derive the formula for the surface area of the ball s=4 r2. First, we can project a sphere onto the xyz coordinate system, which has an area of r2 according to the area formula for a circular surface.

Projecting the sphere onto the xyz coordinate system, since the sphere is three-dimensional, and there are 6 circles on its surface, the surface area of the sphere is the sum of the areas of the 6 circles, i.e., s=6 r2.

Next, let's derive the formula for the surface area of the sphere s=4 r2. Assuming that the radii of the circles are all equal, then the surface area of the sphere can be simplified to s=4 r2. Therefore, we can derive the formula for the surface of the sphere s=4 r2.

Introduction to the sphere

The geometry of space formed by a semicircle rotating around the straight line where the diameter is located is called a sphere, referred to as a sphere, and the radius of the semicircle is the radius of the sphere. A sphere is a three-dimensional figure with and only one continuous surface, and this continuous surface is called a sphere. The orthographic projection of a sphere on any one plane is a circle of equal size, and the diameter of the projected circle is equal to the diameter of the sphere.

Definitions

Definition: The space geometry formed by a semicircle rotating around the straight line where the diameter is located is called a sphere, and the figure shown in Figure 1 is a sphere. The sphere is a three-dimensional figure with a continuous surface, and the geometry enclosed by the sphere is called a sphere.

There is no absolute sphere in the world. Absolute spheres exist only in theory.

But in a weightless environment, such as space, the droplets automatically form absolute spheres. The surface of the ball is a curved surface, and this surface is called a sphere. A ball is similar to a circle in that it also has a center called the center of the ball.

1.The surface area of a sphere refers to the size of the area of the geometry enclosed by the sphere, and is calculated as s is equal to 4 times the square of r, where s is the total area and r is the radius of the sphere.

2.The volume of a sphere refers to the size of the space enclosed between the spheres, and the volume is calculated by the formula V is equal to four-thirds multiplied by the cubic of r, where v represents the total volume and r is the radius.

If the upper hemisphere of a ball with radius r is transversely cut into n parts, each part is of equal height, and each part is regarded as a similar round table, where the radius is equal to the radius of the circle on the top surface of the similar round table, then the side area of the k-like round table from bottom to top: s(k)=2 r(k) h, where r(k) = r 2-(kh) 2], s(k)=2 r(k)h=(2 r 2) n, then s=s(1)+s(2)+s(n)=2 r 2;Multiply by 2 to get the surface area of the entire bend ball 4 r 2.

Definition: 1. The linear line where the diameter of the semicircle is located is the axis of rotation, and the rotating body formed by the rotation of the semicircular surface is called a solid sphere, referred to as a ball. (as defined in terms of rotation).

2. The rotating body formed by rotating the circle surface by rotating 180° with the diameter of the circle as the axis of rotation is called a solid sphere, referred to as a sphere. (From the angle of rotation of the fixed trouble).

3. The set of points whose distance from the fixed point in space is equal to the fixed length is called the spherical surface, that is, the surface of the sphere. This fixed point is called the center of the ball, and the fixed length is called the radius of the ball.