Derivation formula for the area and perimeter of a circle

1.Cut the circle into several equal parts along the radius (the more the better) (into several sectors)2The fan is divided into two parts, corresponding to each other to form an approximate rectangular shape. (The more you are, the closer you are to the rectangle).

3.Area of the rectangle = long multiplication.

Width, the length of this assembled rectangle is the circumference.

2pr), so the length is PR (pi.

I won't hit the symbol, it's represented by p), the width is the radius of the circle r, so the area of the circle is obtained.

is calculated as s=squared).

Derivation of the circumference of the circle.

Find several round objects, measure their circumference and diameter, and calculate the ratio of circumference to diameter. Through trials and statistics, we can know that the circumference of a circle is always a little more than three times its diameter. Well, the ratio of the circumference to the diameter of any circle is a fixed number (pi).

Because the circumference of a circle is always a multiple of its diameter, when we know the diameter or radius of a circle, we can calculate its circumference. Namely. c=d

c=2r.Derivation of the area of a circle:

Draw a circle on cardboard, divide the circle into equal parts, cut it and use these approximate isosceles triangles.

Small pieces of paper can be put together to form an approximate parallelogram.

If you score more points, the more detailed each one will be. The closer the shape will be to the rectangle. The length of the rectangle is equal to half the circumference of the circle, ie.

The width of r is equal to the radius of the circle.

r because of the area of the rectangle.

length and width, so the area of the garden. rr

r i.e. s=r²

Because the area of a circle is seven-ninths of the area of its inscribed square when it is "squared", so"The area s of the circle is equal to seven times the square of one-third of the diameter d"。

Circle area formula: s=7(d3).

Since the ratio of the circumference of the circle to the diameter is 6+2 3 to 3, "the circumference of the circle is equal to (6+2 3) times the diameter d".

Circumference formula: c=d(6+2 3) 3.

To derive the formula for calculating the circumference of a circle, add the six dots on the periphery of the circle surface in "The figure below is a unique method for deriving the area of a circle" (Fig. 4)."Pythagorean theorem"The sum of the 2 and 3 points of the overlapping points is obtained.

The circumference of the circle c=d(6+2 3) 3.

Infinitely replace approximation with a group of inscribed isosceles triangles with vertices at the center of the circle, and the larger the number of small triangles, the closer the result will be.

Since elementary school, we have been exposed to a wide variety of shapes, such as rectangles (rectangles), squares, triangles, trapezoids, parallelograms, circles, etc. Here, we're focusing on circles. , I believe that the first 7 decimal places of pi are ripe for us to memorize, you know, it is an infinite non-cyclic decimal number, that is, an irrational number.

We are also familiar with the formula for the area of a circle and the formula for the circumference of a circle. After college, I studied advanced mathematics.

I'm curious how these two formulas are derived. Following my curiosity, I deduced step by step with the advanced mathematics knowledge I had learned, and finally knew why.

We all know the equation of the circle, and we have been exposed to it in high school math, that is, where a is the radius of the circle, which is greater than zero. Since the circle is a figure that is symmetrical with respect to the x-axis and symmetrical with respect to the y-axis. We only need to be above the x-axis, to the right of the y-axis, that is, the area of the quarter circle where both x and y are greater than or equal to 0.

Then, multiply the area of a quarter circle by 4 to get the circumference and area of the circle.

So, how do you find the circumference and area of a quarter circle? First, since x and y are greater than or equal to 0, we can deform the formula to get a function. where the interval of x is [0,a].

Let's take a look at the derivation of the circumference of a circle. Let's start by finding the derivative of the function. The reasoning for the number of trouser buckets of this function is as follows:

Then, according to the arc length formula of any function, the circumference of a quarter circle can be found, which is derived as follows:

Then, we multiply the circumference of the resulting quarter circle by four times to get the formula for the circumference of the circle. It is.

Looking at the derivation of the area of a circle, we know that the geometric definition of the definite integral is the algebraic sum of the area of the function graph in the [a,b] interval with y=a,y=b and the pure limb of the x-axis collapse, which may be negative. And our quarter circle is above the x-axis and to the right of the y-axis, so we don't consider negative numbers. Our quarter circle is on the interval [0,a], so we know that the area must be positive, and all the area is above the x-axis, so we don't need to consider the part below the x-axis, only the part above the x-axis.

Here's how to find the area of the quarter circle. The derivation is as follows:

Then, we multiply the area of the resulting quarter circle by a factor of four to get the formula for the area of the circle. It is.

Dear friends, have you learned? If there are other ways to derive the area and circumference of a circle using advanced mathematical calculus, you can also tell me. Thank you!

The formula for finding the arc length of the parametric equation is: l = dx dt) 2 + dy dt) 2] (1 2) dt

The parametric equation for a circle of radius 1 is x = cost y = sint (0<= t <=2 r).

Now let's substitute the parametric equation into the arc length gong let vertical opening equation :

dx dt = sint , dy dt = cost to get l = tanru (0 to 2 ) sint) 2 + cost) 2]dt = 0 to 2t) dt = 2 (where the chain r = 1).

The circumference of the circle is 2 vultures

The area of the circle r

Hope it helps, thank you!

Circumference of the circle = pi diameter, i.e. c = d; The circumference of the circle = pi 2 radius, i.e. c=2 r; The area of the circle = the square of the radius of pi, i.e. s = r2, note s: area c: perimeter d = diameter r = radius.

A circle is a type of geometric shape. By definition, a circle is usually drawn with a compass. The diameter of the circle within the same circle, the number of long bridges of the radius are always the same, and the circle has an infinite number of radii and an infinite number of diameters.

A circle is an axisymmetric, center-symmetrical figure. The axis of symmetry is the straight line where the diameter is located. In the same limb, the circle is again a "positive infinite polygon", and "infinity" is just a concept.

When the polygon has more faces, the shape, circumference, and area of the polygon are closer to the circle. So, there is no real circle in the world, and the circle is really just a conceptual figure.

The formula for the circumference of a circle: Diameter , or Radius 2.

The area formula for a circle: radius radius.

The circumference of the circle is 2 vultures

The area of the circle r

Hope it helps, thank you!

d is the diameter, and r is the diameter of half a guess mu.

The circumferential spike of a circle refers to the length formula: l= d=2 r

The area formula of the circle is s= r 2= (d 2) 2

Assuming that the radius of the rounded ridge is r, then:

The circumference of the circle is: c=2 r

The area of the circle is specified as the wild wheel: s = r

Circumference of a circle: <

where the loose coarse r is the radius of the circle and the pi, which is usually derived from the circumference formula of the circle.

Let the parametric equation for the circle be <>

The integral <> of the circumference of a circle in a week

Substitution, you can get <>

Namely.

Pi (Pi) is the ratio of the circumference of a circle to its diameter, generally expressed in Greek letters, and is a mathematical constant that is prevalent in mathematics and physiology.

It is also equal to the ratio of the area of the circle to the square of the radius. It is the key value to accurately calculate the geometric shape of the circumference, area of the circle, and the volume of the sphere. In analytics, it can be strictly defined as the smallest positive real number x satisfying sin x = 0.

The formula for the circumference of a circle: c = 2 r = d (r is the radius, d is the diameter); The formula for calculating the area of a circle: s = r 2.

A closed curve formed by rotating around a point at a distance of a certain length in a plane is called a circle.

In a plane, a circle is a set of points whose distance to a fixed point is equal to a fixed length, called a circle.

The circle has an infinite number of axes of symmetry, and the axis of symmetry passes through the center of the circle.

Circles have rotational invariance.

A circle is a conic curve that is obtained by a planar truncated cone parallel to the bottom surface of the cone.

The circle is prescribed as 360°, which is the ancient Babylonians who moved a position about every 4 minutes when observing the rising sun on the horizon, and 360 positions in 24 hours a day, so the inner angle of a circle was prescribed to be 360°. This ° represents the sun.

A circle is a type of geometric shape. By definition, a circle is usually drawn with a compass.

The diameter and length of the radius of the circle within the same circle are always the same, and the circle has an infinite number of radii and an infinite number of diameters. A circle is an axisymmetric, center-symmetrical figure. The axis of symmetry is the straight line where the diameter is located. At the same time, Bu Qinghuai, the circle is a "positive infinite polygon", and "infinity" is just a concept.

Introduction to the history of the circle:

The circle is a shape that looks simple, but is actually very wonderful. Ancient people first got the concept of circle from the sun and the moon on the fifteenth day of the lunar calendar. Eighteen thousand years ago, cave people used to drill holes in animal teeth, gravel and stone beads, and some of those holes resembled rounds.

By the time of the pottery age, many pottery pieces were round.

Round pottery is made by placing clay on a turntable. When people began to spin threads, they made round stone spindles or pottery spindles. Ancient people also found that rolling around was easier to carry round wood.

Later, when they were carrying heavy loads, they would roll a few pieces of logs under big trees and big rocks, which of course would be much less strenuous than carrying them.

About 6,000 years ago, the Mesopotamians made the world's first wheel, a round wooden disc. About 4,000 years ago, people fixed round wooden discs under wooden frames, which became the first cars.

You can make circles, but you don't necessarily understand the nature of the circle. The ancient Egyptians believed that the circle was a sacred figure given by the gods.

It was not until more than 2,000 years ago that Mozi in China (about 468 B.C. 376 B.C.) gave a definition of the circle: circle, one middle and the same length.

Meaning: The circle has a center, and the length from the center to the circumference of the circle is equal. This definition predates the definition of circle by the Greek mathematician Euclid (c. 330 BC 275 BCE) by 100 years.

Since the ratio of the circumference of the circle to the diameter is: 6+2 3 to 3 (and the ratio of the circumference of the regular n-sided to the diagonal is: ratio 1), the ratio of the circumference c of the circle to the diameter d can only be:

6+2 3) 3(or approximately equal to pi is not a formula for the circumference of a circle: c=d(6+2 3) 3, and the formula for the circumference of a circle is not c=.

Because the area of a circle is seven-ninths of the area of its inscribed square when it is squared, the area s of the circle is equal to seven times the square of one-third of its diameter d. Circle area formula: s=7(d3).

Area s = rr

r is the radius of the circle.

Circumference = 2 r