Why is the area of a circle the largest when the circumference is equal

Equal circumference: The area of the circle is the largest.

For example, triangles, squares, and circles have a circumference of 12

1.Triangle (take equilateral triangle as an example): 3x=12, then the side length is 4, the height is 2 times the root number 3, and the area is 4 times the root number 3

2.Square: The side length is 3, and the area is 9

3.Circle: 2 r=12, then r= 6 parts, then the area is = 36 parts of the circle, therefore: in the case of equal circumference: circle area, square area, triangle area.

A little more complicated.

First of all, it is proved that the area of a regular polygon is the largest when the number of sides is equal - for example, if two adjacent sides are unequal, it is easy to prove that once they are replaced with equal areas while maintaining the same length and unchanged, they are larger than the original area, so the regular polygon with the largest area is the regular polygon. Then prove that the number of sides is about the larger the area, the method is to cut the regular polygon from the center point into a piece of triangle like cutting a cake, the area of each triangle is equal to the side length multiplied by the distance from the center to the edge divided by 2, so the area of the whole polygon is equal to the perimeter multiplied by the distance from the center to the edge divided by 2, when the perimeter is constant, the longer the distance from the center to the edge, the larger the area. It can be proved that the distance from the center to the edge is greater when the edge is longer, because the distance from the center to the edge is cot2pi 2n*c 2n, which is substituted by n and n, respectively'When the edge length tends to be infinite, the distance from the center to the edge tends to be close to the distance from the center to the vertex, and the area is the largest.

You can compare, for example, a square and a circle with equal circumference.

If the circumference is a, then the side length of the square is a 4, and the radius of the circle is a 2 At this time, the area of the square is: a square 16, and the area of the circle is: a square 4, obviously the denominator is 16>4, so the area of the circle is large.

In general, polygons with equal circumferences have the following conclusions:

Rectangle < Square < Pentagon < Hexagon < n Polygon < Circle.

Think about how to find the area of a circle?

If you subdivide a circle into n equal parts and arrange it into a rectangle, the area of the circle is equivalent to half of the circumference multiplied by the radius! The circumference of this rectangle is equivalent to 2 radii than the circumference of a rectangle (the same circumference as the circle)!

Equal circumference: The area of the circle is the largest.

For example, triangles, squares, and circles have a circumference of 12

1.Triangle (take equilateral triangle as an example): 3x=12, then the side length is 4, the height is 2 times the root number 3, and the area is 4 times the root number 3

2.Square: The side length is 3, and the area is 9

3.Circle: 2 r=12, then r= 6 parts, then the area is = 36 parts of the circle, therefore: in the case of equal circumference: circle area, square area, triangle area.

Yes, with a certain circumference, the area of a circle is the largest, and the area of a triangle is the smallest.

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, Mesopotamia.

man, 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 circles. The ancient Egyptians believed that the circle was a sacred figure given by the gods. Until more than 2,000 years ago, Mozi in China.

c. 468-376 B.C.) gave a definition of the circle: circle, one in the same length. Meaning to say:

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

The ratio of the circumference of any circle to its diameter is a fixed number, which we call pi.

Indicated by letters. It is an infinite non-cyclic decimal.

However, in practice, only an approximation of it is taken, i.e. if c is used to represent the circumference of a circle: c= d or c=2 r"Zhou Ji Sutra" said"Wednesday Trail One", which is considered to be 3, but this is only an approximation.

When the Mesopotarayans made the first wheel, they only knew that pi was 3. Liu Hui of the Wei and Jin dynasties found out in 263 A.D. when he annotated the "Nine Chapters of Arithmetic"."Wednesday Trail One"It's just the ratio of the circumference and diameter of a regular hexagon inside the circle. He invented the circumcision technique, which believed that the circumference of the circle was closer to the circumference of the circle as the number of sides of the regular polygram increased infinitely.

He calculates the pi of a 3072-sided circumscribed circumference within the circle, =3927 1250. Liu Hui applied the concept of limit to solving practical mathematical problems, which was also a major achievement in the history of mathematics in the world. Zu Chongzhi.

A.D. 429-500) continued to calculate on the basis of the calculations of his predecessors, and found that pi was between and was the world's earliest exact value of seven decimal places, and he also used two fractional values to express pi: 22 7 is called the approximate rate, and 355 113 is called the dense rate. In Europe, it was not until 1,000 years later, in the sixteenth century, that the Germans Otto (1573 AD) and Antoniz obtained this value.

Nowadays there are electronic computers.

Pi has been calculated to more than three million decimal places.

The area is minimal.

Assuming that the circumference is l, then with a certain circumference, the area of the square in the quadrilateral is up to (l 4) 2, which is (4 l) squared, and the area is 16 l. squared. And the radius of the circle is l 2, then the area of the circle is the square of l divided by 4. 4 is less than 16, so the area of the circle is larger than the square, and when the circumference is fixed, the area of the circle is the largest.

A circle is the closest thing to a spherical shape. Ball area = 3Square of radius = 3

The square of the radius. The circumference of the circle = @2 The radius of the circle = @2r so the area of the circle = @ r2. The circumference is the same, and the circle area is the largest.

Because the circle is the closest sphere, and the area of the sphere is already known, the circle area is the largest.

Because the circle is the closest sphere, the area of the sphere is already known, so the area of the circle is the largest.

Circle area = r2 [Because the circle is the closest approximate sphere, the area of the sphere is already known.]

The area of a regular n-sided is 1 2nr sin = nr tan 2, and the number of internal angles is (n-2)*180 n, so the larger the n, the larger the area.

There is an answer on Zhihu, let's see for yourself.

According to the parallelogram.

The area derivation formula shows that when the circumference is equal, the area of the parallelogram must be smaller than that of the square and the rectangle.

From this, the circle, the square and the rectangle are compared with each other in terms of circumference, which figure has the largest area;

Let the radius of a circle be 1, its circumference is , the area is , and the area of a square equal to its circumference is: (, the area of a rectangle equal to its circumference is: , let the length and width of this rectangle be a and b: respectively

Take some numbers (,,1,,...1), (It can be found that the closer the length and width of the rectangle, the larger the area, when the length and width are equal, that is, it becomes a square, so the area of this rectangle must be smaller than the area of the square.)

So in the case of equal circumference, the area: circle, square, rectangle, parallelogram

When the circumference is equal, the area with the largest area is indeed a circle, and this can be deduced from mathematics.

There is no why, this is the truth!!

The math is there, and there's no need to ask why.

Summary. Triangle.

Reason: Let the radius of a circle be 1, its circumference is , and the area of a square with an area equal to its circumference is:(

The area of a rectangle equal to its circumference is: , let the length and width of this rectangle be a, b and take some numbers (,,1,,...1), (It can be found that the closer the length and width of the rectangle, the larger the area, when the length and width are equal, that is, it becomes a square, so the area of this rectangle must be smaller than the area of the square.)

Why the area of the circle is the largest when the circumference is equal.

Hello, I'm glad to answer for you, and find out that in the case of equal circumference, the closer to the circle, the larger the area of the figure. Triangle Reason: Let the radius of a circle be 1, its circumference is , and the area of a square with an area equal to its circumference is:

The area of a rectangle equal to its circumference is: , let the length and width of this rectangle be a, b and take some numbers (,,1,,...1), (It can be found that the closer the length and width of the rectangle, the larger the area, when the length and width are equal, that is, it becomes a square, so the area of this rectangle must be smaller than the area of the square.)

Hope it helps.

In the case of equal circumference, the area of the figure closer to the circle will be: Circle, Square, Rectangle, Triangle Reason: Let the radius of a circle be 1, its circumference is , and the area of a square with the same circumference is:

The area of a rectangle equal to its circumference is: 6....

The circle has the largest area.

The analysis process is as follows:

Let the length of the wire be 4a.

Then the side length of the square is a, then the length of the rectangle is a+m, the width is a-m, and the area of the square: a*a=a

Area of the rectangle: (a+m)*(a-m)=a-m The circumference of the circle is 4a, 2 r=4a, and r=4a (2). then the area of the circle is 16a (4 )4a

4a²/πa²＞a²-m²。Therefore, the circumference is a figure of 4a, and the area of the circle is the largest.