Circles and rectangles with equal circumferences, why is the area of the circle large?

Let the sides of the rectangle be a and b, and the perimeter l.

2（a＋b）＝l

Area s ab (a b) 2 2 l l 16 circle radius r.

2 r l area s r r l l 4

So the area of the circle is larger.

Circles and squares with equal circumferences are both x

Then: radius of the circle = x (2 ).

Square area = (x 4) = x 16

The area of the circle = *x (2 ) = x 4 So: the area of the circle is large.

Circumference c, then the side length of the square is: c divided by 4, and the area is (c divided by 4) * (c divided by 4) = the square of c divided by (4*4).

The radius of the circle is: c divided by 2

The area is divided by 2) * (c divided by 2) = c squared divided by (4 * contrasted, circles with equal circumference and squares, the area of the circle is greater than the area of the square.

For a rectangle with equal circumference, the closer the length and width are, the larger the area, so the area of a rectangle with equal circumference is less than the area of a square.

So: a circle with an equal circumference and a rectangle, the area of the circle is large.

Even: Of all shapes with equal circumferences, the circle has the largest area.

The principle is the same as earning pockets. If you want the mouth of the pocket to be larger, make the mouth of the pocket as round as possible.

When the circumference is equal, the area of the circle is also larger when calculated using the area formula.

You see, the circumference of the circle is 28 meters, and the area is square meters. The rectangle is 10 meters long and 4 meters wide, with an area of only 40 square meters. You say, is it a rectangle with a large area or a circular area?

Hehe! See you get it.

This question is similar to why 1+1 is equal to 2, that is, it is equal to 2 and why is not specified.

Circle, reason: Let the radius of the circle be r and the side length of the square be a, then:

The square of a = the square of r.

So a=root number r.

Circumference of the circle = 2 r(1).

The circumference of the square = 4a = 4 root number r(2).

1) The ratio of (2) is 1 at the root of 2, so the circumference of the square is larger.

Perimeter formulaCircle: c = d = 2 r (d is the diameter, r is the radius, ) triangle. The circumference of c = a+b+c (abc is the three sides of the triangle) quadrilateral.

c = a + b + c + d (abcd is the side length of the quadrilateral) rectangle: c = 2 (a + b) (a is long, b is wide) square: hood xian xiang c = 4a (a is the side length of the square) polygon contains shape:

c = sum of all edge lengths.

Circumference of the fan: c = 2r + n r 180 (n = central angle.

Angle) =2r+kr (k=radians.

Let the regular polygon have n sides, and the length of the circumference is c, then the length of each edge is c n, and the area of the footprint can be obtained s=(c 2) (4n*tan(pi n)), pi is pi, and it can be proved that this is an increasing function, when n tends to positive infinity, it is a circle, and the area is the largest at this time.

Assuming the perimeters are all 16, then.

s circle = s length = 5x3 = 15

The circumference is equal, and the area of a circle is larger than that of a rectangle.

Let the perimeter = m, the radius of the circle = m 2, the area of the circle = (m 2 ) 2 = m 2 4 , the square side length = m 4, the square area = m 2 4 2, because 4 < 4 2, so the circle area "square area".

Let the circumference be 8, then the square area is 4 and the circle area is 5, so it is correct.