The area formula of a circle (how is the area formula of a circle obtained?)

For any circle, its area s is equal to the product of pi and the square of the radius r 2. In other words, the ratio of the area of any circle to the square of its radius is the same constant—π. So, is this conclusion rigorously proved by mathematics, or is it a mathematical intuition? In fact, the formula for the area of a circle (S = R 2) can be proved strictly mathematically, and it has been proved by mathematicians in ancient China and ancient Greece. There are many ways to prove the circle formula. Here are a few examples.

# (1) Restriction method 1

If a circle is divided into N equal parts, and then stitched into the following quadrilateral:

When n approaches infinity, that is, the circle is divided into infinitely many equal parts, then the quadrilateral becomes a rectangle. Obviously, the length of this rectangle is the circumference (R) of the semicircle, and the width is the radius (R) of the circle. The area of the rectangle is equal to the area of the circle. The formula for the available circle area is: S=r? r=r^2.

But in order to complete this proof, you first need to prove the pi formula (C=2r). Through the principle of similar triangles, it is easy to prove the constant that the ratio of the circumference of a circle to its diameter is equal by geometrical methods, and this constant is pi.

# (2) Limit method 2

Divide the circle into n equal parts, and connect the intersection points of the radius and the circle in each sector. Suppose the central angle of each sector is 2, then 2 = 2=2/n n.

Examine one of the trigonometric OABs, which can be obtained according to trigonometric functions. OC=rcos, AB=2rsin, the area of the triangle OAB is:

S△OAB=1/2ABOC=r^2sincos

When n tends to infinity, the area of the circle can be Expressed as:

S=lim(n→+∞)nS△OAB

According to the limit principle, s = r 2 can be calculated.

# (3) Integral method 1

Strictly speaking, this is also a limit method, but here the area of the circle is strictly according to the equation of the circle (x ^ 2+y ^ 2 = r^ 2) Calculated by:

# (4) Integral method 2

If the circle is divided into countless thin rings with a thickness of dr, then the area of each ring is 2rdr. Through integration, we can get :

In short, the ratio of the area of a circle to the square of its radius is π, which is a strict mathematical proof, not an empirical formula.