How to find the area of ​​a circle by multiplying the sides?

How to find the area of ​​a circle by multiplying the sides?

Some math lovers have shared comments over the phone that it is frustrating not being able to find the area of ​​a circle by multiplying adjacent sides. They also questioned how such a system could not exist.

And they questioned me to the punch with how I could not find the area of ​​a circle by multiplying the sides by saying that they could derive various mathematical facts from a certain mathematical fact.

It is fair for those math lovers to ask. What if there is no side to the circle? If the circle is enclosed in a square, then the side of the circle contained in the square will be found.

If we find the area of ​​the square and then find a ratio to the area of ​​the circle it contains, we can find the area of ​​the circle by multiplying the sides as we think. But we must not forget that here we are taking the sides of the square as the circle has no sides. It must always be remembered that we find a proportionality to the circle contained within the square by so taking it.

Shall we do that now? How? Let's draw a perfect circle touching all four sides of the square. That is, as shown in the figure below,

Is the diameter of the circle we have drawn equal to the side of the square? Yes, what is the doubt?

Is the diameter of a circle twice the radius? Yes, what do you doubt about that? So the diameter of the circle is d = 2r = a units? That means twice the radius of a circle is equal to the diameter and the side of a square? Yes, what do you doubt about that?

Then we are done.

Now what is the area of ​​the square? That is, the area of ​​a square having twice the radius of the circle i.e. 2r as side i.e. a is 2r × 2r = 4r² square units as per side × side formula?

What this 4r² square units means is that there are four squares inside the square with sides of r units. That is, if we put two diameters in the circle in the picture we have drawn, you will understand. See it for yourself in the picture below.

Yeah right you say? Now is the square divided into four equal parts by the radii of the circle? Each congruent is a square of side r. Since there are four smaller squares inside, is it correct that the area of ​​four is 4r² square units?

Only a square is divided into four equal parts? Is it a circle? Areas of four equilateral segments of a circle lie within four equilateral squares. Then its area will be less than 4r² square units. That means something in the ratio between 3r² square units and 4r² square units.

We are going to use the constant π associated with the circle to determine what that ratio is. you understand now? Right, the area of ​​four equilateral circles inside a square is 3.14 × r². That is, if you subtract the area of ​​the square outside the circle, it will be the same size. The area of ​​that subtracted area is approximately 0.86 r².

Oh, you understand now? So 3.14 r² be πr² square units. That's it, we have arrived at the formula for the area of ​​a circle by squeezing the circle inside a square and multiplying the adjacent sides by it.

Aha, isn't there anything wrong with saying that we can derive various mathematical facts from one mathematical fact?

That's it, now I hear you ask, are you done with the area of ​​the circle?

Before that we have only one calculation about the circumference and area of ​​a circle and next we will look at the method of finding the area of ​​a quadrilateral. So I don't hear you saying we'll see about it tomorrow? Well, we'll see tomorrow.

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