How to Remove the Perimeter from a Circle?

He Perimeter of a circle Is the value of its circumference, which can be expressed through a simple mathematical formula.

In geometry, the perimeter is called the sum of the sides of a flat figure. The term comes from the Greek where Peri Means around and meter measure. The circle consists of only one side, having no edges, is known as circumference.

A circle is a defined area of ​​a plane, delimited by a circumference. The circumference is a flat and closed curve, where all its points are at the same distance from the center.

As shown in the picture, this circle is composed of a circle C, which delimits the plane, at a fixed distance from the center point or origin O. This fixed distance from the circumference to the origin, is known as radius.

The image also shows D, which is the diameter. It is the segment that joins two points of the circumference through its center and has an angle of 180 °.

For the calculation of the perimeter of a circle, the function is applied:

• P = 2r · π if we want to calculate it as a function of radius
• P = d · π if we want to calculate it as a function of diameter.

These functions mean that if we multiply the value of the diameter by the mathematical constant π, which has an approximate value of 3.14. We get the length of the circumference.

Demonstration of the circle perimeter calculation

The proof of the calculation of the circumference is made through geometric figures inscribed and circumscribed. We consider that a geometric figure is inscribed within a circumference when its vertices are on the circumference.

The geometric figures that are circumscribed are those in which the sides of a geometric figure are tangent to the circumference. This explanation is much easier to understand visually.

In the figure we can see that the sides of square A are tangent to the circumference C. Also, the vertices of square B are on the circumference C

To continue with our calculation, we need to obtain the perimeter of the squares A and B. Knowing the value of the radius of the circumference, we can apply the geometric rule in which the sum of the square hinges is equal to the square hypotenuse. In this way, the perimeter of the inscribed square, B, would be equal to 2r ² .

To prove this, we consider r as radius and h ₁ , The value of the hypotenuse of the triangle that we form. Applying the previous rule we have h ₁ ² R ² R ² = 2r ² . When we obtain the value of the hypotenuse, we can obtain the value of the perimeter of square B. To facilitate the calculations later, we will leave the value of the hypotenuse as the square root of 2 by r.

For the calculation of the perimeter of the square A the calculations are simpler, since the length of one side is equal to the diameter of the circumference. If we calculate the average length of the two squares, we can make an approximation of the value of the circumference C.

If we calculate the value of the square root of 2 plus 4, we get an approximate value of 3.4142, this is higher than the number π, but because we have only made a simple adjustment to the circumference.

To get closer values ​​and more adjusted to the value of the circumference, we will draw geometric figures with more sides to make it a more correct value. Through octagonal shapes the value is adjusted in this way.

Through the calculations of the sine of α we can obtain b ₁ And b ₂ . Calculating the approximate length of both octagons separately, then we do the average to calculate the one of the circumference. After the calculations, the final value we obtain is 3.3117, which is closer to π.

Therefore, if we continue to make our calculations until we arrive at a figure of n faces, we can adjust the length of the circumference and arrive at an approximate value of π, that makes that the equation Of C = 2π · r.

Example

If we have a circle with a radius of 5 cm, to calculate its perimeter we apply the formulas shown above.

P = 2r · π = 2 · 5 · 3.14 = 31.4 cm.

If we apply the general formula, the result obtained is 31.4 cm for the length of the circumference.

We can also calculate it with the diameter formula, which would be:

P = d · π = 10 · 3.14 = 31.4 cm

Where d = r + r = 5 + 5 = 10

If we do it through the formulas of the squares inscribed and circumscribed, we need to first calculate the perimeter of both squares.

To calculate the square A, the side of the square would be equal to the diameter, as we saw before, its value is 10 cm. To calculate the square B, we use the formula where the sum of the square hinges is equal to the square hypotenuse. In this case:

H ² R ² + R ² = 5 ² +5 ² = 25 + 25 = 50

H = √ 50

If we include it in the formula of averages:

As we can see, the value is very close to that made with the normal formula. If we adjusted through more expensive figures, the value would become closer and closer to 31.4 cm.

References

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