How Many Diameters Does A Circumference Have?

A circle has infinite diameters. It is easy to see that this is the case, if we start from the definition of diameter, which is the segment that passes simultaneously through the center of the circumference and through two points on it.

In the following figure, on the left, the yellow line corresponding to a diameter of the circumference is observed and it divides it into two parts. On the right side, three other diameters were drawn in different colors: blue, green and pink. They all have the same length and satisfy the condition of joining two points of the circumference, passing through the center of it.

The diameter is a distinctive segment, which always passes through the center of the circumference and joining two points of it. On the left, the yellow diameter divides the circumference in two. On the right are other diameters that equally divide the circumference into two halves. Source: F. Zapata.

As you can see, the possibilities for drawing diameters are infinite, since the points that make up the circumference are infinite. The same can be said about the radius, which is the segment that joins any point on the circumference with the center: an infinite number of radii can be drawn.

And by placing two radii opposite each other, a diameter is obtained, since the radius measures half of this.

Circumference diameter, radius and length

Let D be the diameter of any circumference and R its radius. Since the diameter is twice the radius, it can be written:

D = 2 ∙ R

It means that if the radius of a circumference is, for example, R = 5 cm, the diameter of the circumference is D = 2 ∙ 5 cm = 10 cm.

The diameter is also known as the major chord . Chords are lines or segments that are drawn between any two points on the circumference, but do not necessarily pass through the center. Only the diameter has that distinction.

In the following figure you can see the difference and see why, in effect, the diameter (red) is the largest of the chords that can be drawn on the circumference:

Different strings in a circumference: the diameter is the largest of all. Source: F. Zapata.

Of course, the measure of the diameter (and therefore the radius) is the same in a given circumference. By varying, you get a smaller or larger circumference, depending on.

On the other hand, the size of the strings of the same circumference varies, depending on how far or close the points that are joined are. In the example shown, the green “c” string is noticeably shorter than the “a” and “b” strings.

And the number of strings that can be drawn is also infinite.

The perimeter of the circumference

For its part, the length of the circumference is equivalent to its perimeter or contour. It is related to its diameter, since the larger it is, the larger the circumference (its perimeter is greater).

The ratio or quotient between perimeter and diameter is a constant called π (read “pi”). The value of π is 3.141592… The ellipsis indicates the number of decimal places and infinite, which is because pi is an irrational number. However, for practical purposes, pi can simply be rounded to 3.14

If the perimeter is denoted as C and the diameter as D, this ratio is stated as follows:

C / D = π

Therefore, the formula for the length of the circumference is:

C = π ∙ D

Or if you prefer depending on the radius R:

C = 2π ∙ R

Illustrative example

In the image three identical circles are shown, designated by the letters A, B and C. In each one, the ant travels the path on the segments of the blue color, to go from one point to another on the circumference.

The ant moves from one point to another on the circumference through the thinnest blue lines. Source: F. Zapata.

1.- In which of the cases does the displacement occur exactly on the diameter of the circumference?

Answer

Only in case A, since this path passes through the center of the circumference and goes from one point of the circumference to another.

2.- How many radii does the insect travel in each case?

Answer

In any of the three cases the ant travels two radii of the circumference.

3.- Which of the routes corresponds to the greatest distance?

Answer

The path is equal in length in each case, equivalent to two radii of the circumference.

4.- In which case is the ant farthest from its starting point?

Answer

In case A, since it is at the point that is just in front of it, in the remaining cases, the ant is closer to the starting point.

5.- And in which case does the ant end the route being closer to its point of origin?

Answer

In the case C.

6.- If the radius of the circumferences is 20 cm, how many centimeters is each path?

Answer

Since the ant travels a distance equal to two radii, the total distance traveled is 40 cm.

Solved exercises

Exercise 1

Calculate the radius of a circle whose diameter is 4.5 m.

Solution

The radius is half the diameter, if the diameter is 4.5 cm, then the radius R is 2.25 cm.

Exercise 2

Find the perimeter of the circumference in Exercise 1.

Solution

Perimeter is calculated through diameter or through radius, whichever is preferred. In the first case:

C = π ∙ D

Assuming that π = 3.14 to round, substituting values ​​gives the following result:

C = π ∙ D = 3.14 ∙ 4.5 cm = 14.13 cm

Exercise 3

A designer is asked to draw a logo in the shape of a heart, with the dimensions indicated in the figure. The curved parts correspond to exact semicircles.

With this information respond:

  1. What is the radius of the semi-circles?
  2. What is the perimeter of the heart?

Solution to

The diameter of the semicircles equals the side of the square, which is 3 cm. Therefore, its radius, which is half, is 1.5 cm.

Solution b

The perimeter of the heart-shaped figure is the sum of the two outer sides of the square and the perimeter of the semicircles. As they are identical, their perimeter is equivalent to that of a complete circumference of diameter 3 cm:

C = π ∙ D = 3.14 ∙ 3 cm = 9.42 cm

Therefore, the perimeter P of the figure is:

P = 9.42 cm + 3 cm + 3 cm = 15.42 cm

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