- How to get the diameter?
- Constant width figures
- Diameter of a circumference
- - Example 1
- Solution to
- Solution b
- Solution c
- - Example 2
- Solution
- How many diameters does a circumference have?
- References
The diameter is the straight line that passes through the center of a closed flat curve or a figure in two or three dimensions and that also joins its opposite points. It is usually a circle (a flat curve), a circle (a flat figure), a sphere or a right circular cylinder (three-dimensional objects).
Although circumference and circle are usually taken as synonyms, there is a difference between the two terms. The circumference is the closed curve that encloses the circle, which meets the condition that the distance between any of its points and the center is the same. This distance is none other than the radius of the circumference. Instead, the circle is a flat figure bounded by the circumference.
Figure 1. The diameter of bicycle wheels is an important feature in their design. Source: Pixabay.
In the case of circumference, circle and sphere, the diameter is a straight segment that contains at least three points: the center plus two points of the edge of the circumference or circle, or the surface of the sphere.
And as for the right circular cylinder, the diameter refers to the cross section, which together with the height, are its two characteristic parameters.
The diameter of the circumference and the circle, symbolized by ø or simply the letter “D” or “d”, is related to its perimeter, contour or length, which is denoted by the letter L:
L = π.D = π. or
Whenever there is a circumference, the quotient between its length and its diameter is the irrational number π = 3.14159…, in this way:
π = L / D
How to get the diameter?
When you have the drawing of the circumference or the circle, or directly the circular object, such as a coin or a ring for example, it is very easy to find the diameter with a ruler. You just have to make sure that the edge of the ruler touches two points on the circumference and the center of it at the same time.
A caliper, vernier, or caliper is very suitable for measuring external and internal diameters on coins, hoops, rings, nuts, tubes, and more.
Figure 2. Digital vernier measuring the diameter of a coin. Source: Pixabay.
If instead of the object or its drawing we have data such as the radius R, then multiplying by 2 we have the diameter. And if the length or perimeter of the circumference is known, the diameter can also be known, by clearing:
Another way to find the diameter is by knowing the area of the circle, the spherical surface, the cross section of the cylinder, the curved area of the cylinder, or the volumes of the sphere or cylinder. It all depends on what geometric figure it is. For example, diameter is involved in the following areas and volumes:
-Area of the circle: π. (D / 2) 2
-Area of the spherical surface: 4π. (D / 2) 2
-Volume of the sphere: (4/3) π. (D / 2) 3
-Volume of the right circular cylinder: π. (D / 2) 2.H (H is the height of the cylinder)
Constant width figures
The circle is a flat figure of constant width, since wherever you look at it, the width is the diameter D. However, there are other perhaps less known figures whose width is also constant.
First, let's see what is understood by the width of a figure: it is the distance between two parallel lines -support lines-, which in turn are perpendicular to the given direction and which imprison the figure, as shown in the left image:
Figure 3. Width of any flat figure (left) and Reuleaux triangle, a figure of constant width (right). Source: F. Zapata.
Next to the right is the Reuleaux triangle, which is a figure of constant width and which meets the condition specified in the left figure. If the width of the figure is D, its perimeter is given by Barbier's theorem:
L = π.D
The sewers of the city of San Francisco in California are shaped like a Reuleaux triangle, named for the German engineer Franz Reuleaux (1829 - 1905). In this way the lids cannot fall through the hole and less material is used to manufacture them, since their area is less than that of the circle:
A = (1- √3).πD 2 = 0.705.D 2
While for a circle:
A = π. (D / 2) 2 = (π / 4) D 2 = 0.785. D 2
But this triangle is not the only constant width figure. You can build the so-called Reuleaux polygons with other polygons that have an odd number of sides.
Diameter of a circumference
In the next figure are the elements of the circle, defined as follows:
Chord: line segment that joins two points on the circumference. In the figure is the chord that joins points C and D, but infinite chords can be drawn that join any pair of points on the circumference.
Diameter: it is the chord that passes through the center, joining two points of the circumference with the center O. It is the longest chord of a circumference, for that reason it is called the “major chord”.
Radius: line segment that joins the center with any point on the circumference. Its value, like the diameter, is constant.
Circumference: it is the set of all points equidistant from O.
Arc: it is defined as a circumference segment delimited by two radii (not drawn in the figure).
Figure 4. Parts of the circumference, including the diameter, that passes through the center. Source: Wikimedia Commons.
- Example 1
The rectangle shown is 10 inches tall, which when rolled forms a right circular cylinder whose diameter is 5 inches. Answer the following questions:
Figure 5. A rolled rectangle becomes a right circular cylinder. Source: Jiménez, R. Mathematics II. Geometry and trigonometry. 2nd. Edition. Pearson.
a) What is the contour of the tube?
b) Find the area of the rectangle.
c) Find the cross-sectional area of the cylinder.
Solution to
The outline of the tube is L = π.D = 5π in = 15.71 in.
Solution b
The area of the rectangle is base x height, with the base L already calculated and the height is 10 inches according to the statement, therefore:
A = 15.71 in x 10 in = 157.1 in 2.
Solution c
Finally, the requested area is calculated like this:
A = π. (D / 2) 2 = (π / 4) D 2 = (π / 4) x (5 in.) 2 = 19.63 in. 2.
- Example 2
Calculate the shaded area in Figure 5a. The square has side L.
Figure 6. Find the shaded area in the left figure. Jiménez, R. Mathematics II. Geometry and trigonometry. 2nd. Edition. Pearson.
Solution
In figure 5b two identical size semicircles have been drawn in pink and blue, superimposed on the original figure. Between them they make a complete circle. If you find the area of the square and subtract the area of the circle, you make the shaded area in Figure 5b. And looking closely, it turns out that it is half of the shaded area in 5a.
-Square area: L 2
-Diameter of the semicircle: L
-Area of the circle: π. (L / 2) 2 = (π / 4) L 2
-Difference of areas = half of the shaded area =
L 2 - (π / 4) L 2 = L 2 = 0.2146 L 2
-Shaded area = 2 x 0.2146 L 2 = 0.4292L2
How many diameters does a circumference have?
You can draw infinite diameters on a circle, and any of them measure the same.
References
- Antonio. Reuleaux triangles and other constant width curves. Recovered from: divulgators.com.
- Baldor, A. 2002. Plane and Space Geometry and Trigonometry. Patria Cultural Group.
- Jiménez, R. Mathematics II. Geometry and trigonometry. 2nd. Edition. Pearson.
- Wikipedia. Reuleaux triangle. Recovered from: es.wikipedia.org.
- Wolfram MathWorld. Diameter. Recovered from: mathworld.wolfram.com.