- Main differences between a circle and a circumference
- Definitions
- Cartesian equations
- Graphs on the Cartesian Plane
- Dimensions
- Three-dimensional figures that generate
- References
A circle and a circumference are two very similar geometric concepts, however they make mention of two different objects. On many occasions, the mistake of calling a circumference a circle is made and vice versa. This article will mention some differences between these two concepts.
These concepts are different in several aspects such as: their definitions, the Cartesian equations that represent them, the region of the Cartesian plane that they occupy and the three-dimensional figures that they form.
To notice the differences in terms of drawing a circle and a circumference, it is convenient to use colors when drawing them.
Main differences between a circle and a circumference
Definitions
Circumference: a circle is a closed curve such that all the points of the curve are at a fixed distance "r", called the radius, from a fixed point "C", called the center of the circumference.
Circle: it is the region of the plane that is delimited by a circle, that is, they are all the points that are within a circle.
It can also be said that a circle is all the points that are less than or equal to "r" from the point "C".
Here you can see the first difference between these concepts, because a circle is just a closed curve, while a circle is the region of the plane enclosed by a circle.
Cartesian equations
The Cartesian equation that represents a circle is (x-x0) ² + (y-y0) ² = r², where "x0" and "y0" are the Cartesian coordinates of the center of the circle and "r" is the radius.
On the other hand, the Cartesian equation of a circle is (x-x0) ² + (y-y0) ² ≤ r² or (x-x0) ² + (y-y0) ² <r².
The difference between the equations is that in the circumference it is always an equality, while in the circle it is an inequality.
A consequence of this is that the center of a circle does not belong to the circumference, while the center of a circle always belongs to the circle.
Graphs on the Cartesian Plane
Due to the definitions mentioned in item 1, it can be seen that the graphs of a circle and a circle are:
In the images you can see the difference that was mentioned in item 1. In addition, a distinction is made between the two possible Cartesian equations of a circle. When the inequality is strict, the edge of the circle is not included in the graph.
Dimensions
Another difference that can be noticed is with respect to the dimensions of these two objects.
Since a circumference is just a curve, this is a one-dimensional figure, therefore it only has length. A circle, on the other hand, is a two-dimensional figure, therefore it has length and width, so it has an associated area.
The length of a circle of radius "r" is equal to 2π * r, and the area of a circle of radius "r" is π * r².
Three-dimensional figures that generate
If the graph of a circle is considered, and it is rotated around a line that passes through its center, a three-dimensional object will be obtained which is a sphere.
It should be clarified that this sphere is hollow, that is, it is only the edge. An example of a sphere is a soccer ball because inside it there is only air.
On the other hand, if the same procedure is performed with a circle, a sphere will be obtained but it is filled, that is, the sphere is not hollow.
An example of this filled sphere could be a baseball.
Therefore, the three-dimensional objects that are generated depend on whether a circumference or a circle is used.
References
- Basto, JR (2014). Mathematics 3: Basic Analytical Geometry. Grupo Editorial Patria.
- Billstein, R., Libeskind, S., & Lott, JW (2013). Mathematics: A Problem Solving Approach for Elementary Education Teachers. López Mateos Editors.
- Bult, B., & Hobbs, D. (2001). Lexicon of mathematics (illustrated ed.). (FP Cadena, Trad.) AKAL Editions.
- Callejo, I., Aguilera, M., Martínez, L., & Aldea, CC (1986). Maths. Geometry. Reform of the upper cycle of the EGB Ministry of Education.
- Schneider, W., & Sappert, D. (1990). Practical manual of technical drawing: introduction to the fundamentals of industrial technical drawing. Reverte.
- Thomas, GB, & Weir, MD (2006). Calculation: several variables. Pearson Education.