Is a circle a face or a closed curve?

From the perspective of mathematics, a circle refers to a closed curve, that is, the circumference, referred to as a circle. This can be made clear from the following three definitions of a circle:

First, (geometrically speaking) the figure composed of all points on the plane whose distance from the fixed point is equal to the fixed length is called a circle.

The second is (trajectory theory) that the trajectory of a moving point on a plane with a certain point as the center and a certain length of distance is called a circle, or a circle for short.

The third is (set theory) the set of points whose distance to a fixed point is equal to a fixed length is called a circle. In addition, in analytical geometry, circles are also regarded as special conic sections, which can be expressed by equations. It can be seen that a circle is a curve. Primary school students will popularly say "a circle is a circle", but obviously a circle is not a mathematical language, just like a square cannot be said to be a square. In teaching, students must be guided to standardize expressions.

In Chinese, circle also refers to "the plane surrounded by the circumference. Here we will temporarily give it another name as "circle surface". Many disputes about circles are actually because we use the word "circle" "This word represents the result of the two concepts of "circumference" and "circle surface".

Usually the "circle" that people are used to referring to refers to the "circle surface", which is a flat surface. A figure represents a circle and the plane part it encloses. A circle has an area.

When talking about "circle" in primary school mathematics teaching, we often switch between the two concepts of "circumference" and "circle surface", which also generates a lot of controversy.

Therefore, some experts suggest that "the plane surrounded by the circumference (that is, the circular surface tentatively named earlier)" is called "circle". This can make the two concepts clearly distinguished and avoid ambiguity. However, As teachers, we should pay more attention to the growth of students' knowledge, the cultivation of spatial concepts and the formation of abilities in the process of understanding circles, rather than just giving a certain judgment

Extended information

"Circle" is a common geometric figure in life, but in life, "circle" is often not shown in its true colors. For low-level students, they have been told that the sun is round and the earth is round. , and then gradually confused "circle" and "ball". Later, I was told that coins are round, buttons are round, and "slightly flat cylindrical objects" were confused with "circle".

For advanced students, when teaching "understanding of fractions", teachers often hold a circular piece of paper and say: "Divide this circle into equal parts..." At this time, another Confusing "circle surface" and "circle"

The concept of a circle is not directly defined, but an intuitive description is used to directly tell students: a figure like this is a circle directly. To derive geometric figures from common objects, the abstraction process from physical objects to figures is essential.

Especially the coin in this example. Students may have thought that the entire coin was a circle before. After hiding the physical picture and leaving the geometric figures, students can further realize that the outline of the coin is a circle, and thus have a more scientific understanding of the circle.