Circles

A circle is a curve that always has the same distance from a given point, the center of the circle. Circles are also classified as conics.

Description of a circle

Look at the picture of a circle:

A circle is usually denoted by the lower case letter k or l.

• Each circle has a centre, denoted by S.
• All points of a circle have the same distance from the centre S, this is called the radius of the circle and is denoted by r. In the picture it is the line segment AS.
• The line joining two different points on the circle is called the bowstring. In the figure, the line segment FG.
• The chord that passes through the center of S is called the diameter of the circle and is denoted by d. It is true that 2r = d.

Next, we talk about the inner and outer regions of the circle. The inner region is the set of all points that have a distance from the centre less than the radius of the circle. The outer region is all points that have a distance from the center greater than the radius. If we unify the circle and the inner region of the circle, we get a circle.

Perimeter and content

What is the relationship between the diameter of a circle and the circumference of a circle? How many times is the circumference of a circle greater than its diameter? If you take a circle and try to measure its diameter and then its circumference, you will find that it is approximately three times. More precisely 3,1415-times and even more precisely π-times.

The number π (read pi) is called the Ludolph number. It is an irrational number, that is, a number with an infinite decimal expansion with no period. Thus, if you have the diameter of a circle fixed as some integer, you will never find its circumference quite accurately. In practical life, however, it is abundantly sufficient to know some approximate value.

The formula for the circumference of a circle looks like this:

$$o=\pi d=2\pi r$$

The formula for the circumference of a circle:

$$S=\pi r^2$$

The relative positions of two circles

• Concentric circles have a common centre S. Non-concentric circles are then circles that do not have a common centre. The line joining their centres is called the central line.

• Circles k₁ lie in the outer region of k₂, circles have no common point.

• The circles k₁ lie in the outer region of k₂, but they touch at one point.

• The circles intersect at two points.

• The circles k₁ lie in the inner region of k₂ and touch at one point.

• The circle k₁ lies in the inner region of k₂.

The arcs

Any chord that has extreme points A and B, divides the circle into two parts called arcs. Each point on the circle, except the points A and B, is then an interior point of one of the arcs. An arc is then denoted, for example, by AXB, where X is an interior point of an arc.

If the bowstring is also a diameter, it divides the circle into two congruent arcs, which we call semicircles or semicircles. Otherwise, the circle is always divided into two differently sized arcs.

The central angle

Consider a circle k centered at S and an arc AB. Then we call the angle ASB the central angle over the arc AB. Above the smaller arc is a convex central angle (the angle is less than $180^{\circ}$), and above the larger arc is a nonconvex angle (more than $180^{\circ}$).

The circle in the figure has two arcs marked AB. The larger arc has the angle α, the smaller β.

The perimeter angle

Consider the circle k with the centre at S and the arc AB. Next, choose a point V, which does not belong to this arc but is on the circle. Then the angle AVB is called the circumcircular angle.

In the figure there are two perimeter angles, AV₁B and AV₂B. Note that they both have the same magnitude - $60^{\circ}$ - and that this magnitude is half that of the central angle - $120^{\circ}$.