Center and Parts of a Circle

Circles are not polygons. A circle is defined by its center and radius. More specifically, a circle can be defined as the set of all points equidistant to a given point:

Note that a circle only consists of these points. Point B is on the circle, but point A is not. We say point A is in the circle. In this picture, A is the center of the circle. $\overline{AB}$ is a radius of this circle. A radius is a line from the center to any point on the circle. A circle has an infinite number of radii. A chord is any line segment with endpoints on the circle. A diameter is a chord passing through the center. A secant is any line that intersects the circle twice. A tangent is a line that intersects the circle only once. A trenvent intersects the circle three times. Trenvents do not exist. If you see one, report it to the NSA immediately.

We say that a polygon is inscribed in a circle when all its corners are on the circle.

An arc is a closed segment of a circle. It is determined by a pair of points on the circle.

$\theta$ is the measurement of the angle between the two ends of the arc. L is the length of the arc. r is the radius of the circle

A circle is the set of all points that are an equal, constant distance from a central point. Circles are two-dimensional figures that are not polygons.

To construct a circle, two things are needed: a center and a radius. The center is the point from which all of the points on the center are the same distance; keep in mind, however, that the center is not a point on the circle. The radius is the exact distance from the center to any point on the circle.

The circle is then constructed by connecting the points that are a radius length apart from the center.

The perimeter of the circle, called the circumference of the circle, has a special property: it is directly proportional to the radius. In all circles - no matter their size - the circumference is equal to 2$\pi$ times the radius. To learn more about $\pi$, check out the expii topic here.

Finally, let's talk about arcs! An arc is a segment of the circle that is proportional to an angle; check out the animation below to see how the arc (s) is constructed from the radius (r) and angle ($\theta$).

To learn more about arcs and angles, check out the expii topic here.

When studying Geometry it is important to know the properties of the different shapes and figures that you may be studying. In this case we will discuss the properties of a circle. Let's begin with the definition of a circle. A circle is a curved line that creates a closed loop. This line is made up of infinitely many points and each of these points are an equal distance to the middle.

Properties

A circle has many parts and properties which are listed below.

• Center: The center of a circle is defined as the point in the middle of the circle. The points that make up the curve that is the circle are all equidistant from the center point.

• Radius: The radius is the distance from the center point to the circle, this distance is always equal in a circle. It is half the diameter. When working with the radius in geometry it is often denoted by $r$.

• Diameter: The diameter is the distance from one side of the circle to the other side of the circle. This distance is twice the radius. When working with the radius in geometry it is often denoted by $d$. It is important to remember that $d=2r$.

• Circumference: The circumference of a circle is the distance of the line that makes up the circle. Circumference is denoted by $C$ and can be calculated using the following formula: $C=2\pi r$.

• Area: The area of the space enclosed by the circle can be calculated with the following formula: $A_c=\pi r^2$.

• Chord: A chord is a line segment that links any two points on a circle.

• Tangent: A tangent line is a line that touches a circle at one point.

• Secant: A secant line is a line that goes through a circle and intersects the circle at two points.

Similar to a quadratic equation, a specific type of equation will create a circle when graphed. Unlike quadratic equations, the $x$ and $y$ terms are always squared. The equation of a circle is as follows: $$(x-h)^2+(y-k)^2=r^2,$$ where $h$ and $k$ are the $x$ and $y$ coordinates of the center point of the circle.

A circle is the set of all points in a plane equidistant to a given point in that plane.

Consider the figure below. We can clearly see that all points in this particular circle are exactly one unit away from the origin (0, 0).

All circles have a center and a radius. Looking at the figure below, we see the center is shown by the blue point, while the radii are shown by the green lines.

Note that all radii of a circle are congruent.

Now that we know what the center and the radius of circles is, we can define what a diameter is.

The diameter is two radii. All diameters of the same circle must have the same length.

Looking at the figure we've been dealing with, we see that our diameter is $2$. This is because we simply added the radii together $\rightarrow$ $1+1=2$

Now let's consider arcs. An arc is some section of a circle. Arcs are measured in degrees. In the figure below, our arc $AC$ is equal to $\angle{ABC}$ that is $90^\circ$

Putting these terms together, we have a fully labeled circle.

This is a circle (see diagram below)

A circle is always round.

The portion between the magenta points is an arc.

This is an angle. It is made up of two rays with the same starting point.(pretend that there are arrows at the end:) )

Was that hard?

A circle is nothing but a round closed shape consisting of various points on it's circumference, which are equidistant from a fixed point that is its centre point.

(Video) Introduction to Circles

by Art of Problem Solving

Here's a great introductory video to circles from Art of Problem Solving!