Circle Definitions


What is a circle?
A circle is a shape surrounded by a line that is equidistant (at equal distance) from a point within it, that is called the center.
Let us see what this means.
Imagine a straight object (like a ruler) $OA$, and you fix a pin at the end $O$ and fix a pencil at the other end $A$. The pin and pencil should be tightly fitted with the ruler. Now, keeping the pin fixed on a piece of paper you start moving the pencil in the direction show by the blue arrow.


              

             

-----------book page break-----------
You will start seeing a line like this:
             
           
Keep going and your pencil at $A$ will reach the starting point. Now you have a complete circle, like below:

     


-----------book page break-----------
This same thing can be done very easily with a compass and pencil and in that case $O$ will be the pin of the compass and $A$ will be the tip of the pencil.
Whichever way you do it, what you will have is a circle with center as $O$ and radius (plural radii) as $OA$. The whole curved line that you drew starting at point $A$ and coming back to point $A$, is called the circumference.
Now let's take the same circle and understand different parts of it.

Now let's draw some more straight lines, each starting at the center and ending at some point of your choice on the circumference. Somewhat like the dotted lines shown here:


Each of these lines, $OA$, $OB$, $OC$ and $OD$ will have the same length. Why? Because that is the definition of circle. Each point on the circumference is at the same distance from the center $O$.

-----------book page break-----------
Now let us draw one more straight line passing through the circle. Let this line intersect the circumference of our circle at two points $P$ and $Q$ as shown in the figure below:



The line segment $PQ$ is called a chord of the circle. Now let us draw one more chord, but this time we will draw the chord in such a way that it passes through the center $O$. Let's say this chord intersects the circumference at $M$ and $N$.

    

-----------book page break-----------
The line segment $MN$ has a special name, it is called the diameter. Remember diameter is also a chord, but a special chord that passes through the center. Let us try to find out something more about the diameter.
As we saw before, the distance of $M$ from $O$ and the distance of $N$ from $O$ are the same, and each is equal to the radius.
Since $M$, $N$ and $O$ are on the same straight line, the distance of $M$ from $N$ is twice as much as the radius.
So, we can say:
$diameter\ = 2 \times radius$.

Now let us erase part of our circle, as shown below:


Note that the original circle still can be seen as a dashed line, we didn't use a very good eraser, you can say.
But let us focus on the solid, curved line $RS$ which was part of our original circle. This curved line is called a $Minor\ Arc$. The remaining part of the circumference, which is seen in dashed line, is called the $Major\ Arc$
Since the arc is a part of our circle, the center of this arc remains at $O$, and all points on the curved line $RS$ are at the same distance from $O$.

-----------book page break-----------
We draw two radii from the center $O$ to points $R$ and $S$. Now we paint the region within the arc $RS$ and the radii $OR$ and $OS$ blue. If you notice carefully, the blue region looks very similar to a pizza slice.


Now we paint the region within the arc $RS$ and the radii $OR$ and $OS$ blue. If you notice carefully, the blue region looks very similar to a pizza slice.
The blue colored region is called a $Minor\ Sector$. The remaining part of the circle, which is still in white is called a $Major\ Sector$.
In the same diagram above, we draw a straight line joining points $R$ and $S$. The straight line $RS$ is a chord of the circle shown. Now we paint the region between the line $RS$ and the arc $RS$ red, as shown in the following diagram:

-----------book page break-----------


The red colored region above is called a $Minor\ Segment$. The part of the circle outside the red portion, that is the blue and the white portion together, is called the $Major\ Segment$

Things to remember:
  • A circle is a space surrounded by a line around a point called the center.
  • Every point on the circumference is at the same distance from the center. A line drawn from the center of the circle to any point on the circumference is called a radius.
  • A line joining any two points on the circumference is called an arc.
  • A chord passing through the center is called the diameter, and is the largest possible chord for any given circle.
  • The diameter is twice as long as the radius.
  • The region enclosed by any two radii and the corresponding arc is called a sector.
  • The region enclosed by any chord and the corresponding arc is called a segment.