DEFINITION AND CIRCLE CONSTRUCTION
Geometry
 
1. CONSTRUCTING CIRCLES

A circle is a closed plane curve, every point of which is equidistant from a given fixed point, called the centre. The distance from the centre to the edge of the circle is called the radius.

C is the centre of a circle and r is the radius. The following applet shows that the movement of point P is restricted so that:

PC = r
1.- Drag point P carefully and look at the trace that is drawn: you are drawing a circle. 

The clear button will delete the trace without changing the radius.

Use the coloured arrows next to the radius r.

You can also write values for the radius directly into the box and then press the enter key.
The Init button will restore the initial values.

 

2.- Change r to different values between 2 and 6 and draw the corresponding circles.
2. CHANGING THE CENTRE AND THE RADIUS
We can chose any point on the plane as the centre of our circle. The radius can be set at whatever length we want.
3.- Try drawing more circles. This time drag the centre C and then drag point P carefully. 
 

4.- Change r to different values between 2 and 6 and look at the drawings of the different circles.

5.-  Draw four circles with a radius of 2. The centres of the last three circles should be on the circumference of first circle.
3. THE GENERAL EQUATION OF A CIRCLE WITH ITS CENTRE AT THE ORIGIN
The following applet shows a circle whose centre is the origin of the coordinate plane. r is the radius and C (0,0) is the centre. Using points P(x,y) on the circumference of the circle we can say that: x²+y²=r².(Pythagoras' theorem).
Use the red and blue arrows O.x and O.y to move the axes or write the number directly into the white box and press the Enter key.

6.- Change the value for r and watch how the circle and equation change.

Use the zoom key to change the scale and move the axes whenever necessary.

7.- Drag point P with the mouse and watch how its coordinates change and that the circle equation is always true.


8.- Draw circles with the following radii: 2, 3, 4, 5, 5.5 and 6.

 

4. THE EQUATION OF A CIRCLE WITH ANY POINT AS ITS CENTRE: C( h, k)

The following applet shows a circle with its centre at point (h,k) and radius r. The point P(x,y) is on the circumference of the circle and gives us the following equation: (x-h)²+(y-k)²=r² (Pythagoras' theorem).  When k=0 the centre is located on the horizontal axis and when h=0 the centre is on the vertical axis.  

9.- Change the values for r, h and k and look at how the position on the axes and size of the circle produced changes.

To get the equation for each new circle click on the clear button.

10.- Change the parameters to draw the following circles: Radius 3 and centre (4,2). Radius 3 and centre (4,0). Radius 4 and centre (0,2).

11.- Drag point P with the mouse and watch how the point P moves and the circle equation doesn't change.

12.- Change the parameters to draw the following circles:
    Radius 2 and centre (-4,2). Move point P around the circumference.
    Radius 5 and centre (3,0). Move point P around the circumference.
    Radius 4 and centre (0,-2). Move point P around the circumference.

So far we can say that we have obtained a basic knowledge of circles. If you want to know more about circles go on to look at the general equation section and then the section that looks at circles going through three given points. Otherwise, return to the index and go on to the exercises and games or go to the self-evaluation section. Click on the corresponding arrow to move on.

       
           
  Jesús Fernández Martín de los Santos
 
Spanish Ministry of Education. Year 2001
 
 

Licencia de Creative Commons
Except where otherwise noted, this work is licensed under a Creative Common License