AnglEs IN A CIRCLE
Geometry
 
1.  THE CENTRAL ANGLE

The central angle has its vertex in the centre of the circle. The arc AB on a circumference makes an angle with the centre of the circle AOB. The measurement of this angle, subtended at the centre, is the same as the angle of the arc on the circumference.
1.- Find out what the corresponding arc is for a central angle of 90º in the Descartes window. How much of the circumference forms the arc?

2.- What happens if you increase the radius to 5? Repeat the exercise for angles of 180º and 270º.

3.- Also, work out the central angle of a pentagon, a hexagon and dodecagon.

 
2. THE INSCRIBED ANGLE
The inscribed angle is the angle made by a vertex on the circumference and two sides which cut through the circle. The measurement of the inscribed angle APB is half of the measurement of the angle subtended at the centre of the circle AOB. 

 

 

4.- In the following window, if you move point P from one side to another you can see that the inscribed angle always measures half of the angle subtended at the centre.

If you move point P until POB is a straight line you should understand how the angle measurement is calculated. Take into account that the triangle OPA is an isosceles triangle.

5.- Move points A and B until you get a diameter i.e. an angle of 180º. How much does the inscribed angle measure when it is subtended by a diameter?

6.- Move points A and B until you get an angle of 270º and work out the value of the corresponding inscribed angle. Think about what could be the greatest value of an inscribed angle.

 
       
           
  Miguel García Reyes
 
Spanish Ministry of Education. Year 2001
 
 

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