The Pythagorean Theorem  The square of the hypotenuse on a right triangle is  equal to the sum of the squares of the two legs.  a² + b² = c²  Use the java applet to construct the proof of the Pythagorean Theorem.

Pi is a Greek letter which represents a number approximately equal to 3.141592654.  Pi is an irrational number and thus the number stated above is not exact, but an approximation.  To see the number represented by expanded to more decimal places, click here or here (two different versions).

Archimedes, a great ancient mathematician, used a method called "Exhaustion" to try to find a value for .  It behoves us to know what that value represents. 

Quite simply put, represents the ratio between the circumference and the diameter of a circle.  Thus, = c/d.  The method of "Exhaustion" used by Archimedes to find an approximation for can be viewed at a website which can be accessed by clicking here.

We will also do one ourselves.....now .

Angles and Their Measurement

Definition of an angle.

An angle consists of two half lines (rays) with a common initial point called a vertex.  We agree that the terminal side is obtained by rotating the initial side counterclockwise through the angle t.  An angle is in standard position with its vertex at the origin and its initial side along the positive x-axis.

 An angle of positive measure is obtained when the terminal side is rotated counterclockwise and an angle of negative measure is obtained when the angle is rotated clockwise.  Angles obtained by different rotations that have the same initial and terminal sides are called coterminal. 

Measurement of an angle

There is a difference between an angle (a geometric object) and its measure (a number). However, it is correct to say that "an angle is x units" so that our work does not get too tedious. 

Degree 

The classical unit used to measure angles is a degree.  A complete rotation so that the terminal side lies on the initial side is an angle of 360°, so 1° is 1 / 360 of one complete rotation.  Parts of an angle are measured in minutes and seconds. 

Hereafter the following symbols will be used:
° = degrees
' =  minutes
" = seconds

Based on the definition of the degree (in the preceding paragraph), the following equivalencies should make sense:

Now, using  conversions based on the equivalencies noted in the table above, we can change units from degrees, minutes and seconds to decimal equivalencies:

Radians

A more convienient unit to measure angles in applications is a radian. 

Consider a circle with radius 1 unit. Start at the  point ( 1,0 ) and measure 1 unit counterclockwise along  the circumference of the circle.  Mark this point and and join it to the origin to make the terminal side of an angle.  This angle is 1 radian.

 On a circle with radius 1 unit ( unit circle), an angle of 1 radian cuts off an arc of 1 unit on the circumference of the circle. 

The circumference of any circle is 2(radius).
 
2 radians = 360°   or  radians = 180° 
The angle in radians is the ratio of the length of the arc on the circle to the length of the initial side of the angle, so it has no dimensional units. 

Changing units between degrees and radians.  Using the equivalency of radians = 180°
we can write that as a ratio:  radians are to 180° as x radians are to t °
Or mathematically speaking:

radians  =  x radians 
    180°              t ° 

Enter your given angle ( either degrees or radians ) into the proportion and solve for the unknown. 

A.  30° = ? radians   radians =   x rad     180°           30°  180 x = 30  radians         x = /6  radians

B. / 2 radians = ? °  rad =   (/2) rad    180°              t°      2t  = 180°         t = 90°