We have all seen movements that involve turning. When a skater performs a sow chow or a triple lutz we wonder at the beauty and grace of the motion. A gymnast doing a somersault on the beam or a great circle on the uneven bars is equally graceful. While we are watching the amazing performances these athletes have to "spot" the ground and determine how much they have turned. If their estimate of how much and how fast they are turning is incorrect they sit down rather suddenly.
In many situations we need to measure rotation. Sometimes it is easy because we have whole rotations. Sometimes we have fractions of rotations. If we were driving in a park and you saw a deer you might tell your friend that it is at "2 O'clock" from where you are. Your friend would imagine the car to be pointing at 12 O'clock and have to turn a little to the right to see the deer. Notice that you would be using the direction the car is pointing as the initial direction and then the deer would be at 2 O'clock.
An angle is formed when you have two rays with the same endpoint. This common endpoint is called the vertex. One ray forms the initial side and the other is the terminal side.
There are two ways to refer to an angle. We could use three letters; for example the angle in the diagram could be called angle ABC or angle CBA. Notice that the vertex is always in the middle. If it does not cause confusion we can also just use the letter that corresponds to the vertex and call the it angle B, written .
Measuring Angles
We need to be able to say how large an angle is. The ancient Greeks liked multiples of 60 and decided to have 360 degrees in a full circle of rotation. We use the symbol ° for degrees. A half circle of rotation is 180° and a quarter circle 90° and so on. Can you calculate the number of degrees in 1/3 of a rotation? Here is how to think about it. We know that
so if we divide each side of that equation by 3 we get that
In the table below you are given fractions of a rotation.
Calculate the measure of the angle in degrees.
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