Unit 31: Circle Basics

You may be asked to calculate a missing value for part of a circle.  These can take many different forms, but if we fully understand the parts of a circle, they are easy calculations.

Strategy:

  1. All circle problems will be asking you to relate one part of a circle to another (for instance, diameter to circumference).
  2. These problems can be solved 100% of the time by just plugging in to the proper, simple formula.
  3. You must know these formulas.  There’s no tricks or shortcuts here.

There are a couple of basic formulas you should have memorized, but if not, they are in the reference section for the math section.  A common mistake is confusing the two!

The area equals pi times the radius squared.  The circumference equals two times pi times the radius, or pi times the diameter.

They also tell us in the reference:

The number of degrees of arc in a circle is 360
The number of radians of arc in a circle is 2π

So, 360 degrees = 2π radians.  

To convert from degrees to radians, multiply the degrees by (fracpi{180}).  

To convert from radians to degrees, multiply by (frac{180}{mathrmpi}) .  Note the (pi) will always cancel to leave you with just a number.

An arc is the distance between two points on a circle.  It can be expressed in degrees, radians, or units such as inches.  Here, there is an arc between A and B, denoted as arc AB.  

The degrees in the central angle (angle starting from the center of the circle) is equal to the degrees in the arc.  So below, minor arc AB = 45° since the central angle is 45°.

A minor arc is less than a semicircle (180°) and a major arc is more than a semicircle.  If they don’t specify, assume the minor arc.

There is a very important relationship between central angles, sector area and sector arc length. It is easy to understand when you think of eating a slice of pizza.

This concept gives us one of the most important equation for circles:

$$frac{central;angle}{360^circ}=frac{arc;length}{circumference}=frac{area;of;sector}{area;of;circle}$$

Or in terms of pizza:

$$frac{central;angle;of;slice}{360^circ}=frac{slice;crust}{total;crust}=frac{area;of;slice}{area;of;total;pizza}$$

Applying this concept to the above pizza, you can see that one slice of pizza will be 1/6 of the total area of the pizza. Additionally, the crust will be 1/6 of the total crust of the pizza. Finally, since the pizza has 360°, the central angle of the slice of of pizza will be 60°, because 1/6 * 360° is 60°.