Geometry: Circumference and Area: Pi Anyone?
Circumference and Area: Pi Anyone?
One of the unusual things about circles is that they are all similar to each other. I mean this in the official sense. It's not just that circles have the same shape but that everything is proportional as well. It is from proportionalities that the following postulate is presented.
- Postulate 17.3: The ratio of the circumference of a circle to the length of its diameter is a unique positive constant.
This constant of proportionality is represented by the Greek letter ? (pronounced Pi, as in key lime). The reason that a Greek letter is used is because ? is an irrational number. It cannot be written as neatly as 3 or 3.1, or 22/7, even though these numbers are approximations to ?. It is impossible to write down the precise value of ?.
If you denote the circumference of a circle by C and the length of a diameter of the circle by d, then C = ?d. Because the length of a diameter of a circle is twice the length of the radius, if you let r denote the length of the radius of the circle, you have d = 2r and C = 2?r.
Real numbers are classified as follows: the positive, whole numbers are called the natural numbers. The positive and negative whole numbers make up the integers. The collection of the ratios of integers comprises the rational numbers. The real numbers that are not rational numbers are called irrational numbers. An irrational number cannot be written as the ratio of two integers. The Greeks knew that irrational numbers (like ?2 and ?) existed, and they were able to estimate ? very accurately.
Now that you know how to compute the circumference of a circle, it is time to revisit arcs. Remember that there are two measurements associated with arcs: degree measurement and length. It turns out that the lengths of arcs can be found using a proportionality. The ratio of the degree measure m of an arc to 360 (the degree measure of a circle) is the same as the ratio of the length l of the arc to the circumference. The equation is given by
- m/360 = 1/C = 1/2?r.
Rearranging this equation to solve for the arc length, you get
- 1 = m/360C.
Let's do some calculations.
- Example 1: Find the circumference of the circle and the length of ?ABC in Figure 17.6.
- Solution: To find the circumference of the circle, just use the equation that relates the circumference of a circle to the radius of the circle:
- C = 2?r = 2?(5) = 10? inches.
- To find the length of ?ABC , notice that the central angle has measure 125. Using the equation for arc length, you have:
- 1 = m/360 C = 125/360(10?) = 125?/36 inches
Now consider the problem of finding the area A of a circle whose radius has length r. The formula for this is A = ?r2. You can find the area of a sector using proportions, just as you found the length of an arc. If the area of the entire circle is ?r2, the area of a sector with degree measure ? is found by evaluating
- Asector = ??r2/360.
- Example 2: Find the area of a circle whose radius has a length of 15 inches.
- Solution: You can solve this problem easily using the equation for area: A = ?r2 = ?(15)2 = 225? inches2
- Example 3: Find the area of the sector with a radius length of 4 inches and central angle measuring 120.
- Solution: Substituting into our equation, you see that the area is
- ??r2/360 = 120?(4)2/360 = 16?/3 inches2.
Excerpted from The Complete Idiot's Guide to Geometry 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.