Circumference and Area of a Circle Help
Introduction to Circumference and Area of a Circle
A circle is a geometric figure consisting of all points in a plane that are equally distant from some center point. Imagine a flashlight with a round lens that throws a brilliant central beam of light surrounded by a dimmer cone of light. Suppose you switch this flashlight on, and point it straight down at the floor in a dark room. The outline of the dim light cone is a circle. If you turn the flashlight so the entire dim light cone lands on the floor but the brilliant central light ray is not pointed straight down, the outline of the dim light cone is an ellipse .
The circle and the ellipse are examples of conic sections . This term arises from the fact that both the circle and the ellipse can be defined as sets of points resulting from the intersection of a plane with a cone.
A Special Number
The circumference of a circle, divided by its diameter in the same units, is a constant that does not depend on the size of a circle. This fact was noticed by mathematicians thousands of years ago. The value of this number cannot be expressed as a ratio of whole numbers. For this reason, this number is called an irrational number . (“Irrational” means, in this context, “having no ratio.”) If you try to write this number in decimal form, you get a non-terminating, non-repeating sequence of digits after the decimal point. It is a constant called π , and is symbolized π . This is the same π we encountered earlier when defining the radian as a unit of angular measure.
The value of π has been calculated to many millions of decimal places by supercomputers. It’s approximately equal to 3.14159. If you need more accuracy, you can use the calculator function in a personal computer. In a computer that uses the Windows operating system, open the calculator program and set it for scientific mode. Check the box marked “Inv.” Be sure there are black dots in both the “Dec” and “Radians” spaces. Press 1, then the minus key, then 2, and then the equals key so you get −1 on the display. Finally, hit the “cos” button. This will show you the angle, in radians, whose cosine is equal to −1; this happens to be π . A good calculator will display enough digits to make almost anyone happy.
Here are some formulas that can be used to find the perimeters and areas of circles, ellipses, and regular polygons that are inscribed within, or circumscribed around, circles. You don’t have to memorize these (except for the formulas for the perimeter and interior area of a circle, which are worth memorizing), but they can be useful for reference. As with all the formulas in this book, they are straightforward, even if some of them look messy. Using them is simply a matter of entering numbers into a calculator and making sure you hit the correct keys in the correct order.
Perimeter and Area of a Circle
Perimeter Of Circle
Let C be a circle having radius r as shown in Fig. 4-9. Then the perimeter (or circumference), B , of the circle is given by the following formula:
B = 2 πr
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