Circle

The circle is such a common shape that we take it for granted ... but there is more to it that just a common shape. Mathematically, the circle is defined as the set of all points that are the same distance from a given point, called the center. That distance is called the radius. The word radius comes from the Latin word for rod or spoke of a wheel, and the radii, or spokes, radiate out from the center.

Isn't the internet amazing? You can find just about anything you need! This wagon wheel image came from an interesting website called Western Furnishings Inc., at http://www.alpha-omegawf.com/

Using this definition of a circle (the set of points all at the same distance from a given point called the center) we can graph a circle on a coordinate system.

 

The equation of this circle, with its center at the origin, and its radius equal to 3 (the square root of 9) is:

The equation follows directly from the Pythagorean theorem: the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse; the "x-distance" being one leg and the "y-distance" being the other leg of a right triangle with hypotenuse r:

We can also use this definition to draw a circle in the sand with a piece of string, or on paper with a compass. Our word center comes from the Greek word "kentron" meaning sharp point, as in the sharp point of the compass. Each point is equally distant from the axle of the wheel, making the tire roll smoothly. This makes the circle the ideal shape for gears and wheels and anything that rolls. The circle is the same on all sides. Designers choose this shape when they design an object that has no top or bottom, front or back, and can be used equally well from all sides.

When movement occurs uniformly from a central point its form is circular. A tree grows outward from a central point, and every year adds a new growth ring. This picture, showing the growth rings of a tree-ring specimen from Broken Flute Cave in northeastern Arizona, is from a website about tree research: http://tree.ltrr.arizona.edu/~grissino/index.htm

Perimeter is the length around a shape. The length around a circle has a special name, circumference. The word circumference comes from the Latin word "circum" meaning around, and "ferre" meaning to carry. Circumference: to carry around, like a carnival ride . . .

From a website created by a manufacturer of carnival rides: http://www.users.globalnet.co.uk/~iirides/

Centuries ago, in Egypt, it was known that the length around a circle was approximately 3 times the distance across the same circle. This was probably discovered by measuring many circles. You can do this yourself: draw a circle by tying a string to a pencil, then hold the string down at any point and draw a circle with the pencil, keeping the string taught. Now use a long piece of string to "wrap around" the circle, then stretch the string out and measure its length with a ruler. measure the distance across the middle of the circle. Us a calculator to divide the larger number by the smaller number, and you should get a number a little bigger than 3. If you do this with many different sized circles, and then take an average of the quotients, you should get a number that is approximately 3.14 which is the value we often use for a number we call "pi". The Greek letter for "pi" is:

 

This is the Greek letter for "p", and is the first letter of the Greek word for circumference, or perimeter. Pi is a very interesting number, and you will find many websites on the internet about pi. Visit the websites linked below . . . or do a net search for pi, just to see how many references there are!

A history of pi, called "Pi Through the Ages"

http://www-groups.dcs.st-and.ac.uk:80/~history/HistTo pics/Pi_through_the_ages.html

You can see 2000 decimal places of pi at:

http://www-groups.dcs.st-and.ac.uk:80/~history/HistTopics/2000_plac es.html

Every geometric shape has perimeter and area.

The distance around a circle is called the circumference. The formula for finding the circumference of a circle is "circumference of a circle equals pi times the diameter of the circle", and the formula for the area of a circle is: "Area of a circle equals pi times the square of the radius of the circle":

Which shape has the greatest area? We can compare different shapes by choosing a number for perimeter, and keeping that perimeter constant for all the shapes. These three triangles have the same perimeter but the one will all three sides equal has the greater area.

The more symmetrical a figure is, the greater the area, and the more sides it has, the more area contained within. The polygons below all have the same perimeter. With six, eight, ten, then twelve sides, our straight-sided figure increases in area, as the number of sides increases. Also, as the number of sides increases, our polygon begins to look like...a circle! ...

 

Of all these shapes with the same perimeter, the circle has the greatest area.So the circle is the 2 dimensional figure with the largest area, given a certain perimeter (or circumference). In 3 dimensions, the sphere ( a "ball") is the solid shape with the largest volume for the same surface area.This property of a circle and sphere explains why air bubbles, or any kind of bubble, is always circular from all points of view: the forces pull outward equally in all directions forming a perfect sphere:

If you want to know more about bubbles, check out the San Francisco Exploratorium, a wonderful science museum (the source of this beautiful photograph) at:

http://www.exploratorium.edu/ronh/bubbles/bubble_colors.html

The sphere is the most economical container, but is not a practical choice to use for cans and bottles. What would happen if soda came in spherical cans? The cylinder is the second best choice, having the second-largest volume for a given surface area. The more closely the cylinder resembles a sphere, the more volume per surface area it has: so a cylindrical can should be approximately as tall as it is wide to have the greatest volume compared to the amount of metal used to make it.

The circle can be measured ... and it is often used to measure other things. Each time the wheel rolls around once, the track coach knows he's travelled two feet, the circumference of the circular measuring wheel... the same principle is used in the mileage counter of your car.

The circle is the most perfectly symmetrical geometric figure.

In the next page of The Conics, you will find out about another one of The Conics, the Ellipse:

P18Ellipse.htm