Approximate Circumference of an Ellipse
Where:
π = a Greek symbol for pi; 3.14159...
a = half of the width diameter (referred to as semimajor axis)
b = half of the length diameter (referred to as semiminor axis)
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An ellipse is a circle whose dimensions are distorted so that it is wider than it is long (stretched out horizontally.) Mainly, an ellipse is an oval on its side.
The circumference of an ellipse tells how long the line is that makes up the ellipse.
To better explain this, imagine a line that's been curved into the shape of an ellipse. If we take this line apart and straighten it back out, we can measure it with a ruler to see how long it is. However long it is is the ellipse's circumference.
Obviously, the longer the line is, the larger our ellipse will be.
Semimajor and Semiminor Axes
To find an ellipse's circumference without taking it apart and measuring its line, one must know the values of both the semimajor and the semiminor axes. These terms are simply fancy words for the half of the shape's width and length.
Since all ellipses are wider than they are long (considering width refers to horizontal span and length refers to vertical reach,) half of their width is referred to as the ellipse's semimajor axis. Semi- is used to mean "half of". -major is used for the width because the width is always the largest value of the shape.
-minor is used for the length because an ellipse's length (vertical reach) is always the shortest value of the shape.
Wikipedia provided an interesting point about these axes. Think of an ellipse as four different limited parabolas (two forming the top and bottom and two forming the sides.)(If you're not sure what a parabola is, click the link provided.)
When forming an ellipse, the two arms of all these parabolas (which are naturally infinitely long) are forced to stop at a limit. Therefore, although they must be cut off in order to form the ellipse, they still (as Wikipedia puts it) "tend to infinity." This means that they reach as far as they can because it is in their nature to extend forever.
Since the arms of the two parabolas that make up the sides of the ellipse are allowed to extend further than the arms of the two top parabolas, the parabolas on the side (those on the semimajor axis) reach toward infinity faster than those on the top. Therefore, the span of these longer/faster arms of one of the side parabolas is referred to as the semimajor axis. The shorter/slower span of the top or the bottom parabola is called is the semiminor axis.
Thinking of ellipses as parabolas should help you remember that the sides of ellipses are longer than their tops and bottoms. It should also help you understand why semi-axes of ellipses are used for their calculations rather than their full axes.
What is Pi and Why is It in This Equation?
Pi was introduced to mathematics only because people were getting too bored while doing equations and wanted to add some foodly fun to the grand scheme of things. They chose pi to be part of a circle's equation because pies are almost always in the shape of a circle.
Okay, so maybe that wasn't completely right, but at least I gave my insight (oh, geez, now I'm even rhyming...)
Pi was found many, many thousands of years ago to be (as David Blatner puts it) "the ratio of a circle’s circumference to its diameter." This means that for every perfect circle:
In other words, the circumference of any perfect circle ever made or thought of is larger than its diameter by 3.14159...
Of course, ellipses aren't perfect circles; but since their equations contain pi, ellipses are similar to circles.
When using pi, don't go through the trouble of finding all the numbers after the decimal point that you can get your hands on. Most people use 3.14 in place of pi to avoid unnecessary trouble.
The name and symbol for pi came from the Greek alphabet. Pi is the sixteenth letter of this alphabet.**
Fun Facts
The earth is not a true sphere. Its circumference forms the shape of an ellipse rather than a circle. Its approximate circumference (on the equator) is 24,901.55 miles
There are many, many different formulas for calculating the circumference of an ellipse. Many of the more complicated (yet more accurate) ones were given by a mathematician named Srinivasa Aiyangar Ramanujan from India. All formulas give slightly different answers and none really give a completely exact answer. The easier formulas (ours is one of these) are thought to be less accurate by a few decimal points. You can see a large list of some formulas at: http://home.att.net/~numericana/answer/ellipse.htm#ramanju-2.
The exact value of pi will never be known because there is an infinite amount of numbers that come after the 3. There are world records for who has calculated the most of these infinite digits.
In 1995, Simon Plouffe found an equation (called the BBP Formula) which allowed him to calculate any digit after the decimal point of pi. He and his colleagues used it to calculate the ten-billionth digit.
(For more information on pi than you probably even care about, check out the book The Joy of Pi, written by David Blatner, or go to its website: http://www.joyofpi.com/)
Example: Find the circumference of an ellipse whose semimajor axis is 5 feet and semiminor axis is 3 feet.
First, recall our formula:
Multiply our new value by π (3.14):
3.14 (5.8309518948453004708741528775456) =18.309188949814243478544840035493
That gives us:
2 (18.309188949814243478544840035493)
Multiply that answer by 2:
2(18.309188949814243478544840035493) = 36.618377899628486957089680070986
Reduce that down to four decimal points and our answer is: 36.6184 miles.
** Our Calculators Like to Show Off
Although π (pi) has an infinite amount of numbers after the decimal point, people commonly use 3.14 as its value when solving an equation that uses it. Our calculator, however, uses a more accurate version of pi with thirteen digits after the decimal point (3.1415926535898.) This means that if the same values that I use for the example in this help file are plugged into our calculator, the answer will be slightly different (and more accurate.)
So why do I use a less accurate number if it gives a less accurate answer? I use the rounded version because that is what most teachers will have you use for pi in school. In some cases, you may get counted off for giving a different answer than your math book (or your teacher) has.
Also, it is much easier to learn and remember the rounded value. Not to mention, if I used the number with thirteen decimals in each example, it would take much longer to work through and the long string of digits after the decimal might very well become confusing.
Related Websites:
http://en.wikipedia.org/wiki/Ellipse
Works Cited
Blatner, David. "The Joy of π." The Joy of π. 1998. David Blatner. 19 June 2006.
<http://www.joyofpi.com/thebook.html>.
Michon, Gérard. "Final Answers: Perimeter of an Ellipse." Numericana.com. 25
February 2006. Gérard Michon. 3 July 2006.
<http://home.att.net/~numericana/answer/ellipse.htm#ramanju-2>
Weisstein, Eric W. "BBP Formula." MathWorld. 17 January 2004. Wolfram Research,
Inc. 20 June 2006. <http://mathworld.wolfram.com/BBPFormula.html>.
Weisstein, Eric W. "Ellipse." MathWorld. 1999. Wolfram Research, Inc. 30 June
2006. <http://mathworld.wolfram.com/Ellipse.html>.
"Earth." Wikipedia. 3 July 2006. Wikimedia Foundation, Inc. 3 July 2006.
<http://en.wikipedia.org/wiki/Earth>.
"Semi-major axis." Wikipedia. 14 May 2006. Wikimedia Foundation, Inc. 3 July 2006.
<http://en.wikipedia.org/wiki/Semi-major_axis>.
"What is the Circumference of the Earth?" About.com. 2006. About, Inc. 3 July 2006.
<http://geography.about.com/library/faq/blqzcircumference.htm>.
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