Area of a Circle
Where:
r = radius of the circle (half of its diameter)
π = a Greek symbol for pi; 3.14159...
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The area of a circle tells how many small squares fit inside of it. For example, if a circle is 32 in², thirty-two boxes with the area of one inch can fit inside of it. If a circle is 56 ft², fifty-six boxes with the area of one foot can fit inside of it.
To find a circle's area, simply multiply its radius by itself and multiply that value by π (3.14).
Dissecting the Equation
What is Pi?
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... For more information on a circle's circumference, click the link above.
Since the equation above is the unquestionable principle for all circles, when one needs to find a circle's area, he/she simply divides its circumference by its diameter and multiplies that value by its radius squared.
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.
What Exactly is a Diameter and/or a Radius?
A circle's diameter is the value of its height and width. One would find this value by drawing a straight line directly through its center point so that it extended from one end of the circle to the other. It doesn't matter if this line is drawn horizontally, vertically, or diagonally because the distance from any end of a circle to its opposite end is the same no matter what end you start from:
The radius of a circle is exactly half the value of its diameter. This value is found by dividing the diameter by 2 (or multiplying the diameter by .5).
History
Around 300 BC, finding the area of a circle was a very large challenge for many mathematicians.
Archimedes is known to have found the first (and more complicated) formula by putting a square inside of a circle and using the dimensions of the square to calculate the circle's area. He found that a diagonal line drawn through the center of a square inscribed in a circle made up the circle's diameter and that lead him to make further discoveries, leading to a formula.
How Can I Use This Outside of Math Class?
Knowing the area of a circle can be very helpful in a large variety of instances. For example, construction of the Olympic Stadium required finding the area of the circle to see how big to make the foundation of the stadium.
Pizza companies need to know the area of their small, medium, large, etc.-sized pizzas so that boxes of the proper size can be made.
The area of the inside of a tire is often calculated to figure out what size hubcap to place on it.
Knowing the area of a tabletop can help in choosing what size tablecloth to purchase.
I'm sure there are many more instances you can think of.
Fun Facts
Measurements of pizza refer to the pizza's diameter, not its area. So, if I say I made a 12 inch pizza, the diameter of the pizza is 12 inches; the area is not 12 in². (To find its area, divide the diameter by two, square that value, and then multiply that by pi.)
The largest pizza ever made and eaten had a diameter of 122 feet and 8 inches. This pizza was made in Norwood, South Africa.
Circles can be pretty popular; the longest running television commercial was an ad for a tire company called Discount Tire Company. The commercial ran for forty-nine years.
Ever wonder why manholes are round? I have too. I was once asked on a job interview and ever since, it's been bugging me.
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
Shelly needs to know how big her pizza pan is. She measured the diameter of the pan to be 13 inches. What is the area of her pan?
Area = r²(π)
First, we need to find the pan's radius. This won't be hard since we know the diameter and we know that the radius of any circle is half of its diameter.
Why Square Inches?
Note that my answer is in square inches (square in or in²). When calculating the area of a circle, the units of your answer will always be squared (square feet, square inches, etc.) This is because when assigning one number for the measurement of a circle, both the length and the width need to be accounted for.
As was explained before, the value for the area of a circle tells how many single-unit squares fit into the circle. Single unit squares are used because they account for both length and width.
** 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://www.mathgoodies.com/lessons/vol2/circle_area.html
Works Cited:
Blatner, David. "The Joy of π." The Joy of π. 1998. David Blatner. 19 June 2006.
<http://www.joyofpi.com/thebook.html>.
Weisstein, Eric W. "BBP Formula." MathWorld. 17 January 2004. Wolfram Research, Inc.
20 June 2006. <http://mathworld.wolfram.com/BBPFormula.html>.
"Area of a Circle." Edu 2000. 1996. Edu2000 America. 21 June 2006.
http://www.education2000.com/demo/demo/botchtml/areacirc.htm
"How Archimedes Found the Area of a Circle." UBC Undergrad Mathematics Labs.
2002. UBS Mathematic Department. 21 June 2006.
http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/integration/archimedes.html
"Perimeter, Area, Volume." MWCC/Devens Learning Center Home Page. 2 November 2001
Georgette Gagne. 22 June 2006.
http://www.mwcc.mass.edu/HTML/DEVENSLEARNINGCENTER/pav.html
Guinness World Records. 2006. 2006 Guinness World Records, A HIT Entertainment
Ltd Company. 22 June 2006. <http://www.guinnessworldrecords.com/>.
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