If you know either the diameter of the circle or its radius,
then you're as good as done with the problem.
The formula for the area of a circle is
Area = (pi) x (radius²) .
The radius is 1/2 of the diameter.
π (the Greek letter for 'pi') is a slight problem. It's an "irrational"
number, and that means you can never write down its exact value.
As a decimal, it keeps going on and on forever, and as a fraction,
it can't be written at all.
So what are we supposed to do ? How can we ever write down the
exact value for the area of a circle ?
Using digits, we can't ! The only way to write the exact area
of a circle is to leave the letter π in the answer.
For example, if the radius of the circle is 5 ,
then the area is
Area = π R² = 25 π .
If you can only use digits to write the area, then you can never
write the exact number. Anything you write will always be
slightly wrong. BUT ... you can get very very close.
Technically, even though you can never be exactly correct,
you can get AS CLOSE as you want to. In books and online,
you can find π printed out with 1,000 decimal places, and the
more of them you use to calculate the area, the more accurate
your answer will be.
Here are the first 15 decimal places of π .
(These are the only ones I've memorized.)
3. 14159 26535 89793
At the end of 2009, a team got together and ran their computers for
131 days, and calculated π with 2,700,000,000,000 decimal places !
(It still doesn't end.)
So how many decimal places should you use ?
How close does the answer need to be for school ?
To answer that, I'm going to have to reveal the Big Secret
of school to you. Here it is. Please don't spread it around:
In school math, the answer doesn't matter !
The answer is not important, and nobody needs it.
Your teacher doesn't need the answer. If s/he did,
s/he could easily figure it out, and if s/he didn't know
how and had to ask somebody else, s/he certainly
wouldn't ask her students, because they're just now
learning how to do it.
What's important is knowing HOW TO FIND the answer.
The only reason they want to see your answers in
school math is: That's the fastest, easiest way to tell
whether you know HOW TO FIND the answer.
If you can invent a faster, easier way to tell whether
you know HOW TO FIND the answers, then nobody
will ever need to turn in the homework answers again.
I told you that, to tell you this: Your answer for the area doesn't need
to be very close at all. It only needs to be close enough to show that
you knew how to figure it out.
Most of the time in school math, the question will tell you
what number to use for π . Very often, it's 3.14 .
You would think that for a number that has trillions and trillions
of digits and goes on forever, that using only 2 decimal places
would not get you very close to the real answer.
You would be wrong.
-- If you use 3.14 for π , then the answer you get is too small,
but only by about 0.05 % .
-- If you use 3.142 for π , then the answer you get is too big,
but only by about 0.01 % .
-- If you don't like decimals at all, and want a fraction for π ,
then use 22/7 . Then the answer you get is too big again,
but only by about 0.04 % .
Any of these is way closer than you need to be for school math.
They're even closer than you usually need to be in real Engineering.
(Trust me. I know.)
______________________________________
Just one more thing:
About 36 days from today will be the day celebrated by
math people all over the world.
It's called "Pi Day" . It's on March 14th. ( 3-14 ) :-)
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