The Area of a Circle

1
The Area of a Circle

• Too big! Too small! Just right!

2
Recall the story of Goldilocks and the Three
Bears.
3
While they were gone, a girl named Goldilocks
came walking by the Bear house.  She was hungry
and smelled porridge, so she decided to go into
the house. First she tried Papa Bear's porridge,
but it was too hot. Then she tried Mama Bear's
porridge, but it was too cold. Then she tried
Baby Bear's porridge, and it was just right, so
sheate it all up.
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After she had eaten the porridge, Goldilocks
wanted to sit down and take a rest. First, she
sat in Papa Bear's chair, but it was too high for
her. Then she sat in Mama Bear's chair, but was
too soft. Then she sat on Baby Bear's chair, and
it was just right. She rocked back and forth on
the chair, and suddenly it collapsed and broke
into many little pieces. Oops!
5
After Baby Bear's chair broke, Goldilocks was
exhausted, so she decided to take a nap. She went
into the  bedroom and first tried to get into
Papa Bear's bed, but it was too big. Then, she
crawled onto Mama Bear's bed, but it was too
slippery and she slid right off. Finally, she got
into Baby Bear's bed. It was just right. She fell
asleep right away.
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We are going to use this idea of too small,
too big, just right to help us make sense out
of the formula for the area of a circle.
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Lets estimate the area of this circle.
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Although its going to be too much area, lets
find the area of the entire grid.
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If we break the grid into four 3 by 3 squares,
then the area is 4 x 3 x 3 or 36. This is too
much area.
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Although its going to be too little area, lets
find the area inside of the two triangles.
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The area inside of the two triangles is 2 x (3 x
6 ?2) or 18.
3
6
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2 x (3 x 6 ?2) is the same as 2 x 9
3
6
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2 x 9 is the same as 2 x 3 x 3
3
6
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Lets summarize what we have done so far.
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4 x 3 x 3 is too much area.
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2 x 3 x 3 is too little area.
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If 4 x 3 x 3 is too much and 2 x 3 x 3 is too
little, then what should be just right?
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Thats right! 3 x 3 x 3 should be just right for
the area of the circle.
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Now lets figure out what the numbers mean.
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3 appears to be a very popular number in all of
the expressions that we saw.
4 x 3 x 3 Too Much 2 x 3 x 3 Too Little 3 x 3 x
3 Just Right
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3 appears to be a very popular number in all of
the expressions that we saw.
4 x 3 x 3 Too Much 2 x 3 x 3 Too Little 3 x 3 x
3 Just Right
These threes represent the radius of the circle.
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The radius of the circle is 3.
3
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Can you guess what magical number makes the
formula just right ?
4 x 3 x 3 Too Much 2 x 3 x 3 Too Little 3 x 3 x
3 Just Right
What is this special number?
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This special number is pi (3.14)
4 x 3 x 3 Too Much 2 x 3 x 3 Too Little 3.14 x 3
x 3 Just Right
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So you can find the area of a circle, by
multiplying pi times the radius times the radius.
4 x 3 x 3 Too Much 2 x 3 x 3 Too Little 3.14 x 3
x 3 Just Right
? x
r x r