Concentric Circles
by Eugenis, Apr 24, 2015, 11:41 AM
Lemma 1: 
Proof of Lemma 1: Expanding
we yield 
. 
Lemma 2: For two concentric circles
and 
with radii 
and 
, respectively and where 
, the annulus between the two circles is equal to 
.
Proof of Lemma 2: The annulus between the two circles can be expressed as
. Lemma 1 states that . Thus, 
.
Theorem 1: For
concentric circles, with the first circle having radius 
, the second circle having radius 
and the 
th circle having radius 
such that 
for all 
, if the first circle is shaded, then the third, all the day up to the 
circle, the shaded area is equivalent to 
Proof of Theorem 1: The area can be expressed as
. Firstly we can factor out the common factor 
out of all the differences to yield 
. We can factor each of the differences by Lemma 1 as follows 
. Since we can further factor to yield as desired.
Proof of Lemma 1: Expanding
we yield 
. 
Lemma 2: For two concentric circles
and 
with radii 
and 
, respectively and where 
, the annulus between the two circles is equal to 
.Proof of Lemma 2: The annulus between the two circles can be expressed as
. Lemma 1 states that . Thus, 
. Theorem 1: For
concentric circles, with the first circle having radius 
, the second circle having radius 
and the 
th circle having radius 
such that 
for all 
, if the first circle is shaded, then the third, all the day up to the 
circle, the shaded area is equivalent to 
Proof of Theorem 1: The area can be expressed as
. Firstly we can factor out the common factor 
out of all the differences to yield 
. We can factor each of the differences by Lemma 1 as follows 
. Since we can further factor to yield as desired. 
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