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The golden rectangle is inscribed in a circle with a radius of 10cm. Find the area of that rectangle.

Guest Jun 8, 2015

Best Answer 

 #2
avatar+1068 
+10

The golden rectangle is inscribed in a circle with a radius of 10cm. Find the area of that rectangle.

 

\varphi = \frac{1 + \sqrt{5}}{2} = 1.61803\,39887\dots

The ratio of the sides a and b of the golden rectangle is calculated by the upper formula.

Because the ratio of the sides of this rectangle is constant,  so must be the ratio of the angles that these sides form with rectangle's diagonal!

a = 1,   b = 1/1.61803398874989485 = 0.61803398874989484752

tan(B) = 1/ 0.61803398874989484752

B = ≈58.28o        A = ≈31.72o

Circle's radius  r = 10cm,    rectangle's diagonal  d = 20cm      a = ?,   b = ?  and  A = ?

 

cos(A) = a/d             b = sqrt(d2 - a2 )                      A = a*b

a = d*cos(A)            b = sqrt(202 - 17.0132)             A = 17.013*10.515

a = 17.013cm           b = 10.515cm                          A =  ≈178.88cm2

civonamzuk  Jun 8, 2015
 #1
avatar+87309 
+10

Call x one side of the rectangle......then the adjacent side will be (Phi)x

 

And the diameter of this circle will  = 20cm    ......and by the Pythagorean Theorem, we have.......

 

x ^2  +  [(Phi)x]^2  = 400

 

x^2 [ 1 + (Phi)^2]  = 400      and we can note that Phi^2  =  1 + Phi.......so we have

 

x^2 ( 1 + 1 + Phi)  = 400

 

x^2 (2 + Phi )  = 400

 

x^2  =   400 / (2 + Phi)        take the  positive square root of both sides

 

x = 20 / √(2 + Phi) 

 

So....the area of this rectangle  = (20/√(2 + Phi))* (Phi * 20/ √(2 + Phi) ) = 400Phi / (2 + Phi)  = about

178.88 cm^2

 

 

CPhill  Jun 8, 2015
 #2
avatar+1068 
+10
Best Answer

The golden rectangle is inscribed in a circle with a radius of 10cm. Find the area of that rectangle.

 

\varphi = \frac{1 + \sqrt{5}}{2} = 1.61803\,39887\dots

The ratio of the sides a and b of the golden rectangle is calculated by the upper formula.

Because the ratio of the sides of this rectangle is constant,  so must be the ratio of the angles that these sides form with rectangle's diagonal!

a = 1,   b = 1/1.61803398874989485 = 0.61803398874989484752

tan(B) = 1/ 0.61803398874989484752

B = ≈58.28o        A = ≈31.72o

Circle's radius  r = 10cm,    rectangle's diagonal  d = 20cm      a = ?,   b = ?  and  A = ?

 

cos(A) = a/d             b = sqrt(d2 - a2 )                      A = a*b

a = d*cos(A)            b = sqrt(202 - 17.0132)             A = 17.013*10.515

a = 17.013cm           b = 10.515cm                          A =  ≈178.88cm2

civonamzuk  Jun 8, 2015
 #3
avatar+92805 
0

Why is it a golden rectangle?  What is so special about phi ?

Melody  Jun 9, 2015
 #4
avatar
+5

 

http://en.wikipedia.org/wiki/Golden_rectangle

Guest Jun 9, 2015
 #5
avatar
+5

http://en.wikipedia.org/wiki/Golden_ratio

Guest Jun 9, 2015
 #6
avatar+92805 
0

Thanks anon

Melody  Jun 9, 2015

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