Difference between revisions of "2003 AMC 10B Problems/Problem 23"
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A regular octagon <math> ABCDEFGH </math> has an area of one square unit. What is the area of the rectangle <math> ABEF </math>? | A regular octagon <math> ABCDEFGH </math> has an area of one square unit. What is the area of the rectangle <math> ABEF </math>? | ||
− | <asy> unitsize( | + | <asy> unitsize(1cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5); draw(A--B--C--D--E--F--G--H--cycle); label("$A$",A,NNW); label("$B$",B,NNE); label("$C$",C,ENE); label("$D$",D,ESE); label("$E$",E,SSE); label("$F$",F,SSW); label("$G$",G,WSW); label("$H$",H,WNW);</asy> |
<math> \textbf{(A)}\ 1-\frac{\sqrt2}{2}\qquad\textbf{(B)}\ \frac{\sqrt2}{4}\qquad\textbf{(C)}\ \sqrt2-1\qquad\textbf{(D)}\ \frac{1}2\qquad\textbf{(E)}\ \frac{1+\sqrt2}{4} </math> | <math> \textbf{(A)}\ 1-\frac{\sqrt2}{2}\qquad\textbf{(B)}\ \frac{\sqrt2}{4}\qquad\textbf{(C)}\ \sqrt2-1\qquad\textbf{(D)}\ \frac{1}2\qquad\textbf{(E)}\ \frac{1+\sqrt2}{4} </math> | ||
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Area of Octagon: <math> \frac{ap}{2}=1 </math> | Area of Octagon: <math> \frac{ap}{2}=1 </math> | ||
Area of Rectangle: <math> \frac{p}{8}\times 2a=\frac{ap}{4} </math> | Area of Rectangle: <math> \frac{p}{8}\times 2a=\frac{ap}{4} </math> | ||
− | You can see from this that the octagon's area is twice as large as the rectangle's area is <math>\frac{1}{2}</math> | + | You can see from this that the octagon's area is twice as large as the rectangle's area is <math>\boxed{\textbf{(D)}\ \frac{1}{2}}</math> |
+ | |||
+ | ==See Also== | ||
+ | {{AMC10 box|year=2003|ab=B|num-b=22|num-a=24}} |
Revision as of 21:17, 26 November 2011
Problem
A regular octagon has an area of one square unit. What is the area of the rectangle
?
Solution
An easy way to look at this:
Area of Octagon:
Area of Rectangle:
You can see from this that the octagon's area is twice as large as the rectangle's area is
See Also
2003 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |