Try this beautiful problem from Geometry based on the Area of the Octagon from AMC 10A, 2005, Problem 2005
An equiangular octagon has four sides of length 1 and four sides of length \(\frac{\sqrt{2}}{2}\), arranged so that no two consecutive sides have the same length. What is the area of the octagon?
Geometry
Triangle
Octagon
Answer: \(\frac{7}{2}\)
AMC-10A (2005) Problem 20
Pre College Mathematics
We have to find out the equiangular octagon whose four sides of length 1 and four sides of length \(\frac{\sqrt{2}}{2}\),
we join \(AD\),\(HE\),\(BG\) and \(CF\).We assume that side lengths of \(AB=CD=EF=GH=1\) and side lengths of \(AH=BC=DE=GF=\frac{\sqrt{2}}{2}\)( As no two consecutive sides have the same length). Now
Can you now finish the problem ..........
There are 5 squares with side lengths \(\frac{\sqrt{2}}{2}\) and 4 Triangles of side lengths \(1\)
Now area of \(5\) squares=\( 5 \times (\frac{\sqrt{2}}{2})^2\)=\(\frac{5}{2}\) and area of each Triangle is half of the area of a square.so the area of \(4\) Triangles=\(4 \times \frac{1}{2} \times \frac{1}{2}\)=\(1\)
can you finish the problem........
Therefore the Total area of the required octagon=Total area of Five squares + Total areas of Four Triangles=\(\frac{5}{2} +1\)=\(\frac{7}{2}\)
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