$ABCDEFGH$ is an equiangular octagon (that is, an octagon with eight equal interior angles). We know that $AB=EF=1$, $BC=FG=\sqrt 2$, $CD=GH=2,$ and $DE=HA=2\sqrt 2.$ What is the area of the octagon?

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Guest Mar 3, 2021

#1**+1 **

We see that this picture us constructed of squares and 45-45-90 triangles, where AB and FE are the sides of squares with length 1, as well as DC and GH which are sides of squares length 2, and HA and DE are the hypotenuses of 45-45-90 triangles, and CB and GF are hypotenuses of 45-45-90 triangles, so together, we have $(1\cdot1)+(1\cdot1)+(2\cdot2)+(2\cdot2)+(2\cdot2/2)+(2\cdot2/2)+(1\cdot1/2)+(1\cdot1/2)=\boxed{15}$

SparklingWater2 Mar 3, 2021

#1**+1 **

Best Answer

We see that this picture us constructed of squares and 45-45-90 triangles, where AB and FE are the sides of squares with length 1, as well as DC and GH which are sides of squares length 2, and HA and DE are the hypotenuses of 45-45-90 triangles, and CB and GF are hypotenuses of 45-45-90 triangles, so together, we have $(1\cdot1)+(1\cdot1)+(2\cdot2)+(2\cdot2)+(2\cdot2/2)+(2\cdot2/2)+(1\cdot1/2)+(1\cdot1/2)=\boxed{15}$

SparklingWater2 Mar 3, 2021