Question 581522: How many different ways can you find the area of a regular octagon with all sides equal to 3 meters? Describe each method and use it to find the area. Are the answers the same? Should the answers be the same?
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Suppose the vertices of the octagon are A,B,C,...,H in counterclockwise order with AB = 3 m, and O is the center.
1. Divide up the octagon into four triangles and five rectangles (draw lines AF, BE, CH, DG). Find the area of each shape using your knowledge about 45-45-90 triangles.
2. Think of the octagon as a square with four right isosceles triangles cut off. Find the area of the square, then subtract off the areas of the triangles.
3. Divide up the octagon into eight congruent isosceles triangles (OAB, OBC, etc.). Use the area formula A = (1/2)ab sin C where a = b = the radius of the circumscribed circle, and C = 45 deg. You may need additional theorems, such as law of sines.
There are many other ways, but most of them are similar to the above three. You should obtain the same answer in each case.
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