A plant is designed to be in the shape of a regular pentagon with 92.5m on each side. A security fence surrounds the building to form a circle and each corner of the building is to be 25m from the closest point on the fence. How much fencing is required?

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To solve this, we would break the pentagon up into triangle, like one I diagrammed in the attachment.  We need to find the red line I diagrammed.  To find this, we need to consider the pentagon would be broken up into 5 triangles.  That means each central angle is 72...

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To solve this, we would break the pentagon up into triangle, like one I diagrammed in the attachment.  We need to find the red line I diagrammed.  To find this, we need to consider the pentagon would be broken up into 5 triangles.  That means each central angle is 72 degrees.  So, given one half of the triangle, or a right triangle, the top angle is 36 degrees.  So, given that, we can find the red line with trig:

sin 36 = 46.25/x

x = 46.25/sin 36

x = 78.69 m

Given that the fence is 25 meters from this, we would add 25 meters to this value for the radius of the fence:

r = 78.69 + 25 = 103.69 m

So, to find the amount of fencing, we would find the circumference:

C = 2*3.14*103.69 = 651.14 m

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