SOLUTION: Compute the volume of an icosahedron whose edge measures 3 inches.

Question 567895: Compute the volume of an icosahedron whose edge measures 3 inches.
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
The volume of a icosahedron (a regular polyhedron with 20 identical equilateral triangular faces, 12 vertices, and 30 edges) is given by the formula:
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in which V is the volume in cubic units and s is the length of an edge in units matching the cubic units of the volume.
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For this problem you are given that the length of a side is 3 inches. Therefore, the volume calculated by using the formula will in units of cubic inches.
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Start with the formula for the volume:
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Substitute 3 for s to get:
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Replace the with its equivalent value of 27:
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Note that the 12 in the denominator of the leading term can be replaced by 3 times 4 to result in:
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Multiply the leading term by the last term (27) to get:
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Divide the 3 in the denominator of the first term into the 27 in the numerator. This reduces the first term as shown:
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and multiply out the numerator of the first term and you have:
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and if you prefer, you can multiply the right side to have the form of the answer be:
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Then by doing the distributed multiplication in the numerator you get:
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The square root of 5 is an irrational number. Therefore, there is not an exact answer to this problem. However, you can approximate the answer by letting the square root of 5 equal 2.23606798. Substitute that value for the square root of 5 and the formula becomes:
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Add the terms in the parentheses at it simplifies to:
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Multiplying out the numerator (45*5.23606798) and then dividing by the denominator (4) gives the "approximate" answer:
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and its units are cubic inches
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I hope this gives you the information you need to solve this problem.
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