What is the surface area and volume of a pentagonal pyramid with a slant height of 12.5 and a side length of 11?

Question

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Answer 1

The apothem, a,  of a regular pentagon  is the distance from the center of the pentagon to the midpoint of an edge.

Trigonometry can be used to show that a = (s/2) /tan(pi/10),  where s is the length of a side.  This is an exact result. 

 

The area, A, of a regular pentagon is  A = 5 a s/2

 

The volume, V,  of a pyramid is V = (1/3) A h, where h is the height.  

The height can be obtained from the slant height,  SH, by  SH² = h² + a²  

 

The surface area, SA, is given by A + 5 (SH a /2)

 

Taken together, these formulas give an exact answer to the questions. 

Numerical results can be worked out:

 a = (s/2) /.7265425 = 7.5701004

 A = 5 a s/2 =  208.17776

 h = sqrt(SH² - a²) =  9.9470388

 V =  (1/3) h A =  690.25022

 

With the numerical results for a, S and h,   a value for the surface area can also be worked out.