Question 1203310
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if the volume of a hexagonal pyramid is 750 units 3 and the base and the height being 
5 units and 10 units respectively, what is the length of apothem of the pyramid?
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        As the problem is worded,  printed,  posted and presented in this parcel, 

        it is  SELF-CONTRADICTORY  and can not be solved.

        Below I explain it in a comprehensive way,  with all details.



<pre>
The term "apothem" is used in Geometry in combination with the conception of a regular polygon, ONLY.

So, I am forced to assume that the base is a regular hexagon - I have NO other choice.


Then it is clear that three longest diagonals divide the base of the hexagonal regular pyramid 
in 6 congruent equilateral triangles with the side length of a=5 units each.


Hence, the area of each such a triangle is  

    area of a triangle = {{{a^2*(sqrt(3)/4)}}} = {{{5^2*(sqrt(3)/4)}}} square units.


The area of the hexagonal base is 6 times this value

    area of the base = {{{6*25*(sqrt(3)/4)}}} = {{{75*sqrt(3)/2}}} = 64.95191 square units.


The volume of the pyramid is then

    V = {{{(1/3)*10*64.9590528}}} = 216.530176 cubic units,


which is  {{{highlight(highlight(FAR))}}}  from the given value of 750 cubic units.
</pre>


Thus the given data as the input is INCONSISTENT.
The problem CAN NOT be solved in this formulation.


Also, if the length of a base side is given, then the apothem 
can be found in one single and simple line.

Neither volume of a pyramid, nor its height are needed for such calculations.



By creating this nonsense, the problem's composer deserves to be ticketed.