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Expert: - 6/25/2009


Question
I'm trying to calculate the volume of a solid that is similar to a pyramid.  If you picture the base of a pyramid, three of the sides go up at 37 degree angles with the fourth side going up at 90 degrees all meeting at a point directly above the base of the fourth line.

Can you help me determine the volume of this shape?

Answer

two side views of imag
The volume of a pyramid is Ah/3 where A is the area of the base and h is the height.  I had to think about this one for awhile until I realized that it was two shapes with bases that were half of a square.


If you think about all of the walls sloping, you can quickly see that it forms a pyramid with 37 degrees on each wall.

Let the base have a width of 2x.  This makes the area of the
base into A = 4x².  It also makes the h/x = tan(37 degrees),
so h = x tan(37).

If we then step back and remember the peak in the middle
of the four walls at the edge of the pyramid, one is vertical.  
There is a line that goes from the peak in the middle horizontally
over to above the edge of the shape that is directly above the
midpoint of the square on the bottom.

We can cut the pyramid in half, so half of the volume is still in the pyramid and the other half is in a box with a vertical triangular edge.


The volume of the pyramid that is left behind is half of the original, or (Ah/3)/2 = Ah/6.
Putting this in terms of x, we have A = 4x^2 and h = x tan(37 degrees).
This gives (4 x^2)(x tan (37 degrees))/6.
Noting that 4/6 = 2/3 and combining the x’s gives us 2x^3 sin(37 degrees)/3.


The volume of the triangle with width to it is the triangles area times the width.  The area of the triangle is bh/2.  The volume of this section is then the area of that triangle times the width, which is x.

Now the length of the base b was 2x.  The height h was just given as x tan(37 degrees).  The width is x.
Since the volume was bhx/2, we can put in the values.

Thus, the volume of the triangular rectangle is (2x)(x tan(37 degrees))(x/2).
The 2/2 cancels, leaving us (x^3)tan(37 degrees).


Looking at the two volumes, they can be added together.  Both have an x^3tan(37).
The multiples are 1 and 2/3.  The sum of those is 3/3 + 2/3 = 5/3.

So the volume is 5 x^3 tan(37 degrees) / 3.

Scott A Wilson

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