## Thursday, September 17, 2015

### Unizor - Geometry3D - Truncated Pyramid

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Truncated Pyramids

Imagine a pyramid. Pick any point on any of its side edges and draw a plane through this point parallel to a base plane. Cut-off a piece of a pyramid between this new plane and an apex. Whatever is left is called a truncated pyramid.

Let's do a concrete construction of a truncated pyramid.

Assume that the original pyramid was SABCD with apex S and quadrilateral ABCD that belongs to a base plane β.

Pick some point A' on edge SA between apex S and vertex A on the base plane β.

Now draw a plane γ through point A' parallel to base β.

Plane γ intersects edges SB, SC and SD at points B', C' and D' correspondingly, so we have quadrilateral A'B'C'D' within plane γ that bounds our truncated pyramid from above.

We can say now that object ABCDA'B'C'D' is a truncated pyramid.

Since β ∥ γ,

the corresponding edges lying on parallel planes are parallel:

AB ∥ A'B',

BC ∥ B'C',

CD ∥ C'D',

AD ∥ A'D'.

Each side face of a truncated pyramid is, therefore, a trapezois.

Also, from the above property follows similarity of corresponding triangles:

ΔSAB ~ ΔSA'B',

ΔSBC ~ ΔSB'C',

ΔSCD ~ ΔSC'D',

ΔSAD ~ ΔSA'D'.

The center of scaling for all these similarities is apex S and, since pairs of these triangles share sides, they all share the same scaling factor.

That same scaling factor is between the bases of two pyramids - triangles ΔABC and ΔA'B'C'.

Therefore, two pyramids, small pyramid SA'B'C'D' that we cut-off and big original pyramid SABCD are similar with the same scaling center and factor.

Drop a perpendicular from apex S onto base β.

It falls into point H, so SH⊥β.

This same perpendicular is also perpendicular to plane γ since planes β and γ are parallel.

Let the intersection of this perpendicular with γ be point H', so H'∈AH and SH'⊥γ.

It's easy to prove that the same scaling with a center at apex S and the same factor as above for pyramids transforms point H' into H (good exercise for self-study).

We see that similarity between pyramids includes not only all edges but altitudes SH' and SH as well.

Now we can proceed with calculating the volume of a truncated pyramid as a difference between volumes of two pyramids - the original one SABCD and the one we cut-off from it, SA'B'C'D'.

Assume that the factor of scaling we discussed above is f.

Then all linear elements of two pyramids are in this ratio: LenSA/LenSA'=f (edge factor)

(the same for all other edges)

Let the lengths of altitudes SH and SH' be h and h' correspondingly.

Then h/h'=f (altitude factor).

Let the areas of bases of two pyramids, big SABCD and small SA'B'C'D' be s and s' correspondingly.

As was explained in the topic 3-D Similarity, the areas of similar objects are proportional to a square of a scaling factor:

Then s/s'=f².

The volumes of small and big pyramids are:

VolumeSA'B'C'D' = s'·h'/3

VolumeSABCD = s·h/3

Therefore, the volume v of a truncated pyramid ABCDA'B'C'D' equals to

v = s·h/3 − s'·h'/3

This is a good formula, but not good enough. It would be much cleaner to express the volume of an object in terms of elements of this object. In the formula above s and s' are such elements - areas of top and bottom bases of a truncated pyramid, but h and h' are not elements of a truncated pyramid since they involve a distance from apex S, which is outside of our truncated pyramid) to its bases.

Our goal now is to replace dependency on these parameters with a dependency on the height of a truncated pyramid itself - the distance between its bases a.

Let's summarize what we know and what has to be done in terms of elements of a truncated pyramid.

We know that

h/h' = f

h − h' = a

s/s' = f²

We have to express the volume v=s·h/3−s'·h'/3 in terms of s, s' and a.

Consider the top three equations above as a system of three equations with three unknown h, h' and f.

Solving it and substituting the solutions for h and h' into a formula for volume will produce the desired result.

v = (a/3)·(s+√s·s'+s')

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