pyra.html

Pyramids and Prisms

Definition of pyramid Take a polygonal region R and a point V not in the plane of R. The set of points X which are on the line segment from V to some point in R is called the pyramid with vertex V and base R.

pyramidpr

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Definition of prism Take a polygonal region R and a vector V. The set of all points X such that , where P is in the region and is called the prism with base R and height vector V. If the vector V is perpendicular to the base then the prism is called a right prism.

pyramidpr2

The top of the prism with base R and height vector V is defined to be the set of all points X such that , where P is in R. Note that the top of the prism is the translate of the base by adding V. So it is congruent to the base.

Exercise.

Use plots[polygonplot3d] to draw the base and sides of the pyramid with base R= [[0, 0, 0], [1, 0, 0], [2, 1, 0], [1, 2.5, 0], [0, 2, 0], [-1, 1, 0]] and vertex V=[1,2,4]. Then use plots[display] to draw the pyramid.

> base := [[0,0,0],[1,0,0],[2,1,0],[1,2.5,0],[0,2,0],[-1,1,0]];

> V := [1,2,4];

> pl1 := plots[polygonplot3d](base,style=patch,color=magenta):

> sides := NULL:
for i from 1 to nops(base)-1 do
sides := sides,plots[polygonplot3d]([V,base[i],base[i+1]],
style=patch,color=yellow) od:

Color the remaining side.

> sides := sides,plots[polygonplot3d]([V,base[1],base[-1]],
style=patch,color=red):

> plots[display]([pl1,sides]);

More drawing exercises on pyramids.

1. Define a word in Maple, pyramid(R,V), which draws the pyramid with base R and vertex V.

2. Modify your definition of pyramid to include a color option.

3. Another way to draw a pyramid, is to draw the base and then draw the segments from the vertex of the pyramid to each vertex of the base. You can draw a segment in Maple with plots[polygonplot3d]. Modify your definition of pyramid to do this.

Exercises

Exercise. Draw the cylinder with base base R= [[0, 0, 0], [1, 0, 0], [2, 1, 0], [1, 2.5, 0], [0, 2, 0], [-1, 1, 0]] and height vector V=[1,2,3]. Hint: All you need to do is draw the base, the top, and the sides of the prism.

A solution

Exercises on prisms.

Define a Maple word prism(R,V) to draw the prism with base R and height vector V.

Is a cube a prism? What about a tetrahedron?

What is another name for this prism?

prism([[0,0,0],[2,0,0],[2,0,1],[1,0,2],[0,0,1]],[0,2,0])

Cross Sections of pyramids and prisms

Take a pyramid and a plane which cuts the vertex away from the base. The intersection of the plane with the pyramid is called a cross section of the pyramid. Similarly, a cross section of a prism is the intersection of the prism with a plane which cuts the top from the base of the prism. In each case, the plane which is used to form the cross section is called the cross sectioning plane.

pyramidcross1

Theorem about cross sections of a prism. Parallel cross sections of a prism are congruent. (Note: two cross sections are parallel if their cross sectioning planes are parallel.)

pyramidcross3

Proof: Take any side AB(B+v)(A+V) of the prism. The two parallel planes intersect this side in two parallel segments A'B' and A''B'', where A' and A'' are on the segment A(A+V), labelled with A' between A and A'' and B' and B'' are on B(B+V), labelled with B' between B and B''. The segments A'A'' and B'B'' are parallel, since they are both parallel to the vector V. So the quadrilateral A'B'B''A'' is a parallelogram. Let U be the vector from A' to A''. Then since A'B'B''A'' is a parallogram, B'' = B' + U. Continue all the way around the cross section containing A', and use the same argument to show that the vertices on the cross section containing A'' are all of the from X'' = X'+U, where X' is a vertex on the cross section containing A' . This means that the cross section containing A'' is the translate of cross section containing A' by adding U. So these cross sections are congruent by translation. qed

Theorem about cross sections of a pyramid . Parallel cross sections of a pyramid are similar.

pyramidcross2

Proof. We can assume (by translating if need be) that the vertex of the pyramid is at the origin. Take any one of the sides ABV of the pyramid. The two parallel planes intersect this side in two parallel segments A'B' and A1B1, where A' and A1=A" are on the segment AV, labelled with A' between A and A1 and B' and B1=B" are on BV, labelled with B' between B and B''. So the angles VA'B' and VA''B'' are congruent, since they are corresponding angles of the transversal VA of the parallel lines A'B' and A''B''. Then the triangles VA'B' and VA1B1 are similar since corresponding angles are congruent. Hence corresponding sides are proportional. In particular, . Call this constant of proportionality k. Then for each vertex X of the cross section containing A', the point is a vertex of the cross section containing A1. Hence the cross section containing A' is carried onto the cross section containing A1 by the similiarity transformation . qed

Exercises on cross sections of a prism.

Exercise : Show that the converse of the theorem about prisms is false, by finding two congruent cross sections of a cube which are not parallel.

Exercise : Define a maple word crossect(base,vertex,plane) which takes a polygonal base, a height vector, and an equation for a cross sectioning plane and returns the polygonal cross section of the prism.

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