In nearly all of the important formulas for measuring objects, the different dimensions appear either in the exponents or the coefficients. For volumes of cones and pyramids, the dimensions show up in two different ways.
 
The contents of three conical cups exactly fill a cylinder of the same radius and height. 
 
The contents of three pyramids with rectangular bases exactly fill a prism of the same base and height. 
We do not know which ancient artisan first filled a cone three times from the water in a cylinder and thereby observed that the volume of a cylinder is three times the volume of a cone with like base and height. This relationship was later recognized to hold in a wide range of figures: a prism with rectangular sides and square or triangular base has three times the volume of a pyramid with like base and height. Once we observe this relationship, we can express it in formula: the volume of a cylinder or prism is the area of the base multiplied by the height, and the volume of a cone or pyramid is onethird the volume of the corresponding cylinder or prism.
 
Two congruent right triangles meet along a diagonal of a square.  
The relationship between volumes of prisms and volumes of pyramids is analogous to the relationship between areas of rectangles and areas of triangles in the plane. A diagonal line divides a square into two congruent isosceles right triangles. More generally, the area of a rectangle is the length of the base multiplied by the height, and the area of a triangle is onehalf the length of the base multiplied by the height. The pattern lies in the denominatior of the fraction. If we consider a fourdimensional analogue of a pyramid, then its fourdimensional volume should be onefourth the volume of its threedimensional base multiplied by its height in a fourth direction.
Mathematcians are not content merely to observe a pattern: they want to find an argument that shows why the same pattern will always occur. Experiments with waterfilled pyramids do not provide a proof of the volume formula in three dimensions. Fortunately we can establish this relationship by decomposing a cube into three congruent pyramids. Just as a square is decomposed into two congruent triangles which meet along a diagonal line, a cube can be decomposed into three identical pieces which meet along a diagonal of the cube. Each of the pieces is a squarebased pyramid, with its top point directly over one corner of the square base. It follows that the volume of each piece is onethird the volume of the cube. From the relationships in the second and third dimensions, the pattern is already clear. We cannot construct a real model to illustrate the analogous relationship in the fourth dimension, but we can still proceed to formulate hvpotheses. We expect that a fourdimensional hypercube can be decomposed into four congruent offcenter fourdimensional pyramids meeting along the longest diagonal of the hypercube. In general the ndimensioiial cube should be decomposable into n congruent offcenter ndimensional pyramids, each with an (n1)dimensional cube as its base.
The pattern appears both in the exponents and in the denominators of the formulas for area and volume of an offcenter pyramid. The area of an isosceles triangle with side length mis m^{2}/2. The volume of an offcenter pyramid obtained from a cube of side length m is m^{3}/3. The pattern predicts that the fourdimensional volume of an offcenter fourdimensional pyramid obtained from a hvpercube of side length m is m^{4}/4 . The analogous formula for the ndimensional volume of an offcenter ndimensional pyramid is m^{n}/n.
 
Three congruent pyramids meet along a diagonal of a cube. 
The analogy between the decomposition of the square in the plane and the cube in space gave us a way to find the volume formula for an offcenter pyramid with a square base. We would like to prove a formula for a more general class of pyramids with rectangular bases, but unfortunately the analogy between a rectangular region in the plane and a rectangular prism in space is not a perfect one. A square is divided into two congruent triangles by a diagonal, and so is a rectangular region. But even though a cube can be divided into three congruent pieces meeting along a diagonal, it is usually not possible to decompose a rectangular prism into three congruent pieces meeting along a diagonal. However, it is still possible to divide the rectangular prism into three noncongruent pieces of equal volume, and this result relies on a more sophisticated phenomenon, namely the effect of a scale transformation along a single direction.
 
Three noncongruent pyramids with equal volume meet along a diagonal of a rectangular prism. 
We can approximate the volume of a pyramid by stacking thick rectangular cards parallel to the base. If we double the thickness of each card in the stack, then the base stays the same while the height doubles and the weight of the stack (and therefore its volume) also doubles. If we keep the width and the thickness of each card the same and double the length, then the volume also doubles. Doubling any single dimension causes the volume to double, and in general multiplying a single dimension by any number will multiply the entire volume bv that same number.
 
A squarebased pyramid is decomposed into four rectanglebased pyramids having their top points over one corner. 
This procedure enables us to obtain the volume of any pyramid of rectangular base that has its top point directly over a corner of the rectangle. We begin with a cube, having three times the, volume of an offcenter pyramid contained within it. We double the length of one of the edges of the cube to form a rectangular prism, silnultaneously forming an offcenter pyramid with the same rectangular base. As we double the length of the edge of the prism, its volume doubles. But all the slabs approximating the volume of the offcenter pyramid also double, so the pyramid still has a volume onethird the volume of the prism. The basic volume relationship still holds.
What if the top point of the pyramid is located not over the corner but over some other point of the rectangle? We can their divide the pyramid up into four pyramids with the top point over one corner, and the volume of each of these is onethird the area of the base multiplied by their common height. Adding all of these contributions together shows that in general the volume of a pyramid with a rectangular base is onethird the volume of the rectangular prism with the same base and the same height.
 
Thin strips that fill a rectangle approximate the area of a parallelogram. 
It is also possible to deduce this result by applying Cavalieri's principle for shear transformations, which again uses thick slices to approximate areas and volumes: we can fill a parallelogram with the same set of rods that fills a rectangle of the same base and height, and we can fill a slanted box with the same pack of cards that fill a straight box. An offcenter pyramid can be approximated with the same collection of square cards that approximate a centered one.
The formulas for volumes of various kinds of squarebased pyramids can be modified to give formulas for pyramids formed by cutting off a corner of a cube. The dimension of the space again appears in the denominator of the formula, but in a new way. Cutting off one corner of a square produces a triangle with half the area. Cutting off one corner of a cube produces a pyramid with a triangular base. We can obtain this same triangular pyramid by decomposing a squarebased pyramid into four pieces. If we approximate the squarebased pyramid by a stack of square slabs, then the triangular pyramid is approximated by a stack of triangular slabs each with half the volume. It follows that the volume of the triangular pyramid is half the volume of the squarebased pyramid, or onethird the volume of the trianglebased prism, therefore onesixth the volume of the cube. The fourdimensional volume of a corner hyperpyramid of a hypercube will be onefourth the volume of the triangular pyramid which is its threedimensional base, namely ^{1}/_{24} the fourdimensional volume of the hypercube. In general the ndimensional volume of a corner figure of an ncube will be 1/n!, where n! stands for "n factorial," obtained by, multiplying together the first n numbers.

