Monday, June 13, 2011

3D Analogue to the Trapezoid (part 1)

The other day I derived a volume formula that had exactly the result that you would expect if you were to take a guess at it - I didn't even think about it until after I had derived it.

A trapezoid is a quadrilateral (four sided figure) with (at least) one pair of parallel sides.  (If it has two pairs of parallel sides it is a parallelogram, though technically it is still also a trapezoid.)

The area, A, of a trapezoid is given by, A = 1/2(b1 + b2)h, where b1 and b2 are the parallel sides and h is the distance measured perpendicularly between the two parallel sides.  You can think of the formula in words as, “the area of a trapezoid is equal to the average of the lengths of the parallel sides multiplied by the perpendicular distance between them.”

Below is a shape that is similar in some respects, but it’s three dimensional.  Instead of two parallel lines, there are three parallel lines with lengths a, b, and c.  At the base of this solid shape is a triangle.  (It’s not important that the lines all come to a common base – the volume formula I came up with will work anyways.)  What is significant about the triangle B is that it is formed by lines which are all perpendicular to the vertical lines (a, b, c).  I don’t know what this shape is called; let’s call it an irregular triangular prism.

The volume, V, of the above shape is given by V = 1/3(a+b+c)B, where B is the area of the triangle formed by a plane perpendicular to the three parallel sides and a, b, and c are the lengths of the parallel sides.  You can think of this formula in words as, “the volume of an irregular triangular prism is equal to the average of the lengths of the three parallel sides multiplied by the perpendicular cross-sectional area.”

(See proof.  Or, see a different analogous shape in part 2.)

Sorta makes you wonder it there’s a four dimensional analogue…