How does Cavalieri's Principle apply to two pyramids that have the same base area but different heights? Explain.

We know that the formula for the area of a rectangle is "base" times "height". The rectangle at the right has been sliced into 8 congruent sections. The sections have then been pushed to the right to form a slanted figure.

cavstacking1

The slanted figure has the same base, height and area as the rectangle, yet it is not a rectangle. The slanted figure resembles a parallelogram (except for all of the jagged edges). As we saw with the stacking technique, if we had sliced hundreds of very, very very thin sections, instead of 8 thick sections, we would see the parallelogram more clearly. The parallelogram will have the same area as the original rectangle.

Cavalieri described this relationship between figures using parallel lines.

Since parallel lines are everywhere equidistant, his figures ("regions") are of equal height.

Conditions:

• 2 parallel lines (p || q)

• two figures between the parallel lines

• every additional line parallel to p and q,

must cut equal widths on both figures

Conclusion:

• the areas of the two figures are equal

cacstacking2