How is the base of a shape related to its volume?

Question

Aurora Ayala · Accepted Answer

If prism-like then the volume is #"base" xx "height"#

If pyramid-like then the volume is #1/3 xx "base" xx "height"# A solid's volume is determined by multiplying its base area by its height if it has a prism-like shape and size at the top and bottom. A solid's volume is one-third of the product of its base area and height if it has a pyramid-shaped top with sloping sides and a single point. A cube can be divided into six square base pyramids with a common apex in the cube's center to illustrate this. If the length of each side of the cube is #1#, then the base area of each pyramid is #1xx1=1#, its height #1/2# and its volume is #1/6#, being #1/6# of the volume of the #1xx1xx1# cube. So such a square based pyramid has volume #1/3# of the product of its base area and height. If we uniformly stretch or compress the pyramid in any one direction, the formula holds true. If we subject the pyramid to a shear, it keeps its height and base, and the volume also stays constant (imagine a stack of coins). Given any two dimensional shape, we can approximate it arbitrarily closely with a grid of squares, then make square based pyramids on each of those squares up to a common apex. The total volume is the sum of the volumes of the pyramids, which works out the same as #1/3# of the height multiplied by the sum of the areas of the squares. Since the squares approximate the area of the original shape, the volume approximated by the pyramids approximates the volume of a pyramid based on the original shape.

Wyatt Anderson · Answer

undefined The base of a shape is directly related to its volume. In geometric shapes like prisms, cylinders, cones, and pyramids, the volume is determined by multiplying the area of the base by the height of the shape. Essentially, the base provides the foundation upon which the volume is built. A larger base area typically results in a larger volume, assuming other dimensions remain constant. Similarly, a smaller base area usually leads to a smaller volume.