when finding the volume for cones and pyramids, why do you have to multiply it with 1/3 or divide the product by 3?
http://www.shmoop.com/basic-geometry/volume-pyramids-cones.html
http://www.youtube.com/watch?v=ucup8KAZe5I
When finding the volume of a cone or a pyramid, you multiply the base area by the height and divide the product by 3.
The reason for this multiplication by 1/3 or division by 3 is related to the geometry of cones and pyramids. These shapes have a unique property known as "similar figures". Similar figures have the same shape but differ in size. In the case of cones and pyramids, if you take three identical cones or pyramids and stack them together, their combined shape will fill the space of a cylinder or prism that has the same base and height.
By dividing the volume of this combined shape (cylinder or prism) by 3, you are essentially isolating the volume of a single cone or pyramid. This is similar to saying that one-third of the combined shape's volume corresponds to the volume of a single cone or pyramid.
In mathematical terms, the volume formula for a cone or pyramid is derived from the fact that one-third of the volume of a cylinder or prism with the same base and height represents the volume of a cone or pyramid.
When finding the volume of cones and pyramids, you multiply the base area with the height and divide by 3, or multiply by 1/3. This is because the volume formula for cones and pyramids includes a factor of 1/3. Let's break it down step by step:
1. Volume of a Cone:
The volume of a cone is given by the formula: V = (1/3) * base area * height. The base area represents the area of the circular base of the cone, and the height is the perpendicular distance from the base to the apex (top) of the cone.
To understand where the 1/3 factor comes from, imagine cutting the cone into three equal parts. Each of these smaller parts will be a pyramid with a base that is one-third of the original cone's base, and the height remains the same. The volume of each smaller pyramid is obtained by multiplying the base area with the height and dividing by 3. Since we have three of these smaller pyramids when we cut the cone into three parts, the total volume of the cone is the sum of these three pyramid volumes, which simplifies to (1/3) * base area * height.
2. Volume of a Pyramid:
Similarly, for a pyramid, the volume is given by the formula: V = (1/3) * base area * height. Here, the base area represents the area of the polygonal base of the pyramid, and the height is again the perpendicular distance from the base to the apex of the pyramid.
The 1/3 factor in the volume formula for pyramids comes from the same reasoning as for cones. If you imagined cutting the pyramid into three equal parts, each of these parts would form a smaller pyramid. The volume of each smaller pyramid is obtained by multiplying the base area with the height and dividing by 3. Since we have three of these smaller pyramids when we cut the pyramid into three parts, the total volume of the pyramid is the sum of these three pyramid volumes, which simplifies to (1/3) * base area * height.
Thus, multiplying the base area by 1/3, or dividing the product by 3, is essential to account for the specific geometric properties of cones and pyramids and arrive at their accurate volume calculations.