Question

the formula for the volume of a pyramid is $v=\frac{1}{3}bh$. where does the $\frac{1}{3}$ come from? $v = bh$ is the formula for the volume of a prism, and a cube is $\frac{1}{3}$ of the size of a pyramid with the same base and height $v = bh$ is the formula for the volume of a prism, and 3 pyramids fit in a prism with the same base and height. $v = bh$ is the formula for the volume of a sphere, and 3 pyramids fit in a sphere with a diameter equal to the height. $v = bh$ is the formula for the volume of a cylinder, and a pyramid is $\frac{1}{3}$ of the size of a cylinder with the same height.

Explanation:

The formula \(V = Bh\) is for the volume of a prism (or a cylinder). Through experiments or mathematical proofs (such as using integration in calculus for a general case or simple geometric filling for a basic pyramid - prism relationship), it is found that three pyramids with the same base \(B\) and height \(h\) as a prism can fit exactly inside the prism. So the volume of one pyramid is \(\frac{1}{3}\) of the volume of the prism.

Answer:

The second option: \(V = Bh\) is the formula for the volume of a prism, and 3 pyramids fit in a prism with the same base and height.