Question

explain how to get the formula for the volume of a pyramid using the volume of a prism. drag and drop the values to complete the sentences. if you have a prism with a specific base area and height, it would take pyramids with the same base area and height to have the same volume. the formula for the volume of a prism is that means the formula for the volume of a pyramid is ( v = 3bh ) ( v=\frac{1}{3}bh ) ( v=\frac{1}{3}b^{2}h ) ( v = bh ) three

Explanation:

Step1: Volume relationship between prism and pyramid

We know from geometric principles that if a prism and a pyramid have the same base area \(B\) and height \(h\), then the volume of the prism is \(V_{prism}=Bh\). Through experiments (such as filling a pyramid with sand and pouring it into a prism with the same base and height), we find that it takes three pyramids to fill the prism.

Step2: Volume formula of pyramid

Since \(V_{prism}=3V_{pyramid}\) (because three pyramids fill the prism), and \(V_{prism}=Bh\), we can solve for \(V_{pyramid}\). Rearranging the equation \(V_{prism}=3V_{pyramid}\) gives \(V_{pyramid}=\frac{1}{3}V_{prism}\). Substituting \(V_{prism} = Bh\) into the formula for the volume of the pyramid, we get \(V_{pyramid}=\frac{1}{3}Bh\).

Answer:

If you have a prism with a specific base area and height, it would take three pyramids with the same base area and height to have the same volume. The formula for the volume of a prism is \(V = Bh\). That means the formula for the volume of a pyramid is \(V=\frac{1}{3}Bh\).