To determine where the [tex]\(\frac{\pi}{4}\)[/tex] factor comes from in the derivation of the volume of a cone compared to the volume of the pyramid that it fits inside, we need to consider the areas of the cross-sections involved.
1. If you take a cross-section of the cone, you will obtain a circle. Let's denote the radius of this circle as [tex]\(r\)[/tex]. The area of the circle, [tex]\(A_{\text{circle}}\)[/tex], is given by:
[tex]\[
A_{\text{circle}} = \pi r^2
\][/tex]
2. If you take a cross-section of the square pyramid in which the cone fits, you will get a square. Suppose the side length of this square is [tex]\(2r\)[/tex] (i.e., a square that can circumscribe the circle). The area of the square, [tex]\(A_{\text{square}}\)[/tex], is:
[tex]\[
A_{\text{square}} = (2r)^2 = 4r^2
\][/tex]
3. The ratio of the area of the circle to the area of the square is:
[tex]\[
\frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{\pi r^2}{4r^2} = \frac{\pi}{4}
\][/tex]
Therefore, the [tex]\(\frac{\pi}{4}\)[/tex] comes from the ratio of the area of the circle to the area of the square from a cross-section. Thus, the best statement that describes the origin of the [tex]\(\frac{\pi}{4}\)[/tex] factor in the formula derivation is:
It is the ratio of the area of the circle to the area of the square from a cross section.
