To derive the formula for the volume of a cone, we need to consider the relationship between the areas of the base shapes involved in the cross-section. Specifically, we compare the area of the circular base of the cone to the area of a square base of the pyramid that the cone could fit inside.
1. Area of the circle:
- The formula for the area of a circle with radius [tex]\( r \)[/tex] is [tex]\( \pi r^2 \)[/tex].
2. Area of the square:
- If the cone fits inside a pyramid, the base of this pyramid is a square. If we assume the diameter of the circular base of the cone is equal to the side length of the square, then the side length of the square is [tex]\( 2r \)[/tex]. Thus, the area of the square is [tex]\( (2r)^2 = 4r^2 \)[/tex].
3. Ratio of the areas:
- To find the ratio of the area of the circle to the area of the square, we divide the area of the circle by the area of the square:
[tex]\[
\text{Ratio} = \frac{\text{Area of Circle}}{\text{Area of Square}} = \frac{\pi r^2}{4r^2} = \frac{\pi r^2}{4r^2} = \frac{\pi}{4}
\][/tex]
Hence, the correct statement that best describes where the [tex]\(\frac{\pi}{4}\)[/tex] comes from 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."
