The derivation of the formula for the volume of a cone states that the volume of the cone is calculated to be [tex]$\frac{\pi}{4}$[/tex] times the volume of the pyramid that it fits inside.
Which statement best describes where the [tex]$\frac{\pi}{4}$[/tex] comes from in the formula derivation?
A. It is the ratio of the area of the square to the area of the circle from a cross section.
B. It is the ratio of the area of the circle to the area of the square from a cross section.
C. It is the difference of the area of the square and the area of the circle from a cross section.
D. It is the sum of the area of the square and the area of the circle from a cross section.
Answer (1)
Consider a cone inscribed in a pyramid with a square base.
Calculate the area of the square base as s 2 and the area of the inscribed circle as 4 π s 2 .
Determine the ratio of the circle's area to the square's area: 4 π .
Conclude that the 4 π factor represents the ratio of the area of the circle to the area of the square from a cross section. It is the ratio of the area of the circle to the area of the square from a cross section.
Explanation
Problem Analysis Let's analyze the problem. We are given that the volume of a cone is 4 π times the volume of the pyramid that it fits inside. We need to determine where the 4 π factor comes from in the formula derivation. The options relate this factor to the ratio, difference, or sum of the area of a circle and the area of a square from a cross section.
Cone and Pyramid Geometry Consider a cone inscribed within a pyramid such that the base of the cone is inscribed in the base of the pyramid. Assume the base of the pyramid is a square with side length s . Then the area of the square is s 2 . The base of the cone is a circle inscribed in the square. The radius of the circle is r = 2 s . The area of the circle is A = π r 2 = π ( 2 s ) 2 = 4 π s 2 .
Ratio of Areas Now, let's find the ratio of the area of the circle to the area of the square: Area of square Area of circle = s 2 4 π s 2 = 4 π
Other Area Relationships The ratio of the area of the square to the area of the circle is: Area of circle Area of square = 4 π s 2 s 2 = π 4 The difference of the area of the square and the area of the circle is: s 2 − 4 π s 2 = s 2 ( 1 − 4 π ) The sum of the area of the square and the area of the circle is: s 2 + 4 π s 2 = s 2 ( 1 + 4 π )
Conclusion The 4 π factor is the ratio of the area of the circle to the area of the square. Therefore, the correct statement is: It is the ratio of the area of the circle to the area of the square from a cross section.
Examples
Imagine you're designing a circular garden inside a square plot of land. Knowing that the area of the circle is 4 π times the area of the square helps you efficiently plan the garden layout. For instance, if you have a square plot of 100 square meters, the circular garden will occupy approximately 4 π × 100 ≈ 78.54 square meters. This understanding ensures you maximize space while maintaining the desired circular shape within the square boundary.
