Question

deriving a formula for volume of a cylinder
suppose the height of a cylinder is equal to its radius
the cylinder can fit inside a square prism, as shown
below.
the cross - sectional areas are still the same and the ratio
of the area of the circle to the area of the square is still
$\frac{pi}{4}$
complete the derivation of the formula for a cylinder
whose height is equal to its radius
the prisms volume is the area of the base,
$square$, times the height, $square$
since the ratio of the areas is $\frac{pi}{4}$, then the volume of
the cylinder is $square$ times the volume of the prism.
$v = \frac{pi}{4}(4r^{3})$, or $square$

Explanation:

Step1: Find the base area of the prism

The base of the prism is a square with side - length \(2r\). The area of a square \(A = s^2\), so \(A=(2r)^2 = 4r^{2}\).

Step2: Determine the height of the prism

Given that the height of the cylinder (and thus the height of the prism) is \(r\).

Step3: Find the volume of the prism

The volume of a prism \(V_{prism}=A\times h\). Substituting \(A = 4r^{2}\) and \(h=r\), we get \(V_{prism}=4r^{2}\times r=4r^{3}\).

Step4: Relate the volume of the cylinder to the volume of the prism

Since the ratio of the cross - sectional area of the circle (base of the cylinder) to the cross - sectional area of the square (base of the prism) is \(\frac{\pi}{4}\), and the heights are the same, the volume of the cylinder \(V_{cylinder}=\frac{\pi}{4}\times V_{prism}\).

Step5: Simplify the formula for the volume of the cylinder

We know \(V=\frac{\pi}{4}(4r^{3})\). Using the rule \(a\times\frac{b}{a}=b\) (\(a = 4\) in this case), \(V=\pi r^{3}\)

Answer:

The prism's volume is the area of the base, \(4r^{2}\) times the height, \(r\). Since the ratio of the areas is \(\frac{\pi}{4}\), then the volume of the cylinder is \(\frac{\pi}{4}\) times the volume of the prism. \(V=\frac{\pi}{4}(4r^{3})\), or \(\pi r^{3}\)