Question

A cone fits inside a square pyramid as shown. For
every cross section, the ratio of the area of the circle to
Since the area of the circle is the area of the
square, the volume of the cone equals
the area of the square is
or 7
402
o the volume of the pyramid or - Alenchor
arh.
Cross section
r
o the volume of the pyramid or
+(1200)
2n.
o the volume of the pyramid or z { nºm) or
7
2r
2r
22h.
0 Į the volume of the pyramid or = { 29°m) oraz
arh.
m no

Answer 1

The volume of the cone is $\frac{\pi}{4}$ the volume of the pyramid

The ratio of the area is given as:

$R= \frac{\pi r^2}{4r^2}$

or

$R= \frac{\pi}{4}$

The volume of a cone is:

$V_1 = \frac 13 \pi r^2 h$

For a pyramid, the volume is:

$V_2 = \frac 43 r^2h$

Rewrite as:

$V_2 = 4 * \frac 13 r^2h$

Divide by 4

$\frac{V_2}{4} = \frac 13 r^2h$

Substitute $\frac{V_2}{4} = \frac 13 r^2h$ in $V_1 = \frac 13 \pi r^2 h$

So, we have:

$V_1 = \frac{V_2}{4} * \pi$

This gives

$V_1 = \frac{\pi}{4} V_2$

Hence, the volume of the cone is $\frac{\pi}{4}$ the volume of the pyramid

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