Example 4. Six cones have a common vertex. At the same time, each cone has a common tangent line with four other cones. Find the ratio of the total volume of the six cones to the volume of the sphere that is tangent to all the bases of the cones.

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例 4. 六个合同的圆锥具有共同的顶点. 同时每个圆锥与另外四个圆雉各有一条公共的切线。求六个圆锥体积的总和对与所有圆锥底面都相切的球的体欭之比。
To solve this problem, we choose a cube as the structure. The common vertex of the cones is located at the center of the cube, and the base of the cone is inscribed in the face of the cube (Figure 5, only one cone is drawn). At this time, each of the six cones shares a common tangent line with four other cones, and the sphere that is tangent to the base of each cone is the inscribed sphere of the cube. Let the edge length of the cube be $\mathrm{a}$. Then the volume of one cone is
$$
\mathrm{V}=\frac{1}{3} \pi \cdot \frac{\mathrm{a}^{2}}{4} \cdot \frac{\mathrm{a}}{2}=\frac{\pi 1^{3}}{21} .
$$
If the total volume of the six cones is $\frac{\pi a^{3}}{4}$. The volume of the sphere is $\frac{\pi \mathrm{a}^{3}}{6}$. Therefore, the required ratio is $3: 2$.