Revolving to cone

A triangle whose sides are 20 cm, 16 cm, and 12 cm is revolving around its greatest side as axis. What is the volume space it generates after a full revolution in cubic centimeters?

Final Answer:

V =  1930.19 cm³

Step-by-step explanation:

$$\begin{gathered} a=20\text{cm} \\ b=16\text{cm} \\ c=12\text{cm} \\ a>b>c \\ s=\frac{a+b+c}{2}=\frac{20+16+12}{2}=24\text{cm} \\ S=\sqrt{s\cdot (s-a)\cdot (s-b)\cdot (s-c)}=\sqrt{24\cdot (24-20)\cdot (24-16)\cdot (24-12)}=96{\text{cm}}^2 \\ S=\frac{a\cdot r}{2} \\ r=2\cdot S/a=2\cdot 96/20=\frac{48}{5}=9.6\text{cm} \\ {h}_1=\sqrt{b^2-r^2}=\sqrt{16^2-9.6^2}=\frac{64}{5}=12.8\text{cm} \\ {h}_2=\sqrt{c^2-r^2}=\sqrt{12^2-9.6^2}=\frac{36}{5}=7.2\text{cm} \\ {V}_1=\frac{1}{3}\cdot \pi \cdot r^2\cdot h_1=\frac{1}{3}\cdot 3.1416\cdot 9.6^2\cdot 12.8\doteq 1235.3245{\text{cm}}^3 \\ {V}_2=\frac{1}{3}\cdot \pi \cdot r^2\cdot h_2=\frac{1}{3}\cdot 3.1416\cdot 9.6^2\cdot 7.2\doteq 694.87{\text{cm}}^3 \\ V=V_1+V_2=1235.3245+694.87=1930.19{\text{cm}}^3 \end{gathered}$$

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