Cone side

Calculate the volume and area of the cone whose height is 10 cm and the axial section of the cone has an angle of 30 degrees between height and the cone side.

Correct result:

V =  349.0659 cm³
S =  314.1593 cm²

Solution:

$$\begin{gathered} h=10\text{ cm } \\ A=30^{\circ }\to \text{ rad}=30^{\circ }\cdot {\displaystyle \frac{\pi }{180}}=30^{\circ }\cdot {\displaystyle \frac{3.1415926}{180}}=0.5236=\pi \mathrm{/}6 \\ \tan A=r:h \\ r=h\cdot \tan (A)=10\cdot \tan (0.5236)\doteq 5.7735\text{ cm } \\ {S}_1=\pi \cdot r^2=3.1416\cdot 5.7735^2\doteq 104.7198{\text{cm}}^2 \\ V={\displaystyle \frac{1}{3}}\cdot S_1\cdot h={\displaystyle \frac{1}{3}}\cdot 104.7198\cdot 10=349.0659{\text{cm}}^3 \end{gathered}$$

$$\begin{gathered} s=\sqrt{h^2+r^2}=\sqrt{10^2+5.7735^2}\doteq 11.547\text{ cm } \\ {S}_2=\pi \cdot r\cdot s=3.1416\cdot 5.7735\cdot 11.547\doteq 209.4395{\text{cm}}^2 \\ S=S_1+S_2=104.7198+209.4395=314.1593{\text{cm}}^2 \end{gathered}$$

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