Cone side

Calculate the volume and lateral surface area of a cone with a height of 10 cm, given that the axial cross-section has an angle of 30° between the height and the slant side.

Final Answer:

V =  349.0659 cm³
S =  314.1593 cm²

Step-by-step explanation:

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

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

Try another example