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 =  349.066 cm3
S =  314.159 cm2

Solution:

$h=10 \ \text{cm} \ \\ A=30 ^\circ \rightarrow\ \text{rad}=30 ^\circ \cdot \ \dfrac{ \pi }{ 180 } \ =30 ^\circ \cdot \ \dfrac{ 3.1415926 }{ 180 } \ =0.5236=π/6 \ \\ \ \\ \tan A=r:h \ \\ \ \\ r=h \cdot \ \tan(A)=10 \cdot \ \tan(0.5236) \doteq 5.7735 \ \text{cm} \ \\ \ \\ S_{1}=\pi \cdot \ r^2=3.1416 \cdot \ 5.7735^2 \doteq 104.7198 \ \text{cm}^2 \ \\ \ \\ V=\dfrac{ 1 }{ 3 } \cdot \ S_{1} \cdot \ h=\dfrac{ 1 }{ 3 } \cdot \ 104.7198 \cdot \ 10=349.066 \ \text{cm}^3$
$s=\sqrt{ h^2+r^2 }=\sqrt{ 10^2+5.7735^2 } \doteq 11.547 \ \text{cm} \ \\ S_{2}=\pi \cdot \ r \cdot \ s=3.1416 \cdot \ 5.7735 \cdot \ 11.547 \doteq 209.4395 \ \text{cm}^2 \ \\ S=S_{1}+S_{2}=104.7198+209.4395=314.159 \ \text{cm}^2$

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