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.

Result

V =  349.066 cm3
S =  314.159 cm2

Solution:

h=10 cm A=30∘→rad=30∘⋅ π180 =30∘⋅ 3.1415926180 =0.5236 =π/6  tan⁡A=r:h  r=h⋅ tan⁡(A)=10⋅ tan⁡(0.5236)≐5.7735 cm  S1=π⋅ r2=3.1416⋅ 5.77352≐104.7198 cm2  V=13⋅ S1⋅ h=13⋅ 104.7198⋅ 10≐349.0659=349.066 cm3h = 10 \ cm \ \\ A = 30 ^\circ \rightarrow 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 \ cm \ \\ \ \\ S_{ 1 } = \pi \cdot \ r^2 = 3.1416 \cdot \ 5.7735^2 \doteq 104.7198 \ cm^2 \ \\ \ \\ V = \dfrac{ 1 }{ 3 } \cdot \ S_{ 1 } \cdot \ h = \dfrac{ 1 }{ 3 } \cdot \ 104.7198 \cdot \ 10 \doteq 349.0659 = 349.066 \ cm^3
s=h2+r2=102+5.77352≐11.547 cm S2=π⋅ r⋅ s=3.1416⋅ 5.7735⋅ 11.547≐209.4395 cm2 S=S1+S2=104.7198+209.4395≐314.1593=314.159 cm2s = \sqrt{ h^2+r^2 } = \sqrt{ 10^2+5.7735^2 } \doteq 11.547 \ cm \ \\ S_{ 2 } = \pi \cdot \ r \cdot \ s = 3.1416 \cdot \ 5.7735 \cdot \ 11.547 \doteq 209.4395 \ cm^2 \ \\ S = S_{ 1 }+S_{ 2 } = 104.7198+209.4395 \doteq 314.1593 = 314.159 \ cm^2

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