Axial cut

The cone surface is 388.84 cm2, the axial cut is an equilateral triangle. Find the cone volume.

Result

V =  480.661 cm3

Solution:

S=388.84 cm2 s=2r S=πr2+πrs=πr(r+s) S=πr(r+2r)=3 πr2  r=S/(3π)=388.84/(3⋅ 3.1416)≐6.4232 cm s=2⋅ r=2⋅ 6.4232≐12.8464 cm  h2=s2−r2 h=r⋅ 3=6.4232⋅ 3≐11.1253 cm  V=π⋅ r2⋅ h/3=3.1416⋅ 6.42322⋅ 11.1253/3≐480.6611≐480.661 cm3S=388.84 \ \text{cm}^2 \ \\ s=2r \ \\ S=\pi r^2 + \pi r s=\pi r (r+s) \ \\ S=\pi r (r + 2r )=3 \ \pi r^2 \ \\ \ \\ r=\sqrt{ S / (3 \pi) }=\sqrt{ 388.84 / (3 \cdot \ 3.1416) } \doteq 6.4232 \ \text{cm} \ \\ s=2 \cdot \ r=2 \cdot \ 6.4232 \doteq 12.8464 \ \text{cm} \ \\ \ \\ h^2=s^2 - r^2 \ \\ h=r \cdot \ \sqrt{ 3 }=6.4232 \cdot \ \sqrt{ 3 } \doteq 11.1253 \ \text{cm} \ \\ \ \\ V=\pi \cdot \ r^2 \cdot \ h/3=3.1416 \cdot \ 6.4232^2 \cdot \ 11.1253/3 \doteq 480.6611 \doteq 480.661 \ \text{cm}^3

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