Axial section of the cone

The axial section of the cone is an isosceles triangle in which the ratio of cone diameter to cone side is 2: 3. Calculate its volume if you know its area is 314 cm square.

Correct result:

V =  366.43 cm3

Solution:

D:s=2:3 S=312 cm2  2r:s=2:3 r:s=1:3  3r=s  S=πr(r+s) S=πr(r+3r) S=4 πr2  r=S4π=3124 3.14164.9828 cm D=2 r=2 4.98289.9656 cm s=3 r=3 4.982814.9484 cm   Correctness test:  k=D/s=9.9656/14.9484230.6667  h=s2r2=14.948424.9828214.0935 cm  S1=π r2=3.1416 4.98282=78 cm2  V=13 S1 h=13 78 14.0935=366.43 cm3D:s=2:3 \ \\ S=312 \ \text{cm}^2 \ \\ \ \\ 2r:s=2:3 \ \\ r:s=1:3 \ \\ \ \\ 3r=s \ \\ \ \\ S=\pi r(r+s) \ \\ S=\pi r(r+3r) \ \\ S=4 \ \pi r^2 \ \\ \ \\ r=\sqrt{ \dfrac{ S }{ 4 \pi } }=\sqrt{ \dfrac{ 312 }{ 4 \cdot \ 3.1416 } } \doteq 4.9828 \ \text{cm} \ \\ D=2 \cdot \ r=2 \cdot \ 4.9828 \doteq 9.9656 \ \text{cm} \ \\ s=3 \cdot \ r=3 \cdot \ 4.9828 \doteq 14.9484 \ \text{cm} \ \\ \ \\ \text{ Correctness test: } \ \\ k=D/s=9.9656/14.9484 \doteq \dfrac{ 2 }{ 3 } \doteq 0.6667 \ \\ \ \\ h=\sqrt{ s^2 - r^2 }=\sqrt{ 14.9484^2 - 4.9828^2 } \doteq 14.0935 \ \text{cm} \ \\ \ \\ S_{1}=\pi \cdot \ r^2=3.1416 \cdot \ 4.9828^2=78 \ \text{cm}^2 \ \\ \ \\ V=\dfrac{ 1 }{ 3 } \cdot \ S_{1} \cdot \ h=\dfrac{ 1 }{ 3 } \cdot \ 78 \cdot \ 14.0935=366.43 \ \text{cm}^3



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Tips to related online calculators
Check out our ratio calculator.
Tip: Our volume units converter will help you with the conversion of volume units.
Pythagorean theorem is the base for the right triangle calculator.
See also our trigonometric triangle calculator.

 
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