Cone A2V

Surface of cone in the plane is a circular arc with central angle of 126° and area 415 cm2.

Calculate the volume of a cone.

Result

V =  881.1 cm3

Solution:

A=126⋅ π180=126⋅ 3.1416180≐2.1991 rad S=415 cm2  S=πs2⋅ A/(2π)  s=2⋅ S/A=2⋅ 415/2.1991≐19.4274 cm r=A⋅ s/(2π)=2.1991⋅ 19.4274/(2⋅ 3.1416)≐6.7996 cm h=s2−r2=19.42742−6.79962≐18.1986 cm  V=13⋅ π⋅ r2⋅ h=13⋅ 3.1416⋅ 6.79962⋅ 18.1986≐881.1169≐881.1 cm3A=126 \cdot \ \dfrac{ \pi }{ 180 }=126 \cdot \ \dfrac{ 3.1416 }{ 180 } \doteq 2.1991 \ \text{rad} \ \\ S=415 \ \text{cm}^2 \ \\ \ \\ S=\pi s^2 \cdot \ A / (2 \pi) \ \\ \ \\ s=\sqrt{ 2 \cdot \ S/A }=\sqrt{ 2 \cdot \ 415/2.1991 } \doteq 19.4274 \ \text{cm} \ \\ r=A \cdot \ s/(2 \pi)=2.1991 \cdot \ 19.4274/(2 \cdot \ 3.1416) \doteq 6.7996 \ \text{cm} \ \\ h=\sqrt{ s^2-r^2 }=\sqrt{ 19.4274^2-6.7996^2 } \doteq 18.1986 \ \text{cm} \ \\ \ \\ V=\dfrac{ 1 }{ 3 } \cdot \ \pi \cdot \ r^2 \cdot \ h=\dfrac{ 1 }{ 3 } \cdot \ 3.1416 \cdot \ 6.7996^2 \cdot \ 18.1986 \doteq 881.1169 \doteq 881.1 \ \text{cm}^3

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