Calculations at an oblique circular cone. This is a cone, where the apex isn't perpendicular above the base center. The deviation d is the horizontal distance of apex and base center.
Enter the base radius and two of the three values h, d and l. Choose the number of decimal places and click Calculate. Angles are calculated and displayed in degrees, here you can convert angle units. The lateral surface is calculated with an integral and can only be estimated here. The estimation is the better, the larger d is compared to r and h, so the more oblique the cone is.
α = arccos [ ( d² + l² - h² ) / ( 2 * d * l ) ]
β = arccos [ ( (d-r)² + m² - h² ) / ( 2 * (d-r) * m ) ]
if d = r: β = 90°
γ = arccos [ ( (d+r)² + n² - h² ) / ( 2 * (d+r) * n ) ]
L ≈ 2 * r * d, for d / ( r + h ) → ∞
π
L = r *
∫
√ [ r - d * cos(α) ]² + h² dα
0
B = π * r²
V = 1/3 * B * h
Radius, height, deviation and lengths have the same unit (e.g. meter), lateral and base surface have this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). The lateral surface is the curved part of the surface area.
The total surface area would, of course, be the base area plus the lateral surface area. This is not shown here because, although the base area can be calculated exactly, the lateral area is only an estimate. This estimate will vary in quality, as described above, depending on the skewness of the circular cone. With the appropriate mathematical skills, you can obtain a better estimate using the integral. There is probably also special software that can do this work for you.
The three lengths can be calculated using the Pythagorean theorem. The three angles are calculated using the cosine theorem of the triangle. The base area is the area of a circle. Finally, the volume is the same as that of a right circular cone with the same base area and height, according to Cavalieri's principle. For this principle, see the oblique cylinder.