Oblique Cone Section Calculator

Name: Oblique Cone Section - Geometry Calculator
Author: Jürgen Kummer

Calculations for an oblique cone section. When a right circular cone is cut obliquely through its lateral surface such that the cross-section forms an ellipse, two distinct parts are created, an elliptical slant cone and an oblique frustum. The elliptical slant cone contains the apex of the original cone, while the frustum contains its base. The area of the elliptical cross-section can be calculated, as can the volumes of both parts. However, the lateral surface area cannot be calculated using elementary functions. This requires elliptic integrals, which must be evaluated using numerical approximation methods.
Enter the radius and height of the cone, as well as the height of the intersection point, where the cutting plane meets the cone's axis above the base and the angle of the plane. Choose the number of decimal places, then click Calculate. Angles are calculated and displayed in degrees, here you can convert angle units.
The angle of inclination of the cutting plane must not be so steep that the entire elliptical cross-section extends beyond the cone's lateral surface. This angle is limited such that the cross-section does not intersect the base area of the cone. Mathematically, the condition is: |tan(α)|<s/r

Formulas:

a = \frac{r (h - s)}{h \cdot \cos (α) \cdot [1 - \frac{r^{2} \cdot \tan^{2} (α)}{h^{2}}]}

b = \frac{r (h - s)}{h \cdot \sqrt{1 - \frac{r^{2} \cdot \tan^{2} (α)}{h^{2}}}}

E = π \cdot a \cdot b

V_{1} = \frac{π r^{2} \cdot {(h - s)}^{3}}{3 h^{2} \cdot {[1 - \frac{r^{2} \cdot \tan^{2} (α)}{h^{2}}]}^{\frac{3}{2}}}

V_{2} = \frac{π r^{2} h}{3} - V_{1}

Source for the formulas: Frego, M.; Consonni, C.: Volume-Preserving Shear Transformation of an Elliptical Slant Cone to a Right Cone, Axioms 13 (2024), 245
The formulas for a, b, and V₁ were derived from the formulas given in the source for a elliptical slant cone, by converting the parameters c, m, and q into h, r, α, and s. E is derived from the well-known formula for the area of an ellipse. The volume of the frustum V₂ is calculated by subtracting the volume of the elliptical slant cone from the volume of the entire cone.
According to Frego and Consonni, the volume of the elliptical slant cone is equal to the volume of a right cone with the same elliptical base and a height corresponding to the perpendicular distance from the apex to the cutting plane. That is, equivalent to an elliptic cone with the same semi-axes a and b and a height of (h-s)*cos(α).

Radius, heights and axes have the same unit (e.g. meter), the ellipse surface has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter).

Conic sections have been studied since the times of ancient Greeks. The ellipse is the only closed, non-trivial conic section. The two parts produced by such a section are interesting in their own right, owing to their origin in a common geometric shape and their formation via the long-known method of conic sections.

Calculation of volume and other dimensions: oblique cone section

Last updated on 08/25/2026. Author: Jürgen Kummer

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