Oblique Cone

High School

Definition

A cone whose altitude does not intersect with the center of the cone's base. Its altitude is still perpendicular to the base, but it is not in the center. A oblique cone can be summarized as a 3D figure that has a lateral surface with altitude, a base that is circular, and a vertex.

Worked examples

An oblique cone with base radius \(r = 4\) cm and slant height \(l = 10\) cm has lateral area \(A = \pi r l = \pi (4)(10) = 40\pi\) cm².

The lateral area formula works the same way whether the cone is right or oblique.

If the altitude \(h = 6\) cm and base radius \(r = 3\) cm, volume \(V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (3)^2(6) = 18\pi\) cm³.

Volume depends only on altitude and base area, not whether the apex is centered.

Common mistakes

The altitude is the slant distance from vertex to base edge. → The altitude is perpendicular to the base plane, even if it misses the center. Altitude is always the perpendicular height, not the slant height.
Oblique cones have different volume formulas than right cones. → \(V = \frac{1}{3}\pi r^2 h\) for both right and oblique cones. Cavalieri's Principle shows volume depends only on base area and height, not tilt.
An oblique cone has an ellipse as its base. → The base is always circular; the apex is off-center. Oblique refers to where the vertex sits, not the shape of the base.

Where you'll use it next

You'll compare oblique and right cones when studying 3D geometry and solids of revolution, apply Cavalieri's Principle in calculus, and solve real engineering problems where structures tilt.

Found in 1 StudyPug lesson

Surface Area and Volume of a Cone

Geometry

A cone's surface area combines its circular base and unrolled lateral surface; its volume is one-third that of a matching cylinder.

10th Grade10th