Cone (solid)

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Image:Cone.jpg In common usage and elementary geometry, a cone is a solid object obtained by rotating a right triangle around one of its two short sides, the cone's axis. The disk swept by the other short side is called the base, and the endpoint of the axis which is not on the base is the cone's apex or vertex. An object shaped like a cone is said to be conical.

A cone with its apex cut off by a plane parallel to its base is called a frustum. The term cone is sometimes extended to more general shapes, such as the image of a common cone by an affine map — which may turn the basis into an ellipse (producing an elliptical cone) and/or tilt the axis relative to the base (an oblique cone). (See conical surface and conic solid for further generalization.)

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Formulas

The volume, <math>V</math>, of a cone of height, <math>h</math>, and base radius, <math>r</math>, is 1/3 of the volume of the cylinder with the same dimensions, i.e. <math>V = \pi r^2 h/3</math>. The center of mass (assuming the cone is filled with uniform density) is located on the axis, 1/4 of the way from base to apex.

The surface area <math>A</math> is given by <math>A = \pi r (r + s)</math>, where <math>s = \sqrt{r^2 + h^2}</math> is the "slant height" of the cone. The first term in the area formula, <math>\pi r^2</math>, is the area of the base; while the second term, <math>\pi r s</math>, is the area of the curved side surface.

Surface Area = Base Area + Lateral Area

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