# Cone Calculator

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Lets first understand, what is a cone and how we can find different properties of a cone?

Cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.

A cone is formed by a set of lines connecting a common point, the apex, to all of the points on a base that is in a plane that does not contain the apex.

In the case of line segments, the cone does not extend beyond the base.

In the case of lines, the cone extends infinitely far in both directions from the apex, so it is sometimes called a double cone. Either half of a double cone on one side of the apex is called a nappe.

The axis of a cone is the straight line passing through the apex, about which the base has a circular symmetry.

Different types of cone

If the cone is right circular the intersection of a plane with the lateral surface is a conic section.

A cone with a polygonal base is called a pyramid.

Depending on the context, 'cone' may also mean specifically a convex cone or a projective cone.

The perimeter of the base of a cone is called the directrix and each of the line segments between the directrix and apex is a 'generating line' of the lateral surface of the cone.

The 'base radius' of a circular cone is the radius of its base or simply called the radius of the cone.

The aperture of a right circular cone is the maximum angle between two generatrix lines, if the generatrix makes an angle θ to the axis, the aperture is 2θ.

A cone with a region including its apex cut off by a plane is called a 'truncated cone' and if the truncation plane is parallel to the cone's base, it is called a frustum of cone.

A cone with elliptical base is caleed and 'elliptical cone'.

Different equations of cone

Let the Radius of the Cone be \(R\) units

and the Height of the Cone be \(H\) units.

Volume of cone

Volume of the Cone will be \(\dfrac{1}{3}\left(\pi R^2 H \right)\) cubic units.

Surface areas of cone

Total Surface Area of the Cone will be \(\pi R\left(R + \sqrt{R^2 + H^2}\right)\) square units.

Curved Surface Area of the Cone will be \(\pi R \sqrt{R^2 + H^2}\) square units.

Slant height of cone

The slant height of a right circular cone is the distance from any point on the circle of its base to the apex via a line segment along the surface of the cone.

Slant Height of the Cone will be \(\sqrt{R^2 + H^2}\) units.

Center of mass of solid cone

The center of mass of a conic solid of uniform density lies one-quarter of the way from the center of the base to the vertex.

Equation form of cone

The surface of a cone can be parameterized as

$$f(\theta ,h)=(h\hspace{0.1cm}cos\theta,h\hspace{0.1cm}sin\theta,h),$$where \(\theta\hspace{0.1cm}\epsilon\hspace{0.1cm}[0,2\pi)\) is the angle around the cone, and \(h\hspace{0.1cm}\epsilon\hspace{0.1cm}R\) is the height along the cone.

For calculating different properties of the cone using above cone calculator, you have to just write the values in the given input boxes and press the calculate button, you will get the result.