# Cone. Formulas, characterizations and properties of a cone

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Definition of a cone
Elements of a cone
Volume of a cone
Surface area of a cone
Equation of a cone
Basic properties of a cone

Definition.

**Cone**is geometric body which is formed by a set of rays from apex (vertex) point and cross any flat surface, and at the intersection form

**basis**of the cone.

Fig.1 | Fig.2 | |

Fig.3 | Fig.4 |

## Elements of a cone

Definition.

**Apex (vertex)**of a cone is a point (K) of which overlook rays.Definition.

**Base**of a cone is plane is formed as a result of crossing the flat surface and all radiation emanating from the apex cone. In the cone may include a base such as circle, ellipse, parabola and hyperbole.Definition.

**Lateral height**(L) of a right circular cone is the distance from any point on the circle to the apex of the cone.Formula.

**Lateral height**of right circular cone in terms of radius R and height H (by the Pythagorean theorem):L = √R^{2} + H^{2}

Definition.

**Guide curve**of a cone is a curve, which describes the base of circuit cone.Definition.

**Lateral side**of a cone is the set of all generatrix cone. That is, the surface of which is formed by movement of generatrix on the guide curve of a coneDefinition.

**Surface area**of a cone consisting of the lateral surface of cone and bases.Definition.

**Height**of a cone (H) is segment, which comes from the vertex cone and perpendicular to its base.Definition.

**Axis**of a cone (a) is a line passing through the vertex cone and center of the base.Definition.

**Axial section**of the cone is cone-sectional plane that passes through the axis of the cone. This section forms an isosceles triangle whose sides are formed by generatrix and the base of the triangle is a diameter of base of cone.Definition.

**Tangent plan**to cone is a plane passing through generatrix cone and perpendicular to the axial cone section.Definition. Cone based on the circle, ellipse, parabola or hyperbole respectively called

**circular, elliptical, hyperbolic**or**parabolic**cone (the last two have infinite volume).Definition.

**Right**cone is a cone whose axis is perpendicular to the base. In such a cone axis coincides with the height and all generatrix equal.Formula.

where R - base radius, а H - cone height.

**Volume of circular cone**:V = | 1 | πHR^{2} |

3 |

Formula.

**Area lateral side**(A_{l}) of right cone in term of radius R and generatrix length L:A_{l} = πRL

Formula.

**The total surface area**(A_{t}) a right circular in terms of radius R and generatrix length L:A_{t} = πRL + πR^{2}

Definition.

**Oblique**cone is cone whose axis is not perpendicular to the base. In such a cone axis does not coincide with the height.
Formula.

where A - base area and H - cone height.

**The volume of any cone**:V = | 1 | AH |

3 |

Definition.

**Truncated**cone is part of cone which is located between the base cone and the plane of intersection cone.Formula.

where A

**Volume of truncated cone**:V = | 1 | (A_{2}H - A_{1}h) |

3 |

_{1}and A_{2}- area smaller base and larger base respectively, H and h the distance from the top cone (apex) to the center of the lower and upper bases, respectively.## Equation cone

1.

*The equation of the right circular cone with center at the origin the Cartesian coordinate system (x, y, z)*:x^{2} | + | y^{2} | - | z^{2} | = 0 |

a^{2} | b^{2} | c^{2} |

2.

*The equation of the right elliptical cone with center at the origin the Cartesian coordinate system (x, y, z)*:x^{2} | + | y^{2} | = | z^{2} |

a^{2} | b^{2} | c^{2} |

## Basic properties of a cone

1. All generators directly cone are equal.

2. When rotating a right triangle around his cathetus at 360 ° formed right circular cone.

3. When rotating isosceles triangle around its axis 180 ° formed right circular cone.

4. At the intersection of cone plane is parallel to the base cone formed a circle. (See. Truncated cone).

5. If the intersection of the plane is not parallel on cone and not overlap with the base is formed at the intersection ellipse (Fig. 3).

6. If the plane of section passing through the base is formed at the intersection of parabola (Fig. 4).

7. If the plane passes through the top section then the intersection formed an isosceles triangle (see. Axial section).

8. The center of gravity of any cone is one fourth the height of the center of the base.

Geometry formulas
Square. Formulas and Properties of a Square
Rectangle. Formulas and Properties of a Rectangle
Parallelogram. Formulas and Properties of a Parallelogram
Rhombus. Formulas and Properties of a Rhombus
Circle, disk, segment, sector. Formulas and properties
Ellipse. Formulas and properties of ellipse
Cylinder. Formulas and properties of a cylinder
Cone. Formulas, characterizations and properties of a cone
Area. Formulas of area
Perimeter. Formulas of perimeter
Volume. Formulas of volume
Surface Area Formulas

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