Cone

A cone is considered as the three-dimensional shape of geometry which is smoothly tapered from a base that is flat to another point that is called apex or as the vertex.

A cone generally formed with the help of a set of line segments, half-lines that are connecting with a common point, the apex and with all the points present on the base which is plane.

There is a possibility that the base can also be restricted from the formation of a circle which is a one-dimensional quadratic form present on the plane. Which can be any of the closed figures that are one dimensional or it can be any other type of enclosed figure. The cone is considered to be a solid object, also it is considered as a two-dimensional object that is present in the space that is three dimensional. When a solid object is considered, the boundary that is formed due to the lines or due to the partial lines is the lateral surface. In any case, if the lateral surfaces are unbounded then it is called as the conical surface.

When a line segment is considered, the cones do not get extended that extends the base, besides of which when the half-lines are considered it gets extended infinitely. In the case of any of the lines, the cones get extended infinitely in both of the directions started from the apex, in this case, it is also called the double cone. May behalf of a double cone, are present individually on each side of the apex, this is called nappe.

The axis of the cone is also known as the straight line which is passed through the apex in which the base has circular symmetry.

In case of the common usage of the elementary geometry, the cones that are assumed which must be right circular in which the circular is defined as the base which is a circle and the word right defines the axis that is passed through the base center which is the right angle to that of the plane. If the cone is considered as right circular the plane intersects with that of the surface that is lateral which is a conic section. However, if the base is of any shape and the apex lies outwards. The right cones are known as the oblique cones where the axis is passed from the center of the base that is non-perpendicular.

A cone that has a base that has a shape of a polygon is known as a pyramid. Cone can be defined as a context that is described as a convex cone.

Any of the cones that have the section that is right circular is known as the right circular cone. This is a type of circular cone which has the axis that is perpendicular to the base of the cone.

Properties

• The slant height in the case of a right circular cone which has the length of an element. Both of the slant heights and all the elements that are denoted by naming L.
• The right circular cone altitude is considered as the perpendicular drop of a vertex that is to the center of the bus. It also coincides along the axis of any of the right-angled circular cone and this is denoted by naming h.
• If any of the right-angled triangles is revolved on the opposite side one of the leg of the right-angled triangle, in which the solid is formed is known as the right circular cone. Thereafter the surface that is generated due to the hypotenuse of any triangle is described as the lateral area of a right-angled triangle and the area of the base of the cone is considered as the surface that is generated due to the leg that is not considered as the rotation of the axis.
• All the axis of any of the circular cone is always equal.
• Any of the section which is parallel to that of the base is in which case the center is always on the axis of the cone is also considered.
• Any of the sections of a specific right circular cone that contains the two different points and the vertex of the base of the isosceles triangle.

The formula for the Right Circular Cone

Area of the base, A_b

The base or the right circular cone are also called the obvious circles

A _b = π r²

Lateral area, A_L

The lateral area in case of a right circular cone which is always equal to that of the cone which is equal to the one-half of the product of a specific circumference which has the base c and the height of the slant is L.

A_L = (1 / 2) c L

Considering c = 2πr, the formula in case of the lateral area of the circular cone which will increase in a very convenient form.

An _L = π r L

The relationship in between the base radius r, the altitude h, and the slant height L, which are already given

R² + h² = L²

Volume, V

The volume in case of the volume of the cone which is equal to that of the one-third of the product of the area of the base and the altitude.

V = (1 / 3) A _b h

V = (1 / 3) πr²h

Examples

1. Solve the area of the surface of the right cone which has the radius of 6 cm and the height of the plant is 10 cm.

Solution:

Given, Radius (r) = 6 cm

Slant height (s) = 10 cm

Surface area of a cone = ππ r(r + s)

Solving surface area,

SA = 3.14 ×× 6(6 + 10)

SA = 3.14 ×× 6 ×× 16

SA = 301.44

Therefore the surface area of the right angle cone is 301.4 square cm.

1. Solve the volume of right angle cone which has a radius of 6 cm and height 10 cm.

Solution:

Given, Radius = 6 cm

Height = 10 cm

Volume = 1/3 π r² h

Solving Volume,

V = 1/3 ×× ππ ×× (6)² ×× 10

V = 0.333 ×× 3.14 ×× 36 ×× 10

V = 376.42