A cone is a three-dimensional geometric structure with a smooth transition from a flat, usually circular base to the apex or vertex, a point that creates an axis to the base’s centre. The cone can also be described as a pyramid with a circular cross section rather than a pyramid with a triangular cross section. These cones are described as circular cones as well.
A cone is a shape created by connecting the points on a circular base to a common point, known as the apex or vertex, using a series of line segments or lines (which does not contain the apex). The height of the cone is determined by measuring the distance between its vertex and base. The radius of the circular base has been measured. The slant height is the distance along the cone’s circumference from any point on the peak to the base. There is the Cone Formula that may be used to calculate the cone’s surface area and volume based on these numbers.
Right Circular Cone
The Cone Formula is used to find out parameters and area. A cone with a circular base and an axis that runs through the centre of the circle from the vertex of the cone to the base, just above the middle of the circular base is where the cone’s vertex is located. The reason the axis is referred to as being “right” in this context is because it either makes a right angle with the cone’s base or is perpendicular to it. This is the kind of cone that is used in geometry the most frequently.
The term “oblique cone” refers to a cone with a circular base but an axis that is not parallel to the base. This cone’s vertex is not quite above the centre of the circular base. As a result, this cone seems to be skewed or slanted. The Cone Formula is used to find the area.
There are three crucial Cone Formula. These three factors are a cone’s volume, surface area, and slant height. Finding the product of the squares of the radius and the height of the cylinder, as indicated by the Cone Formula below, yields the slant height of a cone. Radius2 + height2 equals slant height2. If the cone’s radius is r, its height is h, and its slant height is l, then l2 = r2 + h2. “L” = r 2 + h 2 is the Cone Formula for the slant height.
The surface area of the cone
The area encircled by a cone’s curved portion is known as the cone’s curved surface area. The curved surface area of a cone with radius “r”, height “h”, and slant height “l” is as follows:
Surface Area of a Curved Surface = πrl square units.
Derivation of the Formula
Take a cone with the dimensions height (h), base radius (r), and slant height (l). Cut the cone open from the centre, which resembles a sector of a circle, in order to calculate the surface area of cone derivation (a plane shape). Total cone surface area is equal to πr2 + πrl = πr (r + l).
The volume of the Cone
The Cone Formula is determined by its volume. A cone’s circular base tapers from a flat base to a point known as the apex or vertex in three dimensions. A cone is made up of a collection of line segments, half-lines, or lines that connect the apex—the common point—to each point on the base, which is in a plane without the apex.
Derivation of the Formula
Take a flask with a conical base and the same height as a cylindrical container. The conical flask should be completely filled with water before proceeding. Start filling the cylindrical container you took with this water. People will see that it does not completely fill the container. If they try to replicate this experiment, they will still see some empty space in the container. Repeat the experiment a second time, and see that the cylindrical container is this time totally filled. Thus, a cylinder with the same base radius and height has a volume that is equal to one-third that of a cone.
Solved Example on Cone Formula
- What is the equation for a cone’s volume?
A cone’s volume can be calculated using the Cone Formula ⅓ 𝜋r2h cubic units, where r is the radius of the cone’s circular base and h is its height.
- What is the equation for a cone’s total surface area?
𝜋r(l + r) square units, where r is the radius of the circular base and l is the slant height of the cone, equals the entire surface area of a cone.