Cone Formula

Cone Formula

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. 

What is Cone ?

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. 

Oblique Cone

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.

Cone Formula

Following are the cone formulas-

The Slang Height of the Cone

There are three important formulas related to a cone. They are the slant height of a cone, the volume of a cone, and its surface area. The slant height of a cone is obtained by finding the sum of the squares of radius and the height of the cylinder which is given by the formula given below. slant height² = radius² + height². If the slant height of the cone is ‘l’ and the height is ‘h’ and the radius is ‘r’, then l² = r² + h². The formula for the slant height of the cone is ‘l’ = $\sqrt{r^2+h^2}$

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.

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. 

V = (πr²h)/3

Solved Example on Cone Formula

Question: Calculate the volume of the cone if radius, r = 2 cm and height, h = 5 cm.

Solution: By the formula of volume of the cone, we get,

V = ⅓ πr²h

V = (⅓) × (22/7) × 2² × 5

V = 20.95 Cubic Cm

Question: What is the total surface area of the cone with the radius = 3 cm and height = 4 cm?

Solution: By the formula of the surface area of the cone, we know,

Area = πr(l + r)

Since, slant height l = √(r²+h²) = √(3²+4²) = √(9+16) = √25 = 5

Therefore,

Area, A = π × 3(5 + 3) = π × 3(8) = π × 24= 24π Cm²