Cone: The Three-Dimensional Shape And Its Total Surface Area
Cone is a unique 3-d shape that has a circular base. The figure at first instance seems as if many non-congruent disks like structures stacked upon each other in a manner that the radius always is constant. The surface area of cone, volume, etc. depends on the base radius, the slant height, and perpendicular height of the figure. A cone comprises a single flat and curved surface.
As per the geometric and mathematical theory, a cone can also be called a pyramid with the base area cross-section. A conical figure is always described in terms of height, slant height, and base.
The surface area of the cone
The extent of the region engaged by the cone is the total surface area of the three-dimensional figure. Collecting numerous triangular-shaped figures rotated around the line of the axis will result in a cone. The point that is vertical to the center of the circular base is the vertex of the same. Since the cone owns a flat base thus, the curved and the total surface area remain the same for the three-dimensional shape.
The formula for the surface area of the conical figure
As discussed earlier a cone comprises a curved surface and a flat surface hence its curved surface area can be said to be the total surface area of the same. These are two types of areas that a cone has and the formulas for the same are:
TSA (total surface area) = πr (r + l)
CSA (curved surface area) = πrl
Application of Pythagorean theorem gives the relation among the TSA of the cone and the perpendicular height of the same. Hence if h is perpendicular height, r stands for radius, and l stands for slant height, the relation would be: l= √(h^2+r^2)
Henceforth, in terms of height total surface area of the cone can be represented as
TSA=πr{r+√(h^2+r^2)} square units
Similarly, the curved surface area CSA=πr {√( h^2+r^2)} square units
The related derivation of the formula for total surface area
For getting to the final formula of the total surface area of the cone we need to add up the circular base area of the figure which is (πr^2) and the curved surface area, which is πrl and this means
The total surface area of any cone= πr^2+πrl
The right circular cone and its surface area
The respective cone is a type with the right circular section as well as an axis that stands perpendicular to the base.
Then Surface area of the right circular cone is given by πr(r+l) and curved surface area of the same would be πrl
The volume of the conical figure
The space/capacity held by the cone is considered as the volume of cone. Finding the volume of a conical figure is an easy task. The same is said to be equal to 1/3rd of the cylindrical capacity having a similar base radius and height, i.e., one third the product of the area of the circular base and the perpendicular height of the conical figure, resulting in the formula to be (1/3) π(r^2) h cubic units.
To be remembered: Volume of cone= 1/3 * volume of the cylinder.
It should be noted that the capacity (volume) for a normal cone or a right circular cone or an oblique cone is always the same or equal.
Cuemath explains the geometric and 3-d concepts well with implied practical examples and details. The topics are easy to understand and apply practically. The experts at the Cuemath website clear the child’s doubts in the best possible way. Download the app for a better understanding and knowledge of the concept.