A well-defined set of lines or segments of a line can join together at some common point, called apex and with all the other points placed on a circular base, forms a cone. Sometimes it is also referred to as a pyramid with a circular cross-section. Thus, a cone is a 3-dimensional geometric shape that widens smoothly as it proceeds from the apex towards the base. It is looked at as a group of non-congruent disks stacked one above the other to maintain a constant ratio of the radius of neighboring disks. The volume of cone relates to the parts and dimensions of the cone-like radius, slant height, and height of the shape. Before jumping straight, let’s first understand the terms in detail.
Any circular cone owns two faces, a vertex, and an edge. Vertex is the corner of the shape where all the other edges meet.
- The perpendicular distance from the vertex to the circular base of the respective shape is called the height, h of the cone.
- The slant distance from the vertex to any point on the circumference of the base is the slant height, l of the cone.
- The radius of the circular base is taken as the radius r of the figure.
Based on the mentioned values, the calculations regarding surface areas and volume of such figures become easy. Even the three of them are themselves related to each other.
All the three, radius, slant height, height form a right-angled triangle, and thus the application of the Pythagoras theorem helps you calculate the missing one when two of them are provided. According to the Pythagoras theorem, l=√(r^2+h^2).
The capacity or space occupied by the conical figure is its volume. The volume of a cone can be given as one-third of the volume of a cylinder.
Now coming to the concept of volume of a cone, you can derive the respective formula by assuming volume as V, the radius of the cone as r, slant height as l, and perpendicular height as h, then the formula becomes,
Volume (V)=1/3πr^2h, where π is a constant value.
It should be noted that volume is always measured in cubic units.
We have two types of cones: the right circular and the oblique cones.
- Right Circular Cone: If the axis of the cone forms a right angle with the circular base of the same then we call it a right circular cone. This is a common geometric shape in a three-dimensional plane.
- Oblique Cone: The oblique cone has a circular base but no perpendicular base. It’s tough to locate the vertex of this figure. This respective cone resembles a slant or a tilted cone.
The sum of the area of four walls (lateral surface area) and they are of the circular base together defines the surface area of cone. Hence, the formula of the same is,
The total surface area of a cone,
where πrl is the area of the four walls and πr^2 is the area of the circular base of the same.
Substituting the values of slant height, radius, and perpendicular height helps us obtain the area of the cone. The surface area is always measured in square units.
Note: You can find out any of the missing values (if all others are provided), by substituting the respective values in an appropriate position.
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