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A cone’s shape is three-dimensional and is used vastly in geometric calculation. It tapers from a smooth flat base (mostly circular) to a point called apex or vertex.
The shape of the cone
A cone consists of lines, half-lines connecting to the apex from the base. The plane is mostly a circular one-dimensional plane. In the case of a solid cone-shaped object, the curved surface is called the ‘lateral surface’. The axis of a 3-dimensional cone is a straight line passing through the center of the base and joining with the vertex. In elementary geometry though comes is assumed to be right circular but in complex geometry, the case is rather different. But this paragraph mostly focuses on the right angular cone.
Formula of cone
Let’s assume the height of the cone is ‘h’ unit, radius be ‘r’ unit, slant height be ‘l’ unit.
The slant height of the cone- The slant height of the right circular cone refers to the distance from the apex to each corner point of the base. The formula of slant height is derived by Pythagoras theorem,
l = (r2 + h2)
In this way, the value of the slant height can be determined if the values of radius and height are known.
The cone’s volume- let’s assume the volume of the right circular cone be v. As mentioned earlier the radius of the cone is r, slant height is l and the height of the cone is h.
The cone’s volume (v) = 1/3 π * r2 * h cubic unit
In this way, the cone’s volume can be calculated if the values of radius and height are known.
The cone’s surface area- The total surface area of a right-angle circular cone is the total area of the lateral surface and the area of the circular plane.
The area of the circular plane is = π * r2 sq. unit.
The area of the lateral surface is = π * r * l sq. unit (where l is the slant height)
So, the total surface area of the right circular cone is equal to = π * r2 + π * r * l sq. units
= π * r * (r + l) sq. units
In this way, the value of a total surface area can be calculated if the values of radius and slant height are known.
Types of cones
There are mainly two types of cones in Geometry, one is a right circular cone and the other is the oblique cone. A cone that has a circular base and the line of axis passes from the vertex through the center of the circular plane is called a right circular cone. The word right is used because the axis makes a right angle with the base. This is the most common cone found and the main emphasis is put on this category in this paragraph. Whereas in a cone where the axis is not perpendicular to the base, but that has a circular base is called an oblique cone. It looks like a tilted cone or a slanting one.
Properties of the cone: A cone only possesses a single face i.e., the circular base of the cone.
A cone has only a single vertex or apex
The cone’s volume is 1/3 π * r2 * h cubic unit.
The right angular cone’s surface area is π * r * (r + l) sq. units.
The cone’s slant height is equals to (r2 + h2) unit.
Mathematical examples regarding cone-
- If the radius of the Cone is, r = 3 cm and the height is, h = 7 cm, then find the cone’s volume.
The cone’s volume is = 1/3 π * r2 * h cubic unit.
= 1/3 * 22/7 * 9 * 7
= 66 cubic cm
So, the cone’s volume is 66 cubic cm.
- If a cone has the radius of = 3 cm and the height of = 4 cm, what is the total cone’s surface area?
Answer. The total surface area of the right circular cone is = π * r * (r + l) sq. units.
= 22/7 * 3 * (3+4)
= 22/7 * 3 * 7
= 66 sq. cm
So, the surface area of the right circular cone is = 66 sq. cm
- Find the volume of a cone that has a base radius of 5 cm and a slant height (l) of 13 cm.
The formula of slant height is = -r2 + h2 unit.
So, the value of height is = -52 + 132 cm
= 12 cm
So, the cone’s volume is = 1/3 π * r2 * h cubic unit
= 1/3 * 22/7 * 25 * 12
= 314.28 cubic cm
So, the volume of the right circular cone is 314.28 cubic cm.
Conclusion
Right angle cone is a very basic and easy-to-grab concept in geometry. It is easy to learn and aspirants must take good care of this topic to score well in government service examinations.
