There are many regular and, at times, not so everyday items we use, which are in the shape of a cone. The caps kids wear on birthdays is the immediate item that comes to mind when thinking about cone-shaped items, but there are others like the cone-shaped injector used to do icing on the cake. Hence, terms like the volume of cone exist in geometry.
So, for example, how much cream can fit into the cone or how much paper would be required to make the birthday caps, we need to know the properties of a cone and then the formula in order to calculate the volume and also the surface area of cone.
The common measures of any shape are the volume and the surface area of that shape, and by knowing these two measures, we are relatively quite well aware of how big or huge the object is. Let us quickly refresh the definition of surface area and volume before we go ahead and see what the surface area and volume exactly mean. The surface area, as the name aptly indicates, is a measure of how much surface area the shape occupies. Do see that objects which have a thickness can sometimes have an inner surface area and an outer surface area, and both can be different. For example, consider a thermos flask of cylindrical shape and thickness 2 inches, then the surface area inside the flask would be lesser than the surface area outside of the flask. The second measure is the volume which determines how much of the volume the object can hold, so taking the example of a cylindrical flask, its volume will determine how much amount of liquid it can hold. Also, notice that while the surface area is in square units, volume, on the other hand, is in cubic units.
Specifically, in the case of a cone which can be viewed as a concentric ring of circles, each smaller in size than the one below, stacked on top of one another till the final ring has zero radii. So, if we need to calculate the surface area, then it would be the addition of the circumference of many circular rings. Let us consider any of the circular rings in the cone of radius ‘x’ then the circumference would be 2 pi x, and to get the surface area of the cone, this 2 pi x needs to be added for values of x ranging from constant say ‘r’ to zero. It can be derived hence that the surface area of the curved part of the cone is pi * radius * length of the slant side of the cone. But the surface area of the cone does not only include the curved part but also the area of the base, which is a circle with radius ‘r’.Thus the surface area of cone with the radius of the base as ‘r’ and length of the slant side as ‘L’ is (r + L).
The volume of the cone can be imagined as the summation of the surface area of the many circular rings which constitute a cone. By making use of integration and summing up for a cone with height ‘h’ will give as the volume of a cone equal to 1/3 r2 h.
Do notice that the volume of a cone is directly proportional to the square of the radius of its base, so a cone with double the radius of its base will have a much greater volume than the cone with double the height but half the radius. Using the formulas discussed above, one can easily determine how much material it would take to make the cone or how much of the material can the conical shape hold. For more facts about cones, visit the Cuemath website.