Cone, Sphere and Cylinder in Detail
Any object that occupies space is known as a solid shape or 3-dimensional object. Almost all the objects around us can be considered a 3D shape. In this article, we will learn about three-dimensional figures. We will discuss the area of the cone cylinder and sphere. We will also learn how to derive the curved surface area as well as the total surface area of these solids.
Volume of Cone and Cylinder
The volume of a cone can be defined as the space or the capacity of the cone. A cone is a three-dimensional geometric shape having a circular base that tapers from a flat base to a point called an apex or vertex.
Let the radius of the circular base be ‘r’ and its vertical height be ‘h’. Then slant height ‘l’ can be calculated using Pythagoras Theorem.
Using Pythagoras Theorem, we can write
h² + r² = l²
l = \[\sqrt{h^{2} + r^{2}}\]
The volume of a cone formula is given as ⅓ πr²h cubic units.
Surface Area of Cone Derivation
Area of a Cone
There can be two types of surface areas that we may be concerned about.
Curved Surface Area - It is defined as the area of the curved portion
Total Surface Area - It is defined as the area of the curved portion and area of the circular base
Take a paper cone and we have to cut it along its slant height. Observe the figure which is formed by the surface of the cone. The figure obtained will be some portion of a circle. The radius of this circular region will be the slant height ‘l’ of the given cone.
If we divide this figure into infinite parts, (by drawing a straight line from the centre to the circumference) each of the parts will be a triangle
Now, the total area of this figure is the sum of all of these individual triangles. Hence,
Area of figure = (1/2) × slant height × (b1 + b2 + b3 + ………….. + bn)
= (1/2)× l × (length of an entire curved boundary)
Now, length of entire curved boundary = circumference of the base
= 2πr
Thus, the area of a figure (CSA of a cone) = (1/2)× l × (2πr)
= πrl
Total Surface Area (TSA) = Curved Surface Area(CSA) + Area of circular base
= πrl + πr²
= πr(l + r)
Cylinder:
A cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, that are at a fixed distance.
The formula for the volume of a cylinder is πr²h cubic units.
Where r is the radius of the cylinder
h is the height of the cylinder.
The Surface Area of Cylinder Derivation:
Suppose, we need to cover the surface of a cylinder using paper. The minimum area of paper needed to cover the curved surface of the cylinder will be the curved surface area of the cylinder. The minimum area of paper needed to cover all the surfaces (curved surface, top and bottom circular part) will be the total surface area of the cylinder.
Let ‘r’ and ‘h’ be the radius and height of the cylinder respectively. To cover the curved surface, a rectangular sheet of paper is needed (see the diagram below). The breadth of rectangular paper will be ‘h’ and its length will be the circumference of the circular part i.e 2πr.
Curved Surface Area (CSA) = Area of rectangular sheet
= length x breadth
= 2πr x h
= 2πrh
Total Surface Area (TSA) = Curved Surface Area(CSA) + Area of 2 circular part
= 2πrh + 2 x (πr²)
= 2πrh + 2πr²
= 2πr(h + r)
Sphere:
The volume of the sphere depends on the diameter of the radius of the sphere since if we take the cross-section of the sphere, it is a circle. The surface area of a sphere is the area or region which is covered by its outer surface. Formula to calculate the sphere volume, whose radius is ‘r’ is given below:
The volume of sphere = 4/3πr³cubic units.
Area of a sphere or total surface area of a sphere is the region covered by a surface of a spherical object in a three-dimensional space. Since the sphere is a complete curved shape therefore the curved surface area is equal to the total area of a sphere. It is also called a lateral surface area.
Area of sphere = 4πr² square unit
Comparison Between Volume of Cone and Volume of Cylinder
The volume formulas for cone and cylinder are almost similar. Let’s compare the volume of a cone and cylinder.
The volume of a cylinder is πr²h
The volume of a cone is ⅓πr²h
Hence the volume of a cone is exactly one-third (⅓) of Volume of the cylinder.
FAQs on Cone, Sphere and Cylinder
1. What are the basic definitions of a cone, a sphere, and a cylinder?
These are three common three-dimensional shapes:
- A cylinder has two identical flat circular ends and one curved side. Think of a can of soup.
- A cone has a circular base and a single point at the top called a vertex. Think of a party hat.
- A sphere is a perfectly round 3D object where every point on its surface is equally distant from the center. Think of a ball.
2. What is the main difference between how a cone and a cylinder are structured?
The key difference lies in their bases and vertices. A cylinder has two circular bases, one at each end, and straight sides connecting them. In contrast, a cone has only one circular base and its side tapers up to a single point called the vertex.
3. Could you explain the difference between Curved Surface Area (CSA) and Total Surface Area (TSA) for these shapes?
Yes, the distinction is simple. The Curved Surface Area (CSA) is the area of only the curved parts of a shape, excluding any flat bases. The Total Surface Area (TSA) is the total area of all surfaces, which includes the CSA plus the area of any flat circular bases.
4. Where can we see examples of cones, spheres, and cylinders in everyday life?
You can find these shapes all around you:
- Cones: Ice cream cones, traffic cones, and funnels.
- Cylinders: Drink cans, water pipes, and batteries.
- Spheres: Footballs, marbles, and planets like Earth.
5. Why does a cone have a 'slant height' but a cylinder just has a 'height'?
This is because of their structure. A cylinder's sides are perpendicular to its base, so its height is a straight line. A cone's surface slopes from the base to the vertex. The slant height (l) is the length of this sloped surface, while the height (h) is the perpendicular distance from the center of the base to the vertex.
6. How are the volumes of a cone and a cylinder related if they have the same base and height?
There's a very direct relationship between them. The volume of a cone is exactly one-third (1/3) the volume of a cylinder that has the same base radius and height. This means you could pour the contents of three identical cones into one cylinder of the same dimensions to fill it up completely.
7. Why doesn't a sphere have a separate 'Curved Surface Area' and 'Total Surface Area'?
A sphere is unique because its entire surface is curved. It has no flat bases or edges. Since there are no flat parts to add, its curved surface area is the same as its total surface area. It is one single, continuous surface.
8. If you slice a sphere perfectly in half, how do you find the total surface area of the new shape?
When you cut a sphere in half, you create a hemisphere. Its total surface area is not just half the sphere's area. You must also include the area of the new flat circular base created by the cut. Therefore, the TSA of a hemisphere is the area of its curved part (2πr²) plus the area of its flat circular base (πr²).
