Key Formulas and Real-Life Applications of Surface Area & Volume
Mathematics has a branch called Geometry which includes shapes and their properties. These shapes can be two-dimensional or three-dimensional. Two-dimensional shapes have two dimensions: Length and Breadth whereas three-dimensional shapes have three dimensions: Length, Breadth, and Height.
Area
The space occupied by any two-dimensional shape is known as Area. And, the space occupied by any three-dimensional shape is known as Surface Area.
For example, sphere is a three-dimensional shape but circle is a two-dimensional shape. Then, space occupied by a sphere, cuboid, cube, cylinder, cone, hemisphere etc. is known as Surface area but the space occupied by the circle, rectangle, square, triangle etc, is known as Area.
Area is measured in Square units.
Total Surface Area
Total surface area is the area of all the surfaces of an object including its base(s).
Curved Surface Area (Lateral Surface Area)
Curved Surface Area is the area of all the curved surfaces of an object excluding its base(s).
Surface areas for some of the three-dimensional geometric shapes are given as:
Cuboid: A Cuboid is the three-dimensional representation of a rectangle.
It has six surfaces (rectangles). The surface area of a cuboid can be found by summing up all the areas of six rectangles. Thus, the surface area of cuboid is .
The curved (lateral) surface area of a cuboid is the area covered by 4 rectangles. Thus, the lateral surface area of a cuboid is .
Cube: A Cube is the three-dimensional representation of a square.
It has six surfaces (squares). The surface area of a cube can be found by summing up all the area of six squares. Thus, the surface area of cube is .
The curved (lateral) surface area of a cube is the area covered by 4 squares. Thus, the lateral surface area of a cube is .
Cylinder: A cylinder is a three-dimensional rectangle with circular bases.
Curved (lateral) surface area of the cylinder is .
The total surface area of the cylinder will be the sum of areas of two bases and the curved surface area. Thus, the total surface area of the cylinder is .
Cone: A Cone is a pyramid with the circular base.
Curved (lateral) surface area of the cone is .
The total surface area of the cone will be the sum of area of the circular base and the curved surface area. Thus, the total surface area of the cone is .
Sphere: A Sphere is a solid shape. It is a three-dimensional circle.
The surface area of the sphere is equal to the area of four circles. Thus, surface area of sphere is .
Curved surface area of sphere will be the same, that is, .
Hemisphere: If we cut the sphere into two halves, then the half shape is known as a Hemisphere. Thus, it is the half part of the sphere.
The curved surface area of the hemisphere will be the half of the curved surface area of the sphere. Thus, the curved surface area of the hemisphere is .
The total surface area of the hemisphere is the sum of the curved surface area and the area of the circle (base). Thus, the total surface area of the hemisphere is .
Volume
The volume is the amount of space a three-dimensional object occupies. In other terms, volume is the capacity of an object or a container.An object can be solid or hollow. In order to determine the volume, the three dimensions: length, breadth and height must be known.The two-dimensional shapes do not have volume. The shapes rectangle, square, triangle, circle etc. do not have volume but cuboid, cube, cone, sphere etc. have volume as these are three-dimensional solid objects.
Volume is measured in cube units.
Volumes for some of the three-dimensional geometric shapes are given as:
Cuboid: A Cuboid is the three-dimensional representation of a rectangle.
A cuboid is made up of rectangular planes till its height.
Volume of cuboid can be found out using the area of the rectangle as the rectangular planes are stacked up to the height of the cuboid.
Since the area of the rectangle is , the volume of the cuboid will be the multiplication of the area of the rectangle and its height. Thus, the volume of the cuboid is .
Cube: A Cube is the three-dimensional representation of a square.
A cube is made up of square planes till its height.
Volume of cube can be found out using the area of the square as the square planes are stacked up to the height (side) of the cuboid.
Since all the sides of the square are equal, the volume of the cube will be the multiplication of the area of the square and its side. Thus, the volume of cube is .
Cylinder: A cylinder is a three-dimensional rectangle with circular bases.
Volume of the cylinder will be the multiplication of the area of the circular base and its height. Thus, the volume of cylinder is .
Cone: Cone is a pyramid with the circular base.
Volume of the cone is one third of the volume of the cylinder. Since the volume of the cylinder is , the volume of cone will be .
Sphere: Sphere is a solid shape. It is a three-dimensional circle.
From the following figure, we can see that the sphere is been inserted within a cylinder. We did so to find out the volume of the sphere.
Here, the height of the cylinder is equal to the diameter of the sphere as the sphere touches the upper and lower bases of the cylinder. Thus,
Volume of the sphere is two third of the volume of the cylinder.Since the volume of the cylinder is , the volume of sphere will be .
Since , then the volume of the sphere will be .
Hemisphere: Hemisphere is the half part of the sphere.
Volume of a hemisphere will be half of the volume of the sphere. Thus, the volume of the hemisphere is .
Below table shows the Surface Area and Volume of the basic geometric objects:
FAQs on Surface Area and Volume Explained for Students
1. What is the fundamental difference between surface area and volume?
The fundamental difference lies in what they measure. Surface area is the total area of all the exposed surfaces of a three-dimensional object, measured in square units (e.g., cm², m²). It's like the amount of wrapping paper needed to cover the object. In contrast, volume is the amount of space the object occupies or its capacity, measured in cubic units (e.g., cm³, m³). It's like the amount of water the object can hold.
2. What is the difference between Total Surface Area (TSA) and Curved Surface Area (CSA)?
Total Surface Area (TSA) is the sum of the areas of all surfaces of a 3D object, including its top and bottom bases. For example, the TSA of a cylinder includes the area of its two circular bases plus its curved side. Curved Surface Area (CSA), also known as Lateral Surface Area (LSA), is the area of only the curved or side faces, excluding the flat bases. For a cylinder, the CSA is just the area of its rectangular-like curved side.
3. What are the essential formulas for surface area and volume for common shapes as per the CBSE syllabus?
Here are the key formulas for common 3D shapes:
- Cube (side 'a'):
TSA = 6a²
Volume = a³ - Cuboid (length 'l', breadth 'b', height 'h'):
TSA = 2(lb + bh + hl)
Volume = l × b × h - Cylinder (radius 'r', height 'h'):
CSA = 2πrh
TSA = 2πr(r + h)
Volume = πr²h - Cone (radius 'r', height 'h', slant height 'l'):
CSA = πrl
TSA = πr(r + l)
Volume = (1/3)πr²h - Sphere (radius 'r'):
Surface Area = 4πr²
Volume = (4/3)πr³ - Hemisphere (radius 'r'):
CSA = 2πr²
TSA = 3πr²
Volume = (2/3)πr³
4. How do you decide whether to use TSA or CSA when solving a real-world problem?
You decide by considering which surfaces are exposed or need to be covered. Use TSA when you need to calculate the area of all surfaces, including the top and bottom. For example, calculating the material needed to build a closed box. Use CSA (or LSA) when you only need the area of the vertical or curved faces, excluding the base(s). For example, finding the cost of painting the four walls of a room (LSA) or the area of a label on a cylindrical can (CSA).
5. How is calculating the surface area of a combination of solids different from just adding their individual surface areas?
When solids are combined, some of their surfaces join and are no longer exposed. Therefore, you cannot simply add their individual total surface areas. To find the surface area of the new solid, you must add the exposed surface areas of each part. For example, for a cylinder with a hemisphere on top, you would add the CSA of the cylinder, the area of the cylinder's base, and the CSA of the hemisphere. The top of the cylinder and the base of the hemisphere are not included as they are joined together.
6. For a tent shaped like a cylinder topped by a cone, which areas are combined to find the total canvas required?
To find the total canvas required for such a tent, you would need to calculate the area of all exposed surfaces. This means you would add the Curved Surface Area (CSA) of the cylindrical part to the Curved Surface Area (CSA) of the conical part. The circular base of the cone and the top circular face of the cylinder are stitched together and thus are not part of the outer canvas area. The floor of the tent is also usually not included unless specified.
7. If a solid, like a metal sphere, is melted and recast into another shape, like a cylinder, what changes and what stays the same?
This is a classic problem involving the conversion of solids. When a solid is melted and recast, its volume remains constant. The amount of material does not change. However, its surface area will almost always change because the new shape will have different dimensions and form. The key principle is to equate the volume of the original solid to the volume of the new solid to find any unknown dimensions (like the height or radius of the new cylinder).
8. Why is understanding volume and surface area important for practical applications, like in packaging or construction?
Understanding these concepts is critical for efficiency and cost-saving. In packaging, surface area helps determine the minimum amount of material needed to create a box, reducing waste and cost. Volume determines how much product can fit inside the package. In construction, volume calculations are essential for ordering the right amount of concrete for a foundation or determining the capacity of a water tank. Surface area is used to calculate the amount of paint or plaster required for walls.
