What is the Difference Between Surface Area and Volume?

The difference between surface area and Volume is that the Surface Area measures the area occupied by the top layer of a surface, or put another way, it is the area of ​​all the shapes/planes that make up the figures/solids. In contrast, the Volume is the measure of the carrying capacity of a figure/shape or the space enclosed within the figure.

Mathematical methods have a wide scope in almost all fields: economics, physics, geography, or any other. Detailed knowledge and the correct surface area and Volume are important to stand out and achieve perfection.

These concepts become significant in solving real-world problems related to measurement and are covered in the Measurement unit. Integration methods find applications in calculating the area and Volume of irregular and complex surfaces.

Comparison table between surface area and volume

Comparison Parameter

Surface Area

Volume

Definition

It is the area of ​​all shapes/planes that make up the top layer of a figure/solid.

It is the space contained in the solid / three-dimensional figure or the amount of air inside it.

Dimension

It is a two-dimensional concept. The answer is always in a unit square like m² or cm².

It is a three-dimensional concept. The answer is always in a cubic unit like m³ or cm³.

Calculated for?

The surface area can be calculated for any figure on the plane or in space.

Volume is calculated for solids only because they have 3 dimensions.

real life examples

We find the surface area to estimate the area of ​​the walls to be painted to calculate the costs.

We find Volume to estimate how many products can be stored in a store.

Methods to calculate

By integration using the arc or arc revolution concept for complex figures/solids. 

Some formulas are predetermined as in: For Square = S * S and Sphere = 4πr²

By integration using the disk method, washer method, or cylindrical shell methods. Some formulas are special cases of the method as in: For cube = S * S * S

What is Surface Area?

The surface area is the total area covered by the Surface. We get the surface area if we convert our Surface to a 2-D plane and then calculate the total area. It can be calculated for any figure. For a one-dimensional line segment, the surface area is zero.

We will always have positive values ​​since the area is a scalar and only has magnitude. Whatever the Surface’s dimension, the area has two dimensions and would therefore have units such as m² or cm² or mm².

It is a concept widely used by architects and is very important and useful even for the common man. For example, estimating the time, speed, or cost of painting walls, putting up fences, delimiting constituencies, etc.

Some formulas:

  1. Square: S*S
  2. Rectangle: L * B
  3. Sphere. : 4πr²
  4. Cone. : πr (l + r)

Several methods have been formulated to find the area of ​​complex figures: The method to find the surface area is to visualize the solid or three-dimensional object as a revolution of a plane curve. For example, we can generate a sphere by rotating a semicircle. In this case, the area is the total of all the curved surface areas of ​​very small cylindrical pieces that can be cut. This is where integration comes into play; the area is equal to the integration of 2πf (x) √ (1+ (f ‘(x)) ²) with respect to x from x = a to x = b.

What is Volume?

Volume is the carrying capacity or amount of air contained within a solid/figure. It can be calculated for figures that have more than 2 dimensions.

We will have positive volume values because it is a scalar and only has magnitude. Volume is three-dimensional and would have units like m³ or mm³, or cm³.

It is widely used in companies to estimate storage capacity and scientific equipment such as beakers, syringes, etc. For example, to store sacks of grain or to measure medicines.

Some formulas:

  1. Bucket: S*S*S
  2. Cuboide: L * B * H
  3. Sphere. : (4/3) πr³
  4. Cono. : (1/3) πr²h

Methods for calculating the volume of complex and irregular figures:

  1.  Volume per slice:If the cross-sectional area of ​​a solid is known, we can find the Volume by integrating the area as a function of one variable for the domain of the variable.
  2.  Volume by disks:Visualizing solids as a revolution of a plane figure. We can then estimate the cross-sectional area of ​​the tiny, tiny pieces of the solid. The Volume would be the integration of π (f (x)) ² concerning x for the domain of x.
  3.  Volume by washers:In this case, our solid of revolution is formed by a region between two planes/curves. The cross-sectional area would be washer-shaped, and the Volume would be the integration of π[(f(x))²- (g(x))²] concerning x for the domain of x.
  4.  Volume by cylindrical shells:We can also solve the above problems without calculating the cross-sectional area by visualizing our solid as a body of very thin cylinders surrounded by. The Volume is the integration of 2πxf (x) with respect to x for the range of x.

Main Difference Between Surface Area and Volume

  • Surface area is a total of the area of ​​the planes that make up a surface/shape, while Volume is the space enclosed within a figure/shape/surface.
  • The Surface is a two-dimensional concept with units m², cm², or mm², while Volume is a three-dimensional concept with m³, cm³, or mm³ as units.
  • Surface area can be found for 2-D shapes like circles, squares, and rectangles, but can’t find Volume for them. While both can be found in 3-D solids/figures like a cube, sphere, cylinder, or cone.
  • The surface area is found to estimate the area of ​​the walls to be painted, while the Volume is found to estimate the storage capacity within the walls.
  • The area is calculated by integrating the arc or the revolution of an arc (according to the figure), while the Volume is calculated by integrating the revolution of a surface. These methods are used considering very complex functions and are part of higher-level studies.

Conclusion 

Everyone needs to distinguish between the concepts. Surface Area is the total area of ​​the topmost layer of a surface or the area of ​​all the planes that make up the figure by their intersection. Volume is the amount of air that can be filled or enclosed within the space between the intersection of these plans.

Courtesy: Queens College of Vocational Education, Melbourne

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