by Mangal Tiwari


The measure of the total area that a surface occupies is known as surface area. It can be classified into three types of areas.

  • Total surface area or TSA: It is the total area of the space occupied by a particular shape/object
  • Lateral surface area or LSA: It is Total Surface Area (TSA) minus the area of the top and the base
  • Curved surface area or CSA: Area of the curved surface of an object

Formulae of surface areas of some of the objects are as follows:

  • l = Length
  • b = breadth
  • h = height
  • L = slant height
  • r = radius
Cube 6l2 4l2
Cuboid 2(lh+bh+lb) 2h(l+b)
Right Circular Cone πr(L+r) πrL
Right Circular Cylinder 2πr(h+r) 2πrh
Sphere 4πr2 4πr2


The three-dimensional space that is enclosed within a closed surface is known as volume. Space refers to any substance such as solid, liquid, or gas that is held within the enclosed surface. For example, if we have a container and we want to know the volume, it generally refers to the amount of fluid that is held by the box.

Volume is assigned to three-dimensional shapes. This can be easily calculated by arithmetic formulae.

Some examples of objects and their volumes are:

Shapes Volumes
Cube l3
Cuboid lbh
Right Circular Cone ⅓ πr2h
Right Circular Cylinder πr2h
Sphere 4/3 πr3h

A complete study of Surface areas and Volumes requires the integration of sister subjects such as calculus, geometry, plane shapes and solid figures. Thus, the process of learning can become complicated and confusing at times. This article details five methods that will help students in grasping the subject and the concepts of this topic.


Generating interest in a subject is often a difficult task. However, this is probably the most crucial step in both the learning and teaching process. One way to build up interest is to introduce the subject in the form of a story. The best way to start a story is with a quick and fun history lesson.

The person who is credited with the formalization of the concepts of surface areas and volumes is Archimedes. He was always very interested in the concept of curves and had already mastered shapes made of straight lines. After that, he wanted to explore other shapes such as circles, ellipses, spheres, etc.

He was the first one to calculate the surface area and volume of a sphere and gave the formulae for it. Eudoxus of Cnidus had developed ‘the method of exhaustion’ that Archimedes used to invent these formulae. He cut the spheres into two hemispheres, as working with those was more manageable, and by doubling the volume, the resultant would be the volume of the entire sphere.

He took a hemisphere and fitted it in a cylinder. The circumference of the cylinder was equal to that of the hemisphere, and the heights of both shapes were equal too. He performed several calculations after that to give the formula of a sphere as we know today.

On the Sphere and the Cylinder is the work that talks about his process of discovery. It consists of two volumes published in 225 BCE. It gives a detailed account of how he founded the formula of a sphere for a cylinder.


Solving questions related to this topic requires visualising the surface we are talking about. For a simple surface with a known formula, plugging in the values and getting the answers is very easy. However, when we come to more complicated surfaces, this can become difficult. Thus, the first step in this process is to develop a student’s perspective of the concerned three-dimensional objects.

If students can virtually paint a picture of the object under consideration, this skill stays with them throughout their lives. Teaching students, with the help of blocks, cones and other solid objects, helps to achieve this.

Another way to teach students is by using a plane figure and a solid shape. For example, you can ask students to cut a rectangular sheet of paper. Make them roll it into a cylinder and ask them to note down their observations. It is an excellent way to teach students about surface areas, volumes and figure transformations.


Everyone, irrespective of age, loves to play games. What better approach than devising games along with learning experience. There are wide-ranging properties of different shapes that are used in the computation of surface areas and volumes. Moreover, there are a lot of axioms and postulates associated with various shapes. It becomes a hard job to learn and retain them in memory. Thus, with the help of a fun game, students will be able to assimilate this information and apply it to problems.

You can follow a simple template as listed below to create a game.

  • Assign a particular topic to an individual. If the class strength is considerable, this can be done in teams.
  • Ask students to research a particular topic as well as include fun facts in the research.
  • Give them a few days to prepare.
  • Then open the gaming session with a discussion of a few facts.
  • Let students quiz each other on the given topic.
  • Reward the student with the fastest and most accurate answers.

Questions can be based on diagonals, sides, a complete study of a particular shape, surface areas, volumes, simple facts, application of concepts and so on. For higher classes, calculus can also be integrated within the topic.


There are two approaches to this method.

Approach 1:

You can pick up an object such as a bottle, phone etc. Students can be asked to find the surface area and volume of this object. Give them specific guidelines and explain to them how the process works. This helps them to associate an academic topic with a real-life example. It works for anyone curious to have strong basics in the subject. It makes the topic more relatable and also challenges the analytical abilities of a person.

Approach 2:

Give real-life examples where this topic is used. This could not only give an idea to students about the industries that use this concept but also provide them with a hint about their probable career prospects. It is good practice to inform students about various career growth opportunities and industries while teaching a particular subject.

Some fields which use surface area and volumes are

Design: An industry that needs to design a particular object, be it clothes or cars, requires the calculation of surface areas. It helps in deciding the aesthetics and calculating the amount of material needed to make the final product.

Architecture: While constructing a building, you have to take into an account the volume and areas of different rooms. The blueprint of the building is also created based on the principles of surface areas.

Chemical Engineering: While conducting experiments, to get accurate results, you have to measure out exact volumes. This may also require shaving of some chemicals to match the surface area requirements.

The possibilities are endless but talking about a few options might inspire students to learn the subject well and further their career paths.


Sometimes it becomes hard to come up with innovative techniques to teach students. Furthermore, the curriculum needs to be well organized to support the learning process. Coming up with so many ideas can be a tedious task. Hence, taking help from an online platform which can supplement your teaching can be beneficial. Cuemath is a Math Program which helps in this process.

It is one of the best teaching websites to turn to. They focus on teaching concepts and building strong foundations. Cuemath has a wide-ranging variety of products that it offers to aid in teaching. Workbooks, worksheets, math boxes and apps help to organize a curriculum for a particular subject and present it in a fun way. With a fair amount of practice questions, students can understand the topic well, and good grades ensue.


Students are young and impressionable. It is essential to teach them topics in a way that instils confidence rather than fear. Using proper teaching techniques, coupled with stellar concept delivery, any teacher will always be successful in their goals. The techniques listed here, work not only for professional teachers but also for students and parents.

Satiate your curious mind and let your imagination run wild with this beautiful and remarkable topic of surface areas and volumes.

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