Formulae to find surface area and volume for cbse 10 students

Formulae to find surface area and volume for CBSE 10 students

Understanding how a formula is formed is the key to remembering it for long

By Topperlearning Expert 01st Jan, 2020 | 02:21 pm

Surface Area and Volume is a Mathematics chapter in CBSE Class 10. It carries a lot of weightage as far as the Maths board exam is concerned. And the smart students in the class know that this chapter is nothing more than applying the correct formula to find the area or volume of the shape given in the question.

Now there are several shapes, and each has three to four formulae which have to be remembered for class tests, the board exam, art class and even project work. If you try to by-heart these formulae, then you may jumble things up and forget them altogether. Mugging them up may also cause you to forget them at the time of answering questions during the exam. Being under tremendous pressure to remember a lot of things makes even your confident classmates to goof up and the Maths whizz to stumble.

So, the best way to remember these formulae is not to learn them by heart but understand how the formula is formed. In this way even if you forget the formula or you are doubtful regarding its correctness, you can always cross check and verify.

Without further ado, let’s demystify the various formulae that we need to remember in the chapter Surface Area and Volume for Class 10.

1. Cuboid and Cube

Both cube and cuboid are related to each other, and as far as their formulae are concerned, if you understand a cuboid, then you can easily derive the formula of a cube by replacing a few terms. In a cuboid, rectangles form the 6 faces. (Notice how it sounds similar to humanoid and human—a realisation which makes you happy, right?) So, a cuboid resembles a cube but with rectangles instead of squares.

Now, the area of a rectangle is length x breadth. To find the area of the lateral surface, you need to find the area of the 4 faces which are rectangles.

On opening up a cuboid, we get the net diagram shown below.

Hence, the lateral surface area is given by 2lh + 2bh.

Therefore, total surface area = 2lh + 2bh + 2lb

The volume of the cuboid is the space occupied by the cuboid. Imagine the volume by stacking rectangular sheets having area (l x b) one above the other till the stack reaches height h.

Volume of a cuboid = lbh

Now to find the respective formulae of the cube, we just have to substitute l = b = h = a in the formulae of the cuboid, since all the faces of a cube are made up of squares.

Formula for	Cuboid	Cube (l = b = h = a)
Lateral surface area	2lh + 2bh	4a²
Total surface area	2lb + 2lh + 2bh	6a²
Volume	lbh	a³

2. Cylinder and Cone

A cylinder has two circular faces, and we know that a circle has an area πr² and circumference 2πr. Now the curved surface area can be viewed as

Surface area of cylinder formula = 2πrxh

The volume can be viewed as the layers of circular disks each having area πr² kept one above the other to a height of h.

∴ Volume = πr²x h

Total surface area of a cylinder formula = Curved surface area + area of top circle + area of bottom circle

= 2πrxh + πr² + πr²

Now, we can deduce the total surface area of cone and the volume of the cone using the above formulae of the cylinder.

Formula for	Cylinder	Cone
Curved surface area	2πrh	πrl (l: slant height, l² = r² + h²)
Total surface area	2πr´h + πr² + πr²	(curved area + bottom circle) πrl + πr²
Volume	πr²h	(one-third of the volume of a cylinder)

3. Sphere and Hemisphere

A simple way to remember the surface area and volume of sphere is to remember this sentence:

‘The surface area of a sphere is 4 times the area of a circle.’

So,

Surface area or total surface area = 4πr²

And

Volume is

( 4 x volume of a cone having h = r)

For a hemisphere,

Surface area = half of the surface area of a sphere = 2πr²

Total surface area = half the surface area of a sphere + area of a circular top

= 2πr² + πr² = 3πr²

Volume = half of the volume of a sphere =

4. Frustum

For a frustum, we can take the reference of the cone to remember the formulae.

Formula for	Cone	Frustum
Curved surface area	Πrl	π(R+r)l (R: outer radius and r: inner radius)
Total surface area	πrl + πr²	(curved area + bottom circle + top circle) π(R+r)l + πr² + πR² (l² = (R - r)² + h²)
Volume

Using the above relationships, you can easily remember the formulae of all the shapes which are in your CBSE Class 10 Maths syllabus. Try this activity too:

Write the formulae we have discussed on a separate sheet of paper and repeat this process every day for 5 days, say Monday to Friday. On Saturday, you should be able to remember all of them.

To continue to have the formulae in your mind so that you can recall them on the day of the exam, you simply need to practise solving either direct questions or problems based on them every weekend!