Surface Areas and Volumes

A Complete Guide On Surface Areas And Volumes

As a student of standard 9, you are already introduced with various geometrical shapes and sizes. Starting from simple two-dimensional shapes like triangle, quadrilaterals, etc., now you are learning about complex three-dimensional shapes such as cone, cube, cylinder, etc.

Initially, you may face little difficulty to understand these shapes and their measurements. However, a thorough understanding of the surface area and volume formula of these shapes can clear all your doubts and concepts. 

Hence, a systematic explanation of geometrical formulas can help you understand the chapter of surface area and volume class 9. Once you adopt all the formulas and theorems of this chapter, it will be easy for you to solve problems from both 9th and 10th standards. 

Thus, let’s start learning!


Volume And Surface Area Of Solids

In this class 9 surface area and volume, you will learn about four shapes. Here are the formulas of volume and surface area of cone, cube, sphere, and cylinder-


Cone

This is a 3D geometric shape created by two lines segments connecting a common point. This point is known as vertex or apex. This shape tapers from a smooth and flat base to the apex. Following is the diagram of a cone.

                                                         (Image to be added soon)

Here are the formulas of volume and surface area of cone-

If the vertical height of the cone = h

Radius of the base = r

Slant height = l

Then, volume = 1/3 π r2 h

Total surface area = πr2 + πrl

Lateral area = πrl


Sphere

It is a geometrical shape in 3D space that is similar to the outer surface of a planet or ball. Mathematically, it can be defined as a set of points that are situated at an equal distance from any given point. You can check the below diagram of a sphere for reference.

                                                  (Image to be added soon)

To solve mathematical problems related to the sphere, you need to understand the formulas of surface area and volume of spheres.

If the radius = r,

Then, volume = 4/3 πr3

Surface area = 4πr2


Cylinder

This is a solid curvilinear shape with the surface formed by equal and fixed-distant points from a given line. For example, you can imagine the cooking gas cylinder and also find the diagram below. The diagram will help you to understand the cylinder area and volume formulas.

                                                    (Image to be added soon)  

Now, the formula of volume and surface area of cylinder-

If the radius of a cylinder = r

Height = h

The lateral surface area without the top and bottom surface = 2πrh

The total surface area including top and bottom surface = 2πr (r + h)

Volume = πr2h


Cube and Cuboid

A cube is formed with six square faces or sides, and each vertex is the meeting point of three sides. However, a cuboid also has six faces, but the cuboid faces can be any quadrilateral. Mostly, cuboid faces are rectangular. 

First, see the diagram of a cube.

                                                     (Image to be added soon)

Now, check the cuboid diagram to understand the difference between these two shapes.

                                                   (Image to be added soon)                              

Following are the formulas of volume and surface area of a cube -

If the length of one side of a cube = a,

Then, surface area = 6a2

Volume = x3

Now, let’s move to the surface area and volume formula of a cuboid-

If the height of a cuboid = h

Length = l

Width = w

Then, surface area = 2 (hl + lw + wh)

Volume = hlw

With the help of these above formulas, you can easily solve volume and surface area problems of cone, cube, cuboid, sphere, cylinder. 

However, if you require any more details on surface areas and volumes formula, you can check our Vedantu website. Also, you can ask your doubts out by attending our online classes where experts will help you with your problems.

Get our Vedantu app now for high-quality materials on class 9 surface area and volume chapter!

FAQ (Frequently Asked Questions)

1. What Are The Formulas Of Surface Area And Volume Of A Cone?

As per the formulas, the volume of a cone = 1/3 π r2 h, the total surface area = πr2 + πrl and the lateral area = πrl.

Let’s explain it with an example. What are the curved surface area and volume of a cone with 21 cm diameter, 10 cm height, and 20 cm slant height?

The radius will be 21/2 = 10.5 cm and l = 20 cm. Hence, the curved surface area is = πrl = (22/7 X 10.5 X 20) cm2 = 660 cm2. Now, the volume = 1/3 π r2 h, so, (1/3 X 22/7 X 10.5 X10.5 X 10) cm3 = 1155 cm3.

2. How Do You Use The Formula Of Surface Area And Volume Of A Cube?

A cube has equal height, length, and width. So, the volume formula is a3, and the surface area formula is 6a2, where a is the length of its sides. For example, if the side of a cube is 5 cm, then, the volume of that cube will be a3 = 53 cm3 = 125 cm3. And, its surface area will be 6 a2 = 6 X 52 cm2 = 6 X 25 cm2 = 150 cm2.

3. How Do You Explain The Formulas Of Surface Area And Volume Of A Cylinder?

Let’s explain these formulas through an example. What will be the diameter and volume of a cylinder which has a lateral surface area of 88 cm2 and height of 14 cm? As per the cylinder formula, lateral surface = 2πrh. Thus, 2πrh = 88 cm2, 2πr X 14 = 88.

From this equation you can find that r = 1 cm. So, the diameter will be = 2 X r = 2 cm. And, the volume will be πr2h = 22/7 X 1 X 1 X14 = 44 cm3

4. What Are The Sphere Volume And Surface Area Class 10 Formulas?

The formulas of sphere volume formula - 4/3 πr3 and surface area formula = 4πr2, where the radius is “r”. Here the use of these formulas in a geometrical problem. What will be the diameter and volume of a sphere with a surface area of 154 cm2? As per the formula 4πr2 = 154, thus, r = 3.5 cm. So, the diameter is 7 cm. And the volume is = 179.66 cm3.