CBSE Notes For Class 9 Maths Chapter 11 Surface Areas and Volumes

1.0Introduction to Surface Areas and Volumes

We often encounter a variety of three-dimensional objects in our day-to-day life such as boxes, spheres, cylinders, and cones that take up space and have surfaces that can be measured. In maths, we measure these quantities and apply them in daily applications, such as when measuring milk or buying wood by surface areas. 

2.0CBSE Class 9 Maths Chapter-11 Surface Areas and Volumes - Revision Notes

What is Surface Area and Volume?

• Surface area measures the total area that the surface of an object covers. It is the sum of the areas of all the outer faces.  
• The curved Surface area measures only the outer sides or curved side of an object but not the base.
• Volume measures the space that an object occupies. It also refers to the capacity of a 3D object. 

There are different objects, and we have different formulas for each object. 

1. Cone 

A cone is a 3-dimensional geometric shape with a circular base and one curved surface connecting the base to a point, called an apex or vertex, above the base. The tip of “h” in the figure.

Formula related to Cone

h = height of cone

L = Slant height of cone

r = Radius of cone 

Slant height 

L² = h² + r²

Curved surface area of cone $=\pi rL$

The total surface area of the cone $=\pi r(L+r)$

The volume of the cone $=\frac{1}{3}\pi r^{2}h$

2. Hemisphere 

A hemisphere is a three-dimensional figure composed of a sphere cut into two halves. Then, it will have one curved surface, namely the outer part of the sphere, and one flat circular base representing half of the volume and surface area of the sphere.

Formulas related to Hemisphere

The curved surface area of the hemisphere $=2\pi {\mathrm{r}}^2$

The Total surface area of the hemisphere $=3\pi r^2$

The Volume of the hemisphere $=\frac{2}{3}\pi r^3$

3. Sphere 

In maths, a sphere is three-dimensional and is a perfectly round shape. Every point on its surface is equidistant from the centre. It has no edges or corners, merely a smooth, curved surface. This is the reason it does not have a curved surface area.

Formulas related to Sphere 

The Total surface area of the sphere $=4\pi {\mathrm{r}}^2$

The Volume of the sphere $=\frac{4}{3}\pi r^3$

3.0How do you solve questions related to Surface areas and Volumes?

Question 1: Find the total surface area, volume, and curved surface area of an inverted hat that is in the shape of a cone, given that its height is 28 cm and radius is 14cm. 

Solution: h = 28, r = 14, L = ? 

L² = 28² + 14²

L² = 784 + 196

L² = 980

L = 31.31cm

Total Surface area of cone $=\pi r(l+r)=\frac{22}{7}\times 14\times (31.31+14)$ = 1994cm²

Curved Surface area of cone $=\pi rl$

$\frac{22}{7}\times 14\times 31.31$

$=1378{\mathrm{cm}}^2$

The volume of the cone $=\frac{1}{3}\pi r^2\mathrm{h}$

$\frac{1}{3}\times \frac{22}{7}\times 14\times 14\times 28$

$=5749{\mathrm{cm}}^3$

Question 2: Find the volume of a sphere whose surface area is 132cm². 

Solution: Surface area of a sphere $=132=4\pi r^2$

$4\times \frac{22}{7}\times r^2=154$

${\mathrm{r}}^2=\frac{154\times 7}{4\times 22}$

$\mathrm{r}=7/2$

The volume of the sphere $=\frac{4}{3}\pi r^3$

$=\frac{4}{3}\times \frac{22}{7}\times \frac{7\times 7\times 7}{2\times 2\times 2}=179.67{\mathrm{cm}}^3$

Question 3: Find the total surface area of the hemisphere, which has a radius of 7cm. 

Answer: Total surface area of hemisphere $=3\pi r^2$

$=3\times \frac{22}{7}\times 7\times 7$

$=464{\mathrm{cm}}^2$

4.0Key Features of CBSE Maths Notes for Class 9 Chapter 11

• The notes are aligned with the latest CBSE curriculum. 
• These notes provide the solved problems to get a better understanding of surface areas and volumes.
• Visual aid is provided to help you understand the figures more clearly. 
• The notes are well-edited and analysed to ensure 100% accuracy.