CBSE Class 9 Maths Ch13 Surface Areas and Volumes Notes PDF Download

Surface Areas and Volumes Class 9 Notes help you understand how to calculate the outer covering and inner capacity of 3D objects like tanks, cones, balls and boxes. This chapter from NCERT focuses on formulas for cuboid, cube, cylinder, cone, sphere and hemisphere.

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These concepts are widely used in real-life problems involving storage, painting and construction. Let’s understand each solid clearly with formulas and diagrams. Make sure to revise from Maths Syllabus Class 9 regularly.

Surface area tells us the total area that covers the outside of a solid object. Volume tells us the space occupied by a solid object.

What is the Surface Area?

The space occupied by 2-D flat surfaces is called the area. It is measured in square units. The area occupied by a 3-dimensional object by its outer surface is called the surface area. It is also measured in square units.

Generally, area can be of two types–total surface area and curved surface area/ lateral surface area.

• Total Surface Area: refers to the area including the base(s) and the curved part
• Curved Surface Area/ Lateral Surface Area: refers to the area of only the curved part of the shape excluding its base(s)

What is Volume?

The amount of space, measured in cubic units, that an object or substance occupies is called volume. 2-D doesn’t have volume but has area only.

Cuboid

A cuboid is a three-dimensional solid figure bounded by six rectangular faces, with opposite faces equal. It has 12 edges and 8 vertices.

Let the length = l, breadth = b and height = h.

(i) Total Surface Area (TSA) of Cuboid

Each cuboid has 6 rectangular faces:

• 2 faces of area = l × b
• 2 faces of area = b × h
• 2 faces of area = l × h

TSA = 2(lb + bh + hl)

(ii) Lateral Surface Area (LSA) of Cuboid

Excluding the top and bottom, only 4 side faces remain:

LSA = 2h(l + b)

(iii) Volume of Cuboid

The volume of a cuboid = base area × height.

V = lbh  

Cube

A cube is a special cuboid where all edges are equal, i.e., l=b=h=al = b = h = a. It has 6 equal square faces.

(i) Total Surface Area (TSA) of Cube

TSA = 6a²

(ii) Lateral Surface Area (LSA) of Cube

LSA = 4a²

(iii) Volume of Cube

V = a³

Right Circular Cylinder

A right circular cylinder is a solid having two identical circular bases connected by a curved surface and the line joining the centers of the bases is perpendicular to them.

Let radius = r, height = h.

(i) Curved Surface Area (CSA) of Cylinder

If the curved surface is unfolded, it becomes a rectangle of length 2πr and height h.

CSA = 2πrh

(ii) Total Surface Area (TSA) of Cylinder

Includes curved surface + 2 circular bases:

TSA = 2πrh + 2πr²

(iii) Volume of Cylinder

V = πr²h

Right Circular Cone

A cone is a solid with a circular base and a curved surface that tapers smoothly to a point called the vertex.

Let radius = r, height = h, slant height = l.

(i) Relation between Height and Slant Height

By Pythagoras theorem:

l² = h² + r²

(ii) Curved Surface Area (CSA) of Cone

CSA = πrl

(iii) Total Surface Area (TSA) of Cone

TSA = πrl + πr²

(iv) Volume of Cone

V = 1/3 πr²h

Sphere

A sphere is a perfectly round 3D solid in which every point on the surface is equidistant from the center. The distance from the center to any point on the sphere is called its radius (r).

(i) Surface Area of Sphere

TSA = CSA = 4πr²

(ii) Volume of Sphere

V = 4/3 πr²

Hemisphere

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(i) Curved Surface Area of Sphere

CSA = 2πr²

(i) Total Surface Area of Sphere

TSA = 3πr²

(ii) Volume of Sphere

V = 2/3 πr²

Combination of Solids

Many real-life objects are combinations of two or more solids. For example:

• A toy made of hemisphere + cone
• A capsule (cylinder + hemispheres)

Steps:

1. Divide into known solids
2. Apply individual formulas
3. Add or subtract accordingly

Volume v/s Capacity

• Volume is the measure of space occupied by a solid (inside space).
• Capacity refers to the amount of liquid or material a container can hold.
• Both are measured in cubic units (e.g., cm³, m³, litres).

Note: 1 litre = 1000 cm³ and 1 m³ = 1000 litres

Conclusion

Surface areas and volumes are fundamental concepts in geometry with wide practical applications. When we know the formulas for cuboids, cubes, cylinders, cones and spheres, it becomes easy to calculate the paint required for objects, the storage capacity of containers and the space occupied by different solids.

This chapter not only helps us in solving mathematical problems but also strengthens our spatial reasoning skills and the ability to relate mathematics to real-life situations.

FAQs

Q1. What is the difference between curved surface area and total surface area?

Ans. Curved surface area includes only the curved part of a solid, while total surface area includes curved area plus base areas.

Q2. What is the formula for the volume of a cone?

Ans. The volume of a cone is ⇒ V = 1/3 πr²h

Q3. What is the surface area of a hemisphere?

⇒ Curved Surface Area = 2πr² and Total Surface Area = 3πr².

Q4. How do you solve a combination of solids problems?

Ans. Divide the figure into basic solids like cylinder, cone or hemisphere. Then apply individual formulas and add or subtract areas/ volumes.

Q5. What is the relation between volume and capacity?

Ans. 1 litre = 1000 cm³; capacity is the amount of liquid a container can hold, while volume is total space occupied.