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CBSE Notes Class 10 Maths Chapter 12 Surface Areas And Volumes
CBSE Notes Class 10 Maths Chapter 12 – Surface Areas and Volumes help you understand how to calculate the surface areas and volumes of different 3D shapes. In this chapter, you will learn formulas for cubes, cuboids, cylinders, cones, and spheres, along with combinations of solids. These CBSE Notes provide simple explanations, important formulas, and step-by-step examples to make revision easy and effective for your Class 10 board exams
1.0Introduction to Surface Area and Volume
This chapter relates to the measurement of the surface area and volume of various geometrical solids, such as cubes, cuboids, cylinders, cones, spheres, and hemispheres. It gives an understanding of how to calculate the surface areas and volumes directly and applied problems for those shapes.
2.0Download CBSE Notes for Class 10 Maths Chapter 12: Surface Area and Volume - Free PDF!!
Download your CBSE Notes for Class 10 Maths Chapter 12: Surface Areas and Volumes in free PDF format for quick and thorough revision. These notes include essential formulas, clear explanations, and solved examples to help you prepare confidently for your board exams.
3.0CBSE Class 10 Maths Chapter 12 Surface Areas and Volume - Revision Notes
Surface Area of Solids
Surface area measures the total area of all the outer surfaces of any 3D object. It is used to describe the surface external to the object, and a measurement is done in square units, that is cm² or m². Surface areas of solids are of two types:
- CSA (Curved Surface Area): It is the area of a 3D figure's lateral or curved surface, excluding the base or top. For example, in a cylinder, CSA means the curved surface area excluding the circular bases.
- TSA (Total Surface Area): It is the sum of all the outer surface areas of the 3D figure, both the curved/lateral surface and the bases. For example, in a cylinder, TSA comprises both the curved side and the area of the two circular bases.
Volume of Solids
Volume is a measure of the amount of space occupied by a 3D object. Volume is measured in cubic units-for example, cm³ or m³, which therefore represent how much capacity or content an object has.
Formulas Related to the Surface Area and Volume of Solids
Shape | Formulas related to the solids |
| Cube: A Cube has six identical square faces. Total Surface Area of cube = 6a2 Curved Surface area of cube = 4a2 Volume of Cube = a3 |
| Rectangle: A Rectangle has six rectangular faces. Total Surface Area of Rectangle = 2(lb+bh+hl) Curved surface area of Rectangle = 2(l+b)h Volume of Rectangle = lbh |
| Cylinder: A Cylinder consists of two circular base joints with a curved surface. The Total Surface Area of the Cylinder Curved Surface area of the cylinder The volume of the cylinder = |
| Cone: In maths, a cone has a circular base and slanted surface. The Total Surface Area of Cone = Curved Surface area of cone = The volume of cone Slant height |
| Sphere: Sphere has only one curved surface in the shape of a ball. The Total Surface Area of the Sphere = The volume of the Sphere |
| Hemisphere: A hemisphere is half of a sphere and consists of one curved surface and one base. The total Surface Area of the Hemisphere Curved Surface area of the Hemisphere The volume of the Hemisphere |
4.0CBSE Class 10 Maths Chapter 12 Surface Areas and Volume - Key Notes
TSA and CSA of various figures
Cuboid
- Volume = lbh
- Lateral/Curved Surface Area = 2lh + 2bh or 2h(l + b)
- Total Surface Area = 2lh + 2bh + 2lb or 2(lh + bh + lb)
Cube
- Volume = a³
- Lateral Surface Area = 4a²
- Total Surface Area = 4a² + 2a² or 6a²
Right circular cylinder
- Volume = πr²h
- Lateral/Curved Surface Area = 2πrh
- Total Surface Area = 2πrh + 2πr² or 2πr(h + r)
Right circular cone
- Volume = 1/3 πr²h
- Lateral/Curved Surface Area = πrl
- Total Surface Area = πrl + πr² or πr(l + r)
Sphere
- Volume = 4/3 πr³
- Surface Area = 4πr²
- Total Surface Area = 4πr²
Hemisphere
- Volume = 2/3 πr³
- Curved Surface Area = 2πr²
- Total Surface Area = 2πr² + πr² or 3πr²
Surface areas and volumes of a combination of solids
Two or more standard solids (Cube, Cuboid, Cylinder, Cone, Sphere and hemisphere) can be combined to form a new solid.
For e.g.
Frustum of a right circular cone
If a right circular cone is cut off by a plane parallel to its base, the portion of the cone between the plane and the base of the cone is called a frustum of the cone.
- Volume of frustum V = 1/3 πh (r₁² + r₁r₂ + r₂²)
- Slant height (l) = √(h² + (r₁ − r₂)²)
- CSA of frustum = πl (r₁ + r₂)
- TSA of frustum = πl (r₁ + r₂) + πr₁² + πr₂²
- Area of the metal sheet used to make a bucket = πl (r₁ + r₂) + πr₂²
5.0Solved Problems
Question 1: A canal is 300 cm wide and 120 cm deep. The water in the canal is flowing at a speed of 20km/h. How much area will it irrigate in 20 minutes if 8 cm of standing water is desired?
Answer: Volume of water flows in the canal in one hour = width of the canal depth of the canal speed of the canal water =31.21000=72000m3
In 20 minutes the volume of
Area irrigates in 20 minutes
.
Question 2: A cone of radius 4 cm is divided into two parts by drawing a plane through the midpoint of its axis and parallel to its base. Compare the volumes of the smaller cone and the bigger cone.
Answer: Let the height of the cone = h Therefore,
Answer: Let the height of the cone = h
As the plane cut through the midpoint of the cone’s axis then,
The height of the smaller cone will be = h/2
Therefore,
Volume of the cone ………….(1)
Volume of the smaller cone ……………..(2)
Comparing equations (1) and (2),
Question 3: A cone of maximum size is carved out from a cube of edge 14 cm. Find the surface area of the cone and the remaining solid after the cone is carved out.
Solution: The maximum-sized cone that can be carved out of the cube has a base radius of 7cm and a height of 14cm. So, the slant height of the cone
The surface area of the remaining solid = Surface area of the cube - the surface area of a cone
Question 4: Three cubes of a metal whose edges are in the ratio 3:4:5 are melted and converted into a single cube whose diagonal is . Find the edges of the three cubes.
Answer: Let the edges of 3 Cubes 3x, 4x, and 5x.
Diagonal of single resultant cube
The side of single resulted in a cube after melting = a = 12cm
The sum of the volume of 3 cubes = Volume of a single cube.
(a1)3+(a2)3+(a3)3 = a3
(3x)3+(4x)3+(5x)3 = 123
27x3 + 64x3 + 125x3 = 1728
216x3 = 1728
x3 = 8
x = 2 cm
Hence, the edges of cubes are 6 cm, 8 cm, and 10 cm.
6.0Key Features of CBSE Maths Notes for Class 10 Chapter 12
- The notes are aligned with the latest pattern of the CBSE Class 10 curriculum.
- Visual aids are provided with every concept to get a better understanding of surface area and volumes of solid.
- The notes are easy to understand, making it ideal for self-learning.