CBSE Class 10th board exams have commenced, and all that students need to do now is revision. Although preparing for the exams begins a year ago, what you revise before the exam has a huge impact on how you perform.
When the subject is Maths, revision before the exam becomes more important. Solving various problems of different types will boost your confidence. However, it is not possible to do that just before the exam. All you need to do before the exam is revise the important formulas that will help you solve the problems that will appear in the exam.
One chapter included in the CBSE class 10th with many formulas to remember is ‘Surface Area and Volumes’. Remembering all the formulas from this chapter will ensure you secure all the marks in the exam. Read through the article to revise the chapter concerning your exam.
|Table of Contents|
|Overview of the chapter|
Overview of the chapter
CBSE class 10th Maths syllabus has 15 chapters in all. A few chapters from the CBSE Class 10 syllabus have many formulas to memorise. One such chapter is Surface area and Volumes. Memorising the formulas from this chapter can secure all the associated marks for this chapter. First of all, take a look at the concepts covered in this chapter.
The chapter deals with the surface areas and volumes of a few three-dimensional geometrical shapes. The chapter briefly describes the formulas of various three-dimensional shapes like cuboids, cubes, cylinders, cones, and spheres.
The chapter discusses the shapes’ total surface area, curved surface area, and lateral surface area, whichever is applicable. Read the article further for detailed notes for each shape individually. For a deep knowledge of the chapter, go through the important concepts of Maths.
2D faces form three-dimensional objects. Therefore, their surface areas are the sum of the areas of all the faces of the figure. Surface areas are generally categorised as:
- Curved surface area: The area of the curved surfaces of the object form the curved surface area.
- Lateral surface area: The area of all the faces of the object, excluding the top and bottom faces, is the lateral surface area.
- Total surface area: The area of all the faces, including the bases, is called the total surface area.
Volume is the space occupied by the three-dimensional object. Volume is usually the product of the three dimensions of the object and is therefore expressed in cubic units. Students should go through the NCERT solutions for the CBSE class 10th Maths to better understand the chapter. Go through the below given important formulas for surface areas and volumes.
A cuboid is a three-dimensional object with a region covered with six rectangular faces.
Surface Area of a Cuboid
Consider a cuboid with dimensions as length l, breadth b and height h. The total surface area is the sum of the areas of all its six faces.
Therefore, the total surface area of a cuboid = 2(l×b) + 2(b×h) + 2(l×h)
= 2(lb + bh + lh)
The lateral surface area of the cuboid = 2(b×h) + 2(l×h)
Length of a diagonal of a cuboid =
√(l2 + b2 + h2)
Volume of a Cuboid
The volume of a cuboid is the space occupied within its six faces.
Volume of a cuboid = (base area) × height
= (lb) × h = lbh
A three-dimensional solid object with six square faces is a cube. It has twelve edges and eight vertices.
Surface area of a cube
The length, breadth and height of a cube are all equal.
Length = Breadth = Height = l
Therefore, the surface area of a cube =
2 × (3l2) = 6l2
The lateral surface area = 2(l × l + l × l) = 4l2
Diagonal of a cube =√3l
Volume of a cube
The volume of a cube = base area × height = l3
A solid object with two circular faces connected with a lateral face forms a cylinder. It, therefore, has three faces.
Surface area of a cylinder
Consider a cylinder of base radius r and height h. If opened along the diameter, the cylinder can be transformed into a rectangle of length 2πr and height h.
Surface area of a cylinder of base radius r and height h =
2π × r × h + (area of two circular bases)
= 2πrh + 2πr2
Volume of a cylinder
Volume of a cylinder = Base area × height
= (πr2) × h = πr2h
A cone is a three-dimensional shape with one circular base that narrows down smoothly from the base to a single point, called a vertex.
Surface area of a cone
Consider a cone with a circular base of radius r, slant length l and height h.
The curved surface area of this right circular cone is πrl.
The total surface area of the cone = Curved surface area + area of the base
= πrl + πr2 = πr(l + r)
Volume of a cone
Three cones of the same size form a cylinder of the same base and height.
Therefore, the volume of a cone is ⅓ that of a cylinder of the same base and height.
The volume of a cone = =(1/3)πr2h
A circular solid is a sphere, and all the points present are equidistant from the centre.
Surface area of a sphere
For a sphere, the total surface area is the same as the curved surface area.
The total surface area of a sphere = 4πr2
Where r is the radius.
Volume of a sphere
Volume of a sphere = (4/3)πr3
Maths is a subject of immense importance considering the CBSE class 10th board exams. The score you get in the subject is critical when deciding a student’s career. It, therefore, becomes necessary to score well in Maths.
Smart work and hard work are the keys to cracking any exam. When preparing for the CBSE class 10th board exams, it is important to consider which chapters can award you maximum marks and focus on them.
The chapter Surface Area and Volume will give you marks easily as only a few formulas have a role to play in solving the questions from this chapter.
It is advisable to memorise all the formulas from this chapter to hit the bullseye.
1. What is the surface area and volume of a hemisphere?
A sphere is a circular solid with points lying equidistant from the centre. A hemisphere is half the sphere with one flat surface. It looks like a closed bowl.
Consider a hemisphere with radius r.
Surface area of a sphere = 4πr2
A hemisphere is half of a sphere. Therefore,
The curved surface area of a hemisphere = 2πr2
Total surface area of the hemisphere = curved surface area + area of the base circle
Volume of a hemisphere is half that of the sphere.
Volume of a sphere 4/3πr3
Therefore, the volume of the hemisphere is 2/3πr.
2. What is the surface area and volume of a frustum of a cone?
When a plane parallel cuts a cone to the base of the cone, a form of solid is formed. This formed solid is called the frustum of a cone. When the cone is cut by a plane, the part with two circular faces is the frustum of the cone.
Consider the two circular faces of the frustum have radii r1 and r2. The height of the frustum is h.
The curved surface area of the frustum
= π (r1+r2) l
where l= √[h2+(r2 – r1)2]
Total surface area = curved surface area + areas of the two circular faces
Volume of the frustum of the cone =(1/3) πh (r12 + r22 + r1r2)
3. How do you find the surface area and volume of a combined figure?
The combination of solids is an important concept with respect to surface area and volume. Solid shapes are many times a combination of two or more different shapes.
Such complicated shapes can be analysed by looking at them as a combination and breaking them into simpler parts that do not need much study.
To find the area of a combination object, you need to break it down into easier known objects for which the surface area is known.
For instance, when two equal cubes are joined together, the result is a cuboid. Knowing the area of one cube and adding up the value will give you the value of the area of the cuboid.
Similarly, calculating the volume of a complex object breaking down the object into simpler shapes and adding them up will do the work.
For example, consider a cone mounted on a hemisphere of both of the same radii. Adding up the volumes of the cone and the hemisphere will give you the volume of the entire solid.
4. What sources can I use for preparing for the CBSE class 10th Maths exams?
The CBSE itself recommends NCERT textbooks for preparation for the board exams. It is also evident from the previous years’ question papers that the questions that appear in the exam are generally from the ones present in the NCERT textbooks exercises. Therefore, students are advised to thoroughly study the NCERT solutions for class 10th Maths.
Solving the sample question papers is also a good idea to get an idea of the paper pattern. F
To get practice with various problems, students can go through other reference books and solutions like the ones mentioned below.
5. How to prepare the chapter ‘Surface area and Volume’ effectively?
The chapter ‘Surface area and Volume’ is one of the easiest chapters in the CBSE class 10th syllabus. All you need to do is memorise the formulas, and you can ace all the questions from this chapter that will appear in the exam.
One of the best ways to memorise all the formulas is to write down the formulas on a sheet and stick them in a place where you can see them frequently. This will help you keep them in your mind. You can also prepare revision notes for the chapter and go through them a day before the exam.
To tackle any complex questions, practice is of utmost importance. Practice the questions from NCERT solutions thoroughly.
You can also refer to other reference books for a variety of questions.
6. What is the difference between a cube and a cuboid?
A cube and a cuboid both are three-dimensional geometrical shapes with a few similarities and differences. First, let us see the similarities.
- A cube and cuboid both have six faces
- They both are designed to have 12 edges.
- They have eight vertices.
Differences between cube and cuboid: Cube: A cube is three dimensional solid with six square-shaped faces. All the squares are of equal dimensions and have an angle of 90° between them. The opposite edges of the cube must always be parallel to each other and equal.
Cuboid: A three-dimensional solid formed by three pairs of rectangular faces is a cuboid. Their opposite faces are equal, but adjacent cannot be equal. A cuboid can also have two squares out of the six faces. It is also known as rectangular parallelepipeds or right prisms.