{"id":248945,"date":"2022-11-25T09:43:08","date_gmt":"2022-11-25T04:13:08","guid":{"rendered":"https:\/\/www.aakash.ac.in\/blog\/?p=248945"},"modified":"2023-04-02T02:05:24","modified_gmt":"2023-04-01T20:35:24","slug":"how-to-calculate-the-area-of-a-sphere-cbse-10th-maths-revision-notes","status":"publish","type":"post","link":"https:\/\/www.aakash.ac.in\/blog\/how-to-calculate-the-area-of-a-sphere-cbse-10th-maths-revision-notes\/","title":{"rendered":"How to Calculate the Area of a Sphere? CBSE 10th Maths Revision Notes"},"content":{"rendered":"

You might have read about different ‘<\/span>Shapes and Solid Figures<\/span><\/i>‘ in Mathematics. Can you answer then what the shape of a cricket ball is? Isn’t your answer sphere shape? Objects like a football, a cricket ball, etc., are said to have the shape of a sphere.<\/span><\/p>\n

A sphere is a real object in many fields of Mathematics. From a bubble of soap to the shape of the Earth, you can easily identify various spherical and hemispherical objects in nature. Because of its smooth-rolling property, most of the balls used in sports are given spherical shapes.<\/span><\/p>\n

Mensuration, the branch of Mathematics, is an important topic for Class 10 students. You can easily spot a weightage of 6 marks from this unit in the CBSE Class 10 Mathematics exam. This unit consists of immediate problems related to the ‘<\/span>Volume and Surface Area<\/span><\/i> chapter.<\/span><\/p>\n

This article will help you quickly revise important topics of the Class 10 Mathematics and calculate the area of a sphere.<\/span><\/p>\n

Syllabus Of Class 10 Mathematics Exam<\/h3>\n

Before moving to revision or preparation of any subject for exams, it is a must to get an update about the exam syllabus. It is the most basic and important step for achieving an exceptional outcome.<\/span><\/p>\n

The recently updated Class 10 Mathematics syllabus includes the following units and their weightage:<\/span><\/p>\n\n\n\n\n\n\n\n\n\n\n\n
Name Of The Unit<\/b><\/td>\nWeightage<\/b><\/td>\n<\/tr>\n
Algebra(Cont.)<\/span><\/td>\n10<\/span><\/td>\n<\/tr>\n
Geometry(Cont.)<\/span><\/td>\n9<\/span><\/td>\n<\/tr>\n
Trigonometry(Cont.)<\/span><\/td>\n7<\/span><\/td>\n<\/tr>\n
Mensuration(Cont.)<\/span><\/td>\n6<\/span><\/td>\n<\/tr>\n
Statistics & Probability(Cont.)<\/span><\/td>\n8<\/span><\/td>\n<\/tr>\n
Total<\/span><\/td>\n40<\/span><\/td>\n<\/tr>\n
Internal Assessment<\/span><\/td>\n10<\/span><\/td>\n<\/tr>\n
Grand Total<\/b><\/td>\n50<\/b><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Exam Pattern Of Class 10 Mathematics<\/h3>\n

COVID-19 brings a lot of changes in the life of students too. CBSE has changed its exam pattern to help students with safety and precaution measures. Now, students of Class 10 and 12 have to face Term 1 and 2 board exams in different patterns with different time slots, two times a year.<\/span><\/p>\n

The recently updated exam pattern of Class 10 Mathematics is as follows:<\/span><\/p>\n\n\n\n\n\n\n\n
Section<\/b><\/td>\nTypes Of Questions<\/b><\/td>\nNumber Of Questions<\/b><\/td>\nWeightage<\/b><\/td>\n<\/tr>\n
Section-A<\/span><\/td>\nVery Short Answer Type Questions<\/span><\/td>\n6<\/span><\/td>\n12<\/span><\/td>\n<\/tr>\n
Section-B<\/span><\/td>\nShort Answer Type Questions<\/span><\/td>\n4<\/span><\/td>\n12<\/span><\/td>\n<\/tr>\n
Section-C<\/span><\/td>\nLong Answer Type Questions<\/span><\/td>\n4<\/span><\/td>\n16<\/span><\/td>\n<\/tr>\n
Total<\/b><\/td>\n14<\/b><\/td>\n40<\/b><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Know About Sphere<\/h3>\n

A sphere is a geometrical object with a three-dimensional analogue to a two-dimensional circle. It is the centre of a point that moves in space so that its distance from a fixed point remains constant. This fixed point is called the<\/span> sphere’s centre<\/span><\/i>, and the constant distance is called the<\/span> sphere’s radius<\/span><\/i>.<\/span><\/p>\n

From a visual perspective, a sphere has a three-dimensional structure that forms by rotating a circular disc with one of the diagonals. The term sphere was the earliest mentioned by ancient Greek mathematicians. The famous Greek Mathematician Archimedes was the first person to determine the volume and area of the sphere.<\/span><\/p>\n

Generally, the sphere’s radius is denoted by ‘r’ and its centre by ‘O’. If a radius is extended through the centre to the opposite side of the sphere, it creates a diameter. And it is denoted as ‘d’.<\/span><\/p>\n

Surface Area Of A Sphere<\/h2>\n

The area covered by the sphere’s outer surface is known as the surface area. A sphere is a three-dimensional form of a circle.<\/span><\/p>\n

Because a sphere is three-dimensional, it does not have an area. However, when depicted in two dimensions, a sphere is circular. Thus, to best understand a sphere in three dimensions, it is often helpful to start by understanding the area of a circle.<\/span><\/p>\n

The area of a circle is found using the following equation:<\/span><\/p>\n

Area of a circle = r\u00b2.<\/span><\/p>\n

Therefore, for a sphere,<\/span><\/p>\n

Area of a sphere = 4r\u00b2<\/span><\/p>\n

The great Mathematician Archimedes, that if the radius of a cylinder and sphere is ‘r’, the surface area of a sphere is equal to the lateral surface area of the cylinder.<\/span><\/p>\n

Hence, the relation between the surface area of a sphere and the lateral surface area of a cylinder is given as<\/span><\/p>\n

Surface Area of Sphere = Lateral Surface Area of Cylinder<\/b><\/h4>\n

As the lateral surface area of the cylinder = 2rh<\/span><\/p>\n

Where r = radius of the circular side and h = height of the cylinder<\/span><\/p>\n

But, in the case of a sphere,<\/span><\/p>\n

Height of a sphere = Radius of a sphere,<\/span><\/p>\n

Therefore,<\/span><\/p>\n

The surface area of a sphere = 4r\u00b2<\/b><\/p>\n

How To Calculate the Surface Area Of A Sphere?<\/h3>\n

The surface area of a sphere is the space occupied by its surface. The surface area of the sphere can be calculated using the formula of the surface area of the sphere. The steps to calculate the surface area are given below:<\/span><\/p>\n

Step 1:<\/b> Write down all the given data in the question.<\/span><\/p>\n

Step 2:<\/b> Write down the formula needed according to the question (whether CSA or TSA).<\/span><\/p>\n

Step 3:<\/b> Put the values and calculate the solution step-by-step.<\/span><\/p>\n

Let’s take an example for a better understanding.<\/span><\/p>\n

Example:<\/b> Find the surface area of a spherical ball with a radius of 9 inches.<\/span><\/p>\n

Solution:<\/b> Given, Radius of a spherical ball, r = 9 inches,<\/span><\/p>\n

We know that<\/span><\/p>\n

CSA = 4r\u00b2.<\/span><\/p>\n

Now,<\/span><\/p>\n

On putting values, we get<\/span><\/p>\n

CSA = 4 x 3.14 x 9 x 9 square inches.<\/span><\/p>\n

CSA = 1017.36 square inches.<\/span><\/p>\n

Curved Surface Area Of A Sphere (CSA)<\/b><\/h4>\n

The curved surface area is the area of all the curved regions of the solid. In the case of a sphere, since there is no flat surface in the sphere, the curved surface area of a sphere is equal to the total surface area of the sphere. Therefore, the formula for the curved surface area of a sphere is expressed as<\/span><\/p>\n

The curved surface area of a sphere = The total surface area of a sphere.<\/b><\/p>\n

Therefore,<\/span><\/p>\n

The curved surface area of a sphere = 4<\/b>r\u00b2,<\/span><\/p>\n

Where r = radius of the sphere<\/span><\/p>\n

Lateral Surface Area Of A Sphere (LSA)<\/b><\/h4>\n

The lateral surface area is the area of all the regions except the bases (i.e., top and bottom). As in the case of a sphere, it has no flat surface, and it is not possible to separate the area of its base from other sides.<\/span><\/p>\n

Therefore, for all spheres,<\/span><\/p>\n

The lateral surface area of a sphere = The total surface area of a sphere = 4<\/b>r\u00b2<\/span><\/p>\n

Total Surface Area Of A Sphere (TSA)<\/b><\/h4>\n

The total surface area is the area of all the sides, top, and bottom of the solid object.<\/span><\/p>\n

In the case of a sphere, it has no flat surface. Hence, the total surface area of the sphere is <\/span>4<\/b>r\u00b2.<\/span><\/p>\n

Some Important Questions Related To The Sphere For The Class 10 Mathematics Exam<\/h3>\n

Some important questions related to the sphere topic from the Mensuration unit are as follows:<\/span><\/p>\n

    \n
  1. The diameter of a copper sphere is 6 cm. The sphere is melted and is drawn into a long wire of a uniform circular cross-section. If the length of the wire is 36 cm, find its radius.<\/span><\/li>\n
  2. A sphere diameter of 12 cm is dropped in a right circular cylindrical vessel, partly filled with water. If the sphere is completely submerged in water, the water level in the cylindrical vessel rises by 3\u00d75\/9 cm. Find the diameter of the cylindrical vessel.<\/span><\/li>\n
  3. There were 504 cones, each 3.5 cm diameter and 3 cm height, melted and recast into a metallic sphere. Find the diameter of the sphere and hence find its surface area.<\/span><\/li>\n
  4. Two spheres of the same metal weigh 1 kg and 7 kg. The radius of the smaller sphere is 3 cm. The two spheres are melted to form a single big sphere. Find the diameter of the new sphere.<\/span><\/li>\n
  5. Calculate the cost required to paint a football in the shape of a sphere with a radius of 7 cm. If the painting cost is INR 2.5\/square cm. (Take \u03c0 = 22\/7)<\/span><\/li>\n
  6. Calculate the curved surface area of a sphere having a radius equal to 3.5 cm(Take \u03c0= 22\/7)<\/span><\/li>\n
  7. A vessel full of water is in the form of an inverted cone of a height of 8 cm and the radius of its top, which is open, is 5 cm. 100 spherical lead balls are dropped into the vessel. One-fourth of the water flows out of the vessel. Find the radius of a spherical ball.<\/span><\/li>\n<\/ol>\n

    Important Topics For The Class 10 Mathematics Exam Preparations<\/h3>\n

    Some important topics for the Class 10 Mathematics exam preparations are given below:<\/span><\/p>\n

    Chapter 4 – Quadratic Equations:<\/b><\/h4>\n