Sphere Formula Square Root Formula

Sphere Formula

A circle is a closed figure that can be drawn using a constant length from a fixed-point center. A sphere is a three-dimensional circle. Circle and sphere both are round and measured using radius. Here we are going to calculate the surface area, the volume of a sphere by different sphere formula.

Sphere Formula

What is the Sphere?

A sphere is a three-dimensional counterpart of a circle, with all its points lying in space at a constant distance from the fixed point or the center, called the radius of the sphere. Radius of the sphere denoted as r.

The diameter of a sphere: The line passing through the center from one end to the other called the diameter of the sphere. The diameter of the sphere denoted as D.

Diameter = 2 times radius of the sphere

D = 2 × r

Circumference of a sphere: Circumference of a sphere is the distance covered around the sphere. The unit of the circumference is the same as the radius.

Circumference = 2×π×r

The surface area of a sphere: The surface area of a sphere is the number of square units that will exactly cover the surface of a sphere. The unit of the surface area of sphere is a square unit m²

The surface area of a sphere is

Surface Area of sphere = 4 times the area of a circle

Surface Area of sphere =4×π×r2

Where

r	the radius of the sphere

The volume of a sphere: Volume of sphere is the number of cubic units that fill a sphere. The unit of volume of the sphere is a cubic unit m³

The volume enclosed by a sphere is

Volume of sphere = 43π r³

Where,

r	the radius of the sphere

The volume of spherical shell: A spherical shell is a region between two concentric spheres having different radii.

Volume of a spherical shell = 43πR3–43πr3

V =43π(R3−r3)

Where,

R	The outer radius of the sphere
r	The inner radius of the sphere

Solved Examples

Q 1: The radius of the sphere is 5 cm. Find the diameter, circumference, surface area and volume of a sphere?

Solution: Given, r = 5 cm

Diameter of a sphere = 2 × r
= 2 × 5

=10 cm

Circumference of a sphere = 2×π×r
= 2×π×5
= 31.41 cm

Surface Area of sphere =4×π×r2

= 4×π×52

= 4××3.14×25

= 314.16 cm²

Volume of a sphere = \(\frac{4}{3}\pi r^{3}\)

= 43π53

= 43×3.14×125

= 523.60 cm³

Q 2: A shopkeeper has one ladoo of radius 10 cm. With the same material used to form a big ladoo, how many ladoos of radius 5 cm can be made.

Solution: Let the number of smaller ladoos that can be made from a bigger ladoo be x.

Here,

The radius of the bigger ladoo (R) = 10 cm

The radius of the smaller ladoo (r) = 5 cm

So, the volume of the big ladoo is equal to the volume of x small ladoos.

Volume of the big ladoo = Volume of the x small ladoos

43πR3 = 43πr3

43π103 = x43π53

103=x53

x = 10353

x = 1000125

x = 8

Therefore, 8 ladoos of 5 cm can be made from one big ladoo of radius 10 cm.

Square Root Formula

A square root is common function in mathematics. A square root is widely used in different applications in different fields of mathematics and physics. Here is a guide to finding the square root of a number by square root formula.

Square Root Formula

What is a Square root?

The square root of the number y whose square is x. The square root is denoted by √

We find the square root of a number by the following methods:

i) By Prime Factorisation

ii) By Long Division

iii) By Repeated subtraction method

1.By Prime Factorisation: Steps to find the square root of a perfect square by using the prime factorization method.
Step I: Obtain the given number.
Step II: Reduce the given number into prime factors by successive division.
Step III: Now make pairs of prime factors in such a way that both the factors in each pair are equal.

Step IV: Take one factor from each pair and find the product of these factors.
Step V: The product obtained by multiplying the factors is the required square root.

2. By Long Division: Steps to find the square root of a number by long division method.

Step 1: Firstly, we place a bar on every pair of digits starting from the unit digit. If the number of digits in it is odd, we put a bar on the single-digit too. For example, we take 729. So 1st bar is on 29 and 2nd bar is on 7

Step 2: Now we find the largest number whose square is less than or equal to the 1st number.
(22<7<32). We take 2 and divide and get the remainder = 3.

Step 3: Now we bring down the next bar number i.e.29. So, the new dividend is 329.

Step 4: For new divisor, we add the divisor 2 and quotient 2 that gives us 4.

Step 5: Number taken is the product of a new divisor and this digit is equal to or less than 329 (new dividend).
In this case,

47 × 7 = 329.

The new digit is 7. We get remainder as 0.
∴ 729−−−√ = 27.

3. Repeated subtraction method: In this method, the given number is subtracted by 1, 3, 5, 7,… at every step till you get zero at the end. The number of steps in the solution is the required square root.

Solved Examples

Q1. In a concert hall, the number of rows is equal to the number of chairs in each row. If the capacity of the concert hall is 2025, find the number of chairs in each row.

Solution: Let the number of chairs in each row in the concert hall be x.

Then, the number of rows = x.

Total number of chairs in the concert hall = (x×x)=x2

But, the capacity of the concert hall = 2025

Therefore, x² = 2025 = (5 × 5 × 3 × 3 × 3\)

x = (5×3×3) = 45

Number of rows in the concert hall is 45.

Q 2: Find the square root of 49.

Solution : (i) 49 -1 = 48
(ii) 48 – 3= 45
(iii) 45 – 5 = 40
(iv) 40 – 7= 33
(v) 33 – 9 = 24
(vi) 24 – 11 = 13
(vii) 13 – 13 = 0
Here, the total number of subtractions is 7.
∴ 49−−√=7