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Maths Formulas

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 = \(2 \times \pi \times r\)

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 =\(4 \times \pi \times r^{2}\)

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 = \(\frac{4}{3}\)π 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 = \(\frac{4}{3}\pi R^{3} – \frac{4}{3}\pi r^{3}\)

V =\(\frac{4}{3}\pi (R^{3}-r^{3})\)

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 = \(2\times \pi \times r\)
= \(2\times \pi \times 5\)
= 31.41 cm

Surface Area of sphere =\(4 \times \pi \times r^{2}\)

= \(4 \times \pi \times 5^{2}\)

= \(4 \times \times 3.14  \times 25\)

= 314.16 cm²

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

= \(\frac{4}{3}\pi 5^{3}\)

= \(\frac{4}{3} \times 3.14 \times 125\)

= 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

\(\frac{4}{3}\pi R^{3}\) = \(\frac{4}{3}\pi r^{3}\)

\(\frac{4}{3}\pi 10^{3}\) = \(x\frac{4}{3}\pi 5^{3}\)

\(10^{3} = x 5^{3}\)

x = \(\frac{10^{3}}{5^{3}}\)

x = \(\frac{1000}{125}\)

x = 8

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

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