Therefore, area of one face = a²    [By area of square formula]

There are total 6 faces. Therefore,

TSA of cube = a² + a² + a² + a² + a² +a²

TSA of cube = 6a²

A cube consists of ‘n’ number of square units. Hence the space covered by these square units on the surface of the cube is the surface area. Basically, the surface area is the sum of all the area of all the shapes that cover the surface of the shape or object. In the case of a cube, there are 6 faces. So the surface area will be sum of all the area of six faces.

Problem 1:
 If the sidewall of a cubic structure have length 7m, then find the total surface area.

Solution: Given, the length of the sidewall = 7m

As per the formula, we know;

TSA = 6a²

TSA = 6 x 7 x 7 = 294 sq.m

VOLUME OF A CUBE

The volume of a cube defines the number of cubic units, occupied by the cube completely. A cube is a solid three-dimensional figure, which has 6 square faces or sides. To calculate the volume we should know the dimensions of the cube.

Volume of a Cube = Length × Width × Height

V = a × a × a

V = a³

Where ‘a’ is the length the side of cube or edges.

Question 1: Find the volume of the cube, having the sides of length 7 cm.

Given, the length of sides of the cube is 7 cm.

We know, Volume of a cube = (length of sides of the cube)³

Therefore, Volume, V = (7 cm)³

V = 343 cm³

The area of a cuboid refers to the surface area as the cuboid is a three dimensional solid. Thus, the area of cuboid can be calculated using the formula of area of rectangle, since the faces of a cuboid are in a rectangular shape.

The surface area of the cuboid can be of two types-

(i) Total Surface Area

(ii) Lateral Surface Area or Curved Surface Area

Before going into the concept of area, let us denote the dimensions of a cuboid, which are,

Length, Width, and Height are represented by l, w, h, respectively.

The Total surface area of a cuboid (TSA) is equal to the sum of the areas of it’s 6 rectangular faces, which is given by:

The above formula gives the total surface area of a cuboid having all six faces.

The lateral surface area of a cuboid is the sum of 4 planes of a rectangle, leaving the top (upper) and the base (lower) surface. Mathematically, the Lateral Surface Area of a cuboid (LSA) is given as:

The dimensions of a cuboid are given as follows:

Length = 4.8 cm

Width = 3.4 cm

Height = 7.2 cm.

Find the Total Surface area and the Lateral Surface area.

The total surface area is given as

TSA = 2 (lw + wh + hl)

=2((4.8 ×3.4) + (3.4×7.2) + (7.2×4.8))

= 2(16.32 +24.48 +34.56)

= 2(75.36) cm²

Therefore, TSA of a cuboid= 150.72 cm

Also, the lateral surface area = 2 h (l + w)

= 2×7.2 (4.8 + 3.4)

= 14.4 (8.2) = 118.08

Therefore, LSA of a cuboid = 118.08 cm²

The volume of cuboid: The volume of a cuboid is given by the product of its dimensions.

The volume of a cuboid of length ‘l’, breadth ‘b’, height ‘h’ = l×b×h cubic units

Example
.Find the volume of a cuboid of length 20 cm, breadth 15 cm and height 10 cm.

Solution:

Length of the cuboid = 20 cm

Breadth of the cuboid = 15 cm 

Height of the cuboid = 10 cm

Therefore, volume of the cuboid = length × breadth × height

= (20 × 15 × 10) cm³ 

= 3000 cm³

https://www.math-only-math.com/worksheet-on-volume-of-a-cube-and-cuboid.html

Surface area =  4 π r² square units

Example 1- 
Calculate the curved surface area of a sphere having radius equals to 3.5 cm(Take π= 22/7)

We know,

Curved surface area = Total surface area = 4 π r² square units

= 4 × (22/7) × 3.5 × 3.5

Therefore, the curved surface area of a sphere= 154 cm<sup>2

VOLUME OF SPHERE</sup>

The volume of sphere is the capacity it has. The shape of the sphere is round and three -dimensional. It has three axes such as x-axis, y-axis and z-axis which defines its shape. All the things like football and basketball are examples of the sphere which have volume.

The volume here depends on the diameter of radius of the sphere since if we take the cross-section of the sphere, it is a circle. The surface area of sphere is the area or region of its outer surface. To calculate the sphere volume, whose radius is ‘r’ we have the below formula:

Solution :

Given: Radius, r = cm

Volume of a sphere = 4/3 πr³ cubic units

V = 4/3 x 3.14 x 3³

V = 4/3 x 3.14 x 3 x 3 x 3

V = 113.04 cm<sup>3

https://www.storyofmathematics.com/volume-of-a-sphere</sup>

The surface area of a cone is the total area occupied by its surface in a 3D plane. The total surface area will be equal to the sum of its curved surface area and circular base area.

Surface area of cone = πr(r+√(h²+r²))
where r is the radius of the circular base
h is the height of coneOrSurface area of cone =πr(r+L)
where L is the slant height of the coneAndCurved Surface area of cone = πrl

Example 1

1. Determine the curved surface area of a cone whose base radius is 7 cm and slant height is 15 cm.

Solution: Curved surface area of a cone = πrl

= (22/7)× 7 ×15

=  330 cm<sup>2

VOLUME OF CONE</sup>

The volume of a cone defines the space or the capacity of the cone
The volume of a cone formula is given as

Where,

‘r’ is the base radius of the cone

‘l’ is the slant height of a cone

‘h’ is the height  of the cone


Example 1

Calculate the volume if r= 2 cm and h= 5 cm.

Given:

r = 2

h= 5

Using the Volume of Cone formula

The volume of a cone = (1/3) πr²h cubic units

V= (1/3) × 3.14 × 2² ×5

V= (1/3) × 3.14 × 4 ×5

V= (1/3) × 3.14 × 20

V = 20.93 cm³
https://www.ask-math.com/volume-of-cone.html

Total Surface Area:
The area of the curved surface and the area of the circle (base) is called the total surface area

From the surface area of a sphere, we can easily calculate the surface area of the hemisphere.

Since hemisphere is half of the sphere

CSA of hemisphere = (1/2)surface area of the sphere

CSA = (1/2)4πr²

CSA = 2πr²

The total surface area of the sphere = Curved surface area of sphere + base area

We know that the base of the hemisphere is circular in shape, use the area of the circle.

TSA = 2πr² + πr² = 3πr²

Therefore,

Where π is a constant whose value is equal to 3.14 approximately.

“r” is the radius of the hemisphere.


Question:

Find the surface area of a hemisphere whose radius is 4 cm?

Solution:

Given:

Radius, r = 4 cm

The curved surface area = 2πr² square units.

The total surface area = 3πr² square units

Substitute the value of r in the formula.

(i) CSA of the hemisphere= 2 × 3.14 × 4 × 4

CSA = 3.14 × 32

CSA = 100.48 cm²

(ii) TSA of the hemisphere = 3 × 3.14 × 4 × 4

TSA = 3.14 × 48

TSA = 150.72 cm²

Therefore, the curved and the total surface area of the hemisphere are 100.48 and 150.72 cm^2, respectively.

Volume of hemisphere

We can easily find the volume of the hemisphere since the base of the sphere is circular. The volume of the hemisphere is derived by Archimedes.

Where π is a constant whose value is equal to 3.14 approximately.

“r” is the radius of the hemisphere.

Question:

Find the volume of the hemisphere whose radius is 6 cm.

Solution:

Given:

Radius, r = 6 cm

The volume of a hemisphere = (2/3)πr³ cubic units.

Substitute the value of r in the formula.

V = (2/3) × 3.14 × 6 × 6 × 6

V = 2× 3.14 × 2 × 6 × 6

V = 452.16

Therefore, the volume of the hemisphere is 452.16 cubic units.

https://www.vedantu.com/maths/volume-of-hemisphere
https://www.google.com/amp/s/www.embibe.com/exams/volume-of-hemisphere/

Where,
a – apothem length of the prism.
b – base length of the prism.
l – base width of the rectangular prism.
h – height of the prism.


Question 1:
What will be the surface area of a triangular prism if the apothem length, base length and height are 7 cm, 10 cm and 18 cm respectively ?
Solution:

 
Given,
a = 7 cm
b = 10 cm
h = 18 cm
Surface area of a triangular prism
= ab + 3bh
= (7 cm $\times$ 10 cm) + (3 $\times$ 10 cm $\times$ 18 cm)
= 70 cm² + 540 cm²
= 610 cm<sup>2

Volume of Prism</sup>
The volume of a prism is defined as the total space occupied by the three-dimensional object. Mathematically, it is defined as the product of the area of the base and the length.

Therefore,

The measurement unit used to represent the volume of a three-dimensional object is cubic units. 
https://www.onlinemathlearning.com/volume-prism-1.html

https://www.varsitytutors.com/high_school_math-help/how-to-find-the-volume-of-a-prismath-help/how-to-find-the-volume-of-a-prism

The total area of a cylinder is based on:

Curved Surface Area

The curved surface area of a cylinder (CSA) is defined as the area of the curved surface of any given cylinder having base radius ‘r’, and height ‘h’, It is also termed as Lateral Surface Area (LSA). The formula for a curved area or lateral area is given by:

Base Area of Cylinder

The base of the cylinder is in a circular shape. Hence, by the formula of area of the circle, we know,

Area of the circular bases of cylinder = 2 (πr²)  [Since the cylinder has two circular bases]

Total Surface Area of Cylinder

The total surface area of a cylinder is equal to the sum of areas of all its faces. The total surface area with radius ‘r’, and height ‘h’ is equal to the sum of the curved area and circular areas of the cylinder.

Solution, Given, diameter = 28 cm, so radius = 28/2 = 14 cm

and height = 15 cm

By the formula of total surface are, we know;

TSA = 2πr (h + r) = 2x 22/7 x 14 x (15 + 14)

TSA = 2 x 22 x 2 x 29

TSA = 2552 sq.cm

Hence, the total surface area of container is 2552 sq.cm.

Volume of a cylinder


Volume of a Cylinder Formula

A cylinder can be seen as a collection of multiple congruent disks, stacked one above the other. In order to calculate the space occupied by a cylinder, we calculate the space occupied by each disk and then add them up. Thus, the volume of the cylinder can be given by the product of the area of base and height.

For any cylinder with base radius ‘r’, and height ‘h’, the volume will be base times the height.

Therefore, the cylinder’s volume of base radius ‘r’, and height ‘h’ = (area of base) × height of the cylinder

Since the  base is the circle, it can be written as

Volume =  πr² × h

Therefore, the volume of a cylinder = πr²h cubic units.

Problem

Calculate the volume of a given cylinder having height 20 cm and base radius of 14 cm. (Take pi = 22/7)

Given:

Height  = 20 cm

radius = 14 cm

we know that;

Volume, V = πr²h  cubic units

V=(22/7) × 14  × 14  × 20

V= 12320 cm³

Therefore, the volume of a cylinder = 12320 cm³

https://courses.lumenlearning.com/prealgebra/chapter/finding-the-volume-and-surface-area-of-a-cylinder/

https://www.mathwarehouse.com/solid-geometry/cylinder/formula-volume-of-cylinder.php