Surface Area and Volume of Cube and Cuboid - Surface Areas and Volumes, Class 9, Mathematics PDF Download

Surface Areas and Volumes

INTRODUCTION
Uptill now we have been dealing with plane figures that can be drawn on the page of our notebook or on the blackboard. In this chapter, we shall study about some solid figures like cuboid, cube, cylinder and sphere. We shall also learn to find the surface areas and volumes of these figures.

CUBOID

A rectangular solid bounded by six rectangular plane faces is called a cuboid. A match box, a tea-packet, a brick, a book, etc.,are all examples of a cuboid.

A cuboid has 6 rectangular faces, 12 edges and 8 vertices.

The following are some definitions of terms related to a cuboid:

(i) The space enclosed by a cuboid is called its volume.

(ii) The line joining opposite corners of a cuboid is called its diagonal. A cuboid has four diagonals.
A diagonal of a cuboid is the length of the longest rod that can be placed in the cuboid.

(iii) The sum of areas of all the six faces of a cuboid is known as its total surface area.

(iv) The four faces which meet the base of a cuboid are called the lateral faces of the cuboid.

(v) The sum of areas of the four walls of a cuboid is called its lateral surface area.
For a cuboid of length = units, breadth = b units and height = h units, we have :

REMARK : For the calculation of surface area, volume etc. of a cuboid, the length, breadth and height must be expressed in the same units.

CUBE

A cuboid whose length, breadth and height are all equal is called a cube.
Ice-cubes, Sugar cubes, Dice, etc. are all examples of a cube. Each edge of a cube is called its side.
For a cube of edge = a units, we have;

Diagonal of Cuboid = √3a units

Volume of cube = a³ cubic units

Total Surface Area of cube = 6a² sq. units

CROSS SECTION
A cut which is made through a solid perpendicular to its length is called its cross section. If the cut has the same shape and size at every point of its length, then it is called uniform cross-section.
Volume of a solid with uniform cross section = (Area of its cross section) × (length).
Lateral Surface Area of a solid with uniform cross section = (Perimeter of cross section) × (length).

SOLVED EXAMPLES

Ex 1. Find the surface area of a cube whose edge is 15 cm.

Sol. The edge of the cube = 15 cm, i.e., a = 15 cm.
Surface area of the cube = 6a² = 6 × (15)² = 1350 cm².

Ex 2. A small indoor greenhouse is made entirely of glass sheets (including the base) held together with tape. It is 40 cm long, 30 cm wide and 30 cm high. Find
(i) the area of the glass sheet required and
(ii) the total length of the tape required for all the 12 edges.

Sol. The dimensions of the greenhouse are as under :

Length () = 40 cm, Width (b) = 30 cm, Height (h) = 30 cm
The area of the glass sheet required

Length of the tap required = Sum of the length of the 12 edges.

Hence, 400 cm of the tape is required.

Ex 3. A wall of length 10 m was to be built across an open ground. The height of the wall is 4 m and thickness of the wall is 24 cm. If this wall is to be built up with bricks of dimensions 24 cm × 12 cm × 8 cm, then find the number of bricks which are required.

Sol. We know that, the volume of the wall and the sum of the volumes of the required number of bricks is same.
Length of the wall = 10 × 100 cm = 1000 cm
Breadth or the thickness of the wall = 24 cm
Height of the wall = 4 × 100 cm = 400 cm
The wall is in the shape of a cuboid and its volume = 1000 × 24 × 400 cm³
Now, a brick is also a cuboid having length = 24 cm, breadth = 12 cm and height = 8 cm.
Volume of one brick = 24 × 12 × 8 cm³
The required number of bricks = Volume ofthe wall
Volume of onebrick = = 4166.6
Hence, the required number of bricks = 4167.

Ex 4 . Aakriti playing with plastic building blocks which are of identical cubical shapes. She makes a structure as shown in fig. If the edge of each cube is 5 cm, then find the volume of the structure built by Aakriti.

Sol.