MensurationChapter 4, Unit 1

Maths – 8th Grade

Introduction

• What are Solid Objects?

– Look at objects around you. Table, chair, classroom. They have the following properties:

• Occupy fixed amount of space.• Have fixed size and shape.• Can contain other items.• Eg: pencil box contains pencils,rubbers.• Eg: classroom contains tables, chairs, students.• All these objects are called three dimentional (3D) objects

Introduction

• What are the shapes of these solid objects?

– Many shapes exist:

• Cuboid• Cube• Cylinder• Cone• Sphere• Triangular Prism

Identify shapes around you

• Look around you. You will find solid objects with different shapes. Identify these solid shapes:

Cuboid

• A cuboid is a box-shaped solid object.• It has six flat sides and all angles are right angles.• And all of its faces are rectangles.• It has three important measures:

• Height (h)• Length (l)• Breadth (b)

Cube

• A box-shaped solid object.• It has six identical square faces.• A cuboid whose length, breadth and height are all equal is

called a cube. (h = l = b)

Number of Vertices : 8

Vertices in Cube/Cuboid

.

Number of Edges : 12

Edges in Cube/Cuboid

Number of faces: 6

Faces in Cube/Cuboid

Différence Between LSA and TSA

• LSALateral surface area(LSA) in a solid is the sum of the surface areas of

all its faces excluding the base and top of the solid.(e.g. Area required for painting a room(cuboid))• TSATotal surface area (TSA) in a solid is the sum of the surface areas of

all its faces including the base and top of the solid.(e.g. Area required for covering/wrapping a cuboid)

1. Consider the Cuboid ABCDEFGH

2. Open the cuboid along DH

3. For LSA we need not to consider area of II and VI

4. Lateral Surface Area (lh+bh+lh+bh) 5. Lateral Surface Area 2(lh+bh)

TSA and LSA of Cuboid

.

6. Total Surface Area (lh+bh+lh+bh+lb+lb)

7. Total Surface Area 2(lh+bh+lb)

References : http://www.youtube.com/watch?v=R_cl1HHygGw

TSA and LSA of Cuboid

1. A cube is a simple cuboid where l = b = h

2. LSA for the cube = 2(a*a + a*a) = 4a2

3. TSA for the cube = 2(a*a + a*a + a*a) = 6a2

TSA and LSA of Cube

Excercies

1. How many tiles each of 30cm*20cm is required to to cover a hall of dimensions 15m*12m.Ans. 3000(How)

Here hall dimensions are given in meters so dimensions are 1500cm*1200cmSo hall area=1500*1200cm2

Area of a single tile=30*20cm2

So number of tiles=(1500*1200)/(30*20)=3000

Excercies

2. Find the length of each side of a cube having the total surface area as 294cm2.Ans. 7 cm. (How)

Here surface area is 294cm2.So TSA of a cube is 6a2

So a2=294/6=49cm2

a=7cm

Volume

• Amount of space occupied by a three dimentional solid object is called its volume.

• It can be thought of as capacity of a solid object.• Interesting story:

• At a conference in USA, all scientists and mathematicians were present. One guy took up a hollow container of distorted (arbitrary) shape and asked what would its volume be?

• All the mathematicians started applying formulae but couldn’t get the right answere because the shape of the container was not proper.

• At the time, a small boy came, took the container, filled it with water. Then, he poured the water in a measuring cylinder and got the exact volume of the container!

• Everybody started clapping seeing the young boy’s presence of mind!

Volume of a cube

• Suppose we have 100 cubes of side length 1 cm. each.• Q. How many such cubes should we use to make a bigger

cube of side length 2 cm.?• Take a cube of 1 cm.

• Dimentions are 1x1x1

• Increase its height to 2 cm. by putting another such cube below it.• Dimentions are 2x1x1.. Not a cube!!

• Now increase its length to 2 cm. by putting 2 such cubes next to it.• Dimentions are 2x2x1.. Not a cube!!

• Now increase its breadth to 2 cm. By putting 4 such cubes next to it.• Dimentions are 2x2x2.. Now it becomes a cube!

Volume of a cube

• So, we need 8 cubes to make a cube of side length 2 cm.• i.e its volume is 8 cubic units

• Volume of cube = side x side x side = (side)3

• Eg: for cube of side 2 cm. volume = 2 cm x 2 cm x 2 cm = 8 cm3

• Check “Try it yourself section” on this link. (link)

Volume of a cuboid

• Suppose we have 24 cubes of side length 1 cm. each.• Q. Arrange them to form a cuboids of different dimentions.

• Observe the following table :

• We can see that using 24 cubes, we can arrange them in:

Volume of a cuboid

• So, we see all the 3 cuboids in prev. slide have the same volume i.e 24 cubic units.

• Volume of a cuboid = length x breadth x height• Volume of a cuboid = base area x height (area = length x

breadth)

• Eg: Volume of cuboid of dimentions 2 cm. x 3 cm. x 4 cm. = 24 cm3

• Eg: Volume of cuboid of dimentions 3 m. x 5 m. x 10 m. = 150 m3

Excercies

2. A match box measure 4cm*2.5cm*1.5cm.What will be the volume of packet containg 12 such boxes.Ans. 180 cm3. (How)

Here matchbox measure is 4cm*2.5cm*1.5cm.So Volume of a match box is 4*2.5*1.5=15cm3

So Volume of a packet conating 12 boxes is 15*12=180cm3

Excercies

2. A match box measure 4cm*2.5cm*1.5cm.What will be the volume of packet containg 12 such boxes.Ans. 180 cm3. (How)

Here matchbox measure is 4cm*2.5cm*1.5cm.So Volume of a match box is 4*2.5*1.5=15cm3

So Volume of a packet conating 12 boxes is 15*12=180cm3

Summary Topic Formula

LSA of a cuboid 2(lh+bh)

TSA of a cuboid 2(lh+bh+lb)

Volume of cuboid l*b*h (base area*height)

LSA of a cube 4a2

TSA of a cube 6a2

Volume of a cube a3

For cuboid, l=length, b=breadth, h=heightFor cube, a=side of cube

•Area is measured in square units whereas volume is measured in cubic units.