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**Chapter 4, Unit 1 Maths – 8th Grade**

Mensuration Chapter 4, Unit 1 Maths – 8th Grade

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**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

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**Introduction What are the shapes of these solid objects?**

Many shapes exist: Cuboid Cube Cylinder Cone Sphere Triangular Prism

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**Identify shapes around you**

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

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**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)

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**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)

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**Vertices in Cube/Cuboid**

Number of Vertices : 8

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Edges in Cube/Cuboid . Number of Edges : 12

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Faces in Cube/Cuboid Number of faces: 6

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**Différence Between LSA and TSA**

Lateral 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)) TSA Total 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)

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**TSA and LSA of Cuboid (lh+bh+lh+bh) 2(lh+bh)**

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)

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**TSA and LSA of Cuboid Total Surface Area (lh+bh+lh+bh+lb+lb)**

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

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**TSA and LSA of Cube l = b = h 1. A cube is a simple cuboid where**

2. LSA for the cube = 2(a*a + a*a) = 4a2 3. TSA for the cube = 2(a*a + a*a + a*a) = 6a2

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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*1200cm So hall area=1500*1200cm2 Area of a single tile=30*20cm2 So number of tiles=(1500*1200)/(30*20)=3000

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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

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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!

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**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!

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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)

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**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:

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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

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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

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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

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**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=height For cube, a=side of cube Area is measured in square units whereas volume is measured in cubic units.

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