In the previous section we completed the discussion on Absolute value. In this chapter we will see details about a few solids.
Fig.23.1 below shows a rectangular block.
Let us try to understand it's features:
1. It sits on a base. This base is a rectangle
2. The top surface of the block is also a rectangle
3. The 'base rectangle' and the 'top surface rectangle' are exactly the same
4. The side faces of the block are all rectangles
5. All the 'side face rectangles' have the same height
Now consider the blocks shown in fig.23.2 below.
Let us first take the block in fig.a. We will write it's features:
1. It sits on a base. This base is a triangle
2. The top surface of the block is also a triangle
3. The 'base triangle' and the 'top surface triangle' are exactly the same
4. The side faces of the block are all rectangles
5. All the 'side face rectangles' have the same height
Block in fig.b
1. It sits on a base. This base is a quadrilateral
2. The top surface of the block is also a quadrilateral
3. The 'base quadrilateral' and the 'top surface quadrilateral' are exactly the same
4. The side faces of the block are all rectangles
5. All the 'side face rectangles' have the same height
Block in fig.c
1. It sits on a base. This base is a polygon (When we say 'polygon', it can have any number of sides)
2. The top surface of the block is also the same polygon
3. The 'base polygon' and the 'top surface polygon' are exactly the same
4. The side faces of the block are all rectangles
5. All the 'side face rectangles' have the same height
• All the side faces are rectangles
♦ All these rectangles have the same height
■ All solids which have the above features come under the category of Prisms
• If the base of a prism is a rectangle, it is called a rectangular prism
• If the base of a prism is a pentagon, it is called a pentagonal prism
• If the base of a prism is a hexagon, it is called a hexagonal prism
So on ...
■ The polygons at the top and bottom are called the bases
■ The rectangles on the sides are called the lateral faces
■ The bases and lateral faces are together called faces
• We can derive the formula for volume of any prism, based on the rectangular prism. So we will first see the 'volume of rectangular prism' in detail.
In the fig.23.3 (a) below, a rectangular prism sits on it's base. The base is a rectangle of length 7 cm and width 3 cm.
So we can write:
Length of the rectangular prism = 7 cm
Width of the rectangular prism = 3 cm
Height of the rectangular prism = 20 cm
We know that volume of such a solid is length width height
So the volume of the prism in fig.a is 7 × 3 × 20 = 420 cm3
■ Now consider fig.b. It shows the same prism that we saw in fig.a. But it's seating has changed. Now it sits on a rectangular base, whose length is 20 cm and width is 7 cm. So we can write:
Length of the rectangular prism = 20 cm
Width of the rectangular prism = 7 cm
Height of the rectangular prism = 3 cm
So the volume of the prism in fig.b is 20 × 7 × 3= 420 cm3
■ Now consider fig.c. It shows the same prism that we saw in figs.a and b. But it's seating has changed. Now it sits on a rectangular base, whose length is 20 cm and width is 3 cm. So we can write:
Length of the rectangular prism = 20 cm
Width of the rectangular prism = 3 cm
Height of the rectangular prism = 7 cm
So the volume of the prism in fig.b is 20 × 3 × 7= 420 cm3
• In all the three cases, the quantity inside the brackets is 'area of the base of the prism'.
• The quantity outside the brackets is 'height of the prism'
■ So we get a formula for finding the volume of a rectangular prism. We can write:
Volume of a rectangular prism = Area of the base × height
We saw that, all types of prisms (triangular, quadrilateral, hexagonal so on...) will have a base and a height. Can we apply the same formula for these prisms also?
Let us find out:
• Fig.23.4 (a) shows a right triangular prism. [Note the term 'right triangular'. It means that, the base and top surface are right triangles]. We want to find the volume of this prism.
• Let us make an exact replica of this prism. It is shown in fig.b.
• If we join the two prisms together as shown in fig.c, we will get a rectangular prism
• The final rectangular prism is shown in fig.d
Now we can try to find the volume
1. Let the 'area of the base' of the triangular prism in fig.a be 'a' cm
2. Then the 'area of the base' of the final rectangular prism in fig.d = 2a
3. Let the height of the triangular prism in fig.a be 'h' cm
4. Then the height of the final rectangular prism in fig.d is also 'h' cm
5. From (2) and (4) we get:
Volume of the rectangular prism in fig.d = 2a × h = 2ah cm3
6. But this rectangular prism is made up of two identical triangular prisms.
So volume of one triangular prism = volume of the triangular prism in fig.a = half of 2ah = ah cm3
7. But 'ah' is the product of the area of the base of the triangular prism and it's height. So we can write:
Volume of a right triangular prism = Area of the base × height
■ Thus we have the same formula for:
• A rectangular prism
• A right triangular prism
Now let us check whether it will work for ‘any triangle’:
1. Fig.23.5 (a) shows a triangular prism. It’s base is shown separately in fig.b.
2. In the fig.b, the triangle is split into two right triangles. [Note that, any triangle can be split into two right triangles. All we have to do is, drop a perpendicular from the top vertex to the base]
3. So the prism in fig.a can also be split into two. Each will have a right triangular base. This splitting is shown in fig.c
4. Let in fig.b,
• area of one right triangle = b cm2
• area of the other right triangle = c cm2
• Let the total area of the triangle in fig.b = a cm2
• Then a = b + c
5. We can calculate the volume of the two prisms in fig.c separately. Because, each of them have right triangular base. Thus:
• Volume of one triangular prism in fig.c = area height = b × h = bh
• Volume of the other triangular prism in fig.c = area height = c × h = ch
• Volume of the triangular prism in fig.a = Total volume of the two prisms in fig.c = bh + ch = (b+c)h
6. But from (4) we have (b+c) = a
7. So we get: Volume of triangular prism in fig.a = ah
Thus we find that, the formula can be applied to any triangular prism.
■ So we have the same formula for:
• A rectangular prism
• A triangular prism [The base can be any triangle]
Fig.23.1 below shows a rectangular block.
 |
| Fig.23.1 |
1. It sits on a base. This base is a rectangle
2. The top surface of the block is also a rectangle
3. The 'base rectangle' and the 'top surface rectangle' are exactly the same
4. The side faces of the block are all rectangles
5. All the 'side face rectangles' have the same height
Now consider the blocks shown in fig.23.2 below.
 |
| Fig.23.2 |
1. It sits on a base. This base is a triangle
2. The top surface of the block is also a triangle
3. The 'base triangle' and the 'top surface triangle' are exactly the same
4. The side faces of the block are all rectangles
5. All the 'side face rectangles' have the same height
Block in fig.b
1. It sits on a base. This base is a quadrilateral
2. The top surface of the block is also a quadrilateral
3. The 'base quadrilateral' and the 'top surface quadrilateral' are exactly the same
4. The side faces of the block are all rectangles
5. All the 'side face rectangles' have the same height
Block in fig.c
1. It sits on a base. This base is a polygon (When we say 'polygon', it can have any number of sides)
2. The top surface of the block is also the same polygon
3. The 'base polygon' and the 'top surface polygon' are exactly the same
4. The side faces of the block are all rectangles
5. All the 'side face rectangles' have the same height
So, even though the solids are of different shapes, we see some common features:
• The base and top surface are always identical• All the side faces are rectangles
♦ All these rectangles have the same height
■ All solids which have the above features come under the category of Prisms
If we are given a prism, we can further classify it. This classification is based on the shape of it's base. Thus we get:
• If the base of a prism is a triangle, it is called a triangular prism• If the base of a prism is a rectangle, it is called a rectangular prism
• If the base of a prism is a pentagon, it is called a pentagonal prism
• If the base of a prism is a hexagon, it is called a hexagonal prism
So on ...
■ The polygons at the top and bottom are called the bases
■ The rectangles on the sides are called the lateral faces
■ The bases and lateral faces are together called faces
Now we will try to calculate the volume of a prism:
• Of all the different types of prisms available to us, the 'rectangular prism' is the easiest one, as far as the 'calculation of volume' is concerned.• We can derive the formula for volume of any prism, based on the rectangular prism. So we will first see the 'volume of rectangular prism' in detail.
In the fig.23.3 (a) below, a rectangular prism sits on it's base. The base is a rectangle of length 7 cm and width 3 cm.
 |
| Fig.23.3 |
Length of the rectangular prism = 7 cm
Width of the rectangular prism = 3 cm
Height of the rectangular prism = 20 cm
We know that volume of such a solid is length width height
So the volume of the prism in fig.a is 7 × 3 × 20 = 420 cm3
■ Now consider fig.b. It shows the same prism that we saw in fig.a. But it's seating has changed. Now it sits on a rectangular base, whose length is 20 cm and width is 7 cm. So we can write:
Length of the rectangular prism = 20 cm
Width of the rectangular prism = 7 cm
Height of the rectangular prism = 3 cm
So the volume of the prism in fig.b is 20 × 7 × 3= 420 cm3
■ Now consider fig.c. It shows the same prism that we saw in figs.a and b. But it's seating has changed. Now it sits on a rectangular base, whose length is 20 cm and width is 3 cm. So we can write:
Length of the rectangular prism = 20 cm
Width of the rectangular prism = 3 cm
Height of the rectangular prism = 7 cm
So the volume of the prism in fig.b is 20 × 3 × 7= 420 cm3
We find that the volume is the same in all the three cases. We can write this:
(7 × 3) × 20 = (20 × 7) × 3 = (7 × 3) × 20 = 420 cm3.• In all the three cases, the quantity inside the brackets is 'area of the base of the prism'.
• The quantity outside the brackets is 'height of the prism'
■ So we get a formula for finding the volume of a rectangular prism. We can write:
Volume of a rectangular prism = Area of the base × height
We saw that, all types of prisms (triangular, quadrilateral, hexagonal so on...) will have a base and a height. Can we apply the same formula for these prisms also?
Let us find out:
• Fig.23.4 (a) shows a right triangular prism. [Note the term 'right triangular'. It means that, the base and top surface are right triangles]. We want to find the volume of this prism.
 |
| Fig.23.4 |
• If we join the two prisms together as shown in fig.c, we will get a rectangular prism
• The final rectangular prism is shown in fig.d
Now we can try to find the volume
1. Let the 'area of the base' of the triangular prism in fig.a be 'a' cm
2. Then the 'area of the base' of the final rectangular prism in fig.d = 2a
3. Let the height of the triangular prism in fig.a be 'h' cm
4. Then the height of the final rectangular prism in fig.d is also 'h' cm
5. From (2) and (4) we get:
Volume of the rectangular prism in fig.d = 2a × h = 2ah cm3
6. But this rectangular prism is made up of two identical triangular prisms.
So volume of one triangular prism = volume of the triangular prism in fig.a = half of 2ah = ah cm3
7. But 'ah' is the product of the area of the base of the triangular prism and it's height. So we can write:
Volume of a right triangular prism = Area of the base × height
■ Thus we have the same formula for:
• A rectangular prism
• A right triangular prism
Now let us check whether it will work for ‘any triangle’:
1. Fig.23.5 (a) shows a triangular prism. It’s base is shown separately in fig.b.
2. In the fig.b, the triangle is split into two right triangles. [Note that, any triangle can be split into two right triangles. All we have to do is, drop a perpendicular from the top vertex to the base]
3. So the prism in fig.a can also be split into two. Each will have a right triangular base. This splitting is shown in fig.c
4. Let in fig.b,
• area of one right triangle = b cm2
• area of the other right triangle = c cm2
• Let the total area of the triangle in fig.b = a cm2
• Then a = b + c
5. We can calculate the volume of the two prisms in fig.c separately. Because, each of them have right triangular base. Thus:
• Volume of one triangular prism in fig.c = area height = b × h = bh
• Volume of the other triangular prism in fig.c = area height = c × h = ch
• Volume of the triangular prism in fig.a = Total volume of the two prisms in fig.c = bh + ch = (b+c)h
6. But from (4) we have (b+c) = a
7. So we get: Volume of triangular prism in fig.a = ah
Thus we find that, the formula can be applied to any triangular prism.
■ So we have the same formula for:
• A rectangular prism
• A triangular prism [The base can be any triangle]
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