ICSE Class 6 Maths Chapter 24 Solids

Chapter 24: Solids (Volume and Surface Area)

24.1 Solid

An object that occupies space is called a solid. A book, a brick, a ball, etc. are some examples of a solid.

A thin straight line drawn on paper, i.e. a line drawn on a plane, has only length. Thus we say that straight line has only one dimension, namely, a length.

A rectangle, drawn on paper has length and breadth. Thus we say that the figure (rectangle) has two dimensions, namely, length and breadth. In fact, each and every figure drawn on a plane is a two-dimensional figure.

Solids have length, breadth and height. For this reason, every solid is a three-dimensional figure.

Teacher's Note

Understanding dimensions helps students recognize that objects around them - like their classroom, books, and sports equipment - are three-dimensional solids occupying real space.

24.2 Recognizing Faces, Edges and Vertices (Corners) of Some Solids

Prism

The adjoining figure shows a prism with:

Three side faces, namely, AA'C'C, ABB'A' and BB'C'C; each of these three faces is a parallelogram (or rectangle).

This prism also has two congruent end-faces, i.e. bases, namely, triangles ABC and A'B'C'.

The two end-faces (bases) are always parallel to each other.

Nine edges, namely, AB, AC, BC, A'B', A'C', B'C', AA', BB' and CC'.

Of these nine edges, AA', BB' and CC' are parallel to one another, AB is parallel to A'B', BC is parallel to B'C' and AC is parallel to A'C'.

Six vertices, namely, A, B, C, A', B' and C'.

Thus, a prism is a solid, whose side-faces are parallelograms (or rectangles) and whose end-faces, i.e. bases, are two parallel and congruent polygons.

The adjoining figure shows a prism with:

Five side-faces, namely, ABB'A', BCC'B', CDD'C', DEE'D' and AEE'A', each of which is a rectangle.

The prism also has two end-faces, ABCDE and A'B'C'D'E', which are congruent and parallel to each other.

Fifteen edges, AA' \(\parallel\) BB' \(\parallel\) CC' \(\parallel\) DD' \(\parallel\) EE', AB \(\parallel\) A'B', BC \(\parallel\) B'C', CD \(\parallel\) C'D', DE \(\parallel\) D'E' and AE \(\parallel\) A'E'.

Ten vertices, A, B, C, D, E and A', B', C', D', E'.

Pyramid

A pyramid is a solid whose base is a plane rectilinear figure, such as a triangle, a quadrilateral, and whose side-faces are triangles with a common vertex. This common vertex must lie outside the plane of the base.

The adjoining figure shows a pyramid with triangular base ABC and side-faces (each of which is also a triangle) PAB, PBC and PAC. Point P is the common vertex of the side-faces.

Since, the base of this pyramid is a triangle, it is called a triangular pyramid.

In the same way, the adjoining figure shows a pyramid whose base is a quadrilateral ABCD and side-faces are \(\triangle\)PAB, \(\triangle\)PBC, \(\triangle\)PCD and \(\triangle\)PDA. Clearly, P is the common vertex of the side-faces and it does not lie on the plane of the base.

Since the base of this pyramid is a quadrilateral, it is called a quadrilateral pyramid.

Cuboid (A Rectangular Solid)

It is a three-dimensional solid all of whose sides are not necessarily equal. That is, in general, a cuboid has length, breadth and height of different values (sizes).

The figure given alongside shows a cuboid. It is clear from the figure that a cuboid has:

Six faces, namely, ABCD, ABFE, AEHD, CGHD, CGFB and EFGH.

Each face of a cuboid is a rectangle.

Twelve edges, namely, AB, BC, CD, DA, AE, EH, HD, EF, FG, GH, BF and CG.

Eight vertices (corners), namely, A, B, C, D, E, F, G and H.

Also,

Length (l) of the cuboid = AE = DH = CG = BF

Breadth (b) of the cuboid = AB = DC = HG = EF

Height (h) of the cuboid = AD = BC = EH = FG

Cube

A cube is a cuboid with all sides equal, i.e. length = breadth = height.

The adjoining figure shows a cube.

Since a cube is a cuboid, it also has:

Six faces: ABCD, ABFE, AEHD, CGHD, CGFB and EFGH.

Twelve edges: AB, BC, CD, DA, AE, EH, HD, EF, FG, GH, BF and CG.

Eight corners: A, B, C, D, E, F, G and H.

Each face of a cube is a square in shape, and all the six faces of a cube are congruent (equal).

Teacher's Note

Recognizing the properties of cubes and cuboids helps students understand how buildings, storage boxes, and everyday packaging are structured with specific faces, edges, and corners.

24.3 Volume and Surface of Cuboid and Cube

Volume

The volume of a solid is the measure of the space occupied by it.

Volume of a Cuboid

Its length \(\times\) breadth \(\times\) height,

i.e. \(V = l \times b \times h\) unit\(^3\)

Volume of a Cube

Since a cube is a cuboid in which length = breadth = height = say, a units,

Volume of cube = length \(\times\) breadth \(\times\) height = \(a \times a \times a = a^3\) cubic unit (unit\(^3\))

The formula \(V = l \times b \times h\) for the volume of a cuboid can be re-written as:

Length of the cuboid \(l = \frac{V}{b \times h}\)

Breadth of the cuboid \(b = \frac{V}{l \times h}\) and

Height of the cuboid \(h = \frac{V}{l \times b}\)

When the dimensions of a cuboid or a cube are in centimetre (cm), the volume is in cubic centimetre (cm\(^3\)).

Similarly, when the dimensions of a cuboid or a cube are in metre (m), the volume is in cubic metre (m\(^3\)) and so on.

\(1 \text{ m} = 100 \text{ cm}, 1 \text{ m}^2 = 100 \times 100 \text{ cm}^2\) and \(1 \text{ m}^3 = 100 \times 100 \times 100 \text{ cm}^3\)

\(1 \text{ cm} = \frac{1}{100} \text{ m}, 1 \text{ cm}^2 = \frac{1}{100 \times 100} \text{ m}^2\) and \(1 \text{ cm}^3 = \frac{1}{100 \times 100 \times 100} \text{ m}^3\)

Surface Area

The surface area of a solid is the sum of the areas of all its faces.

A cuboid has six faces in which opposite faces are equal in area. Thus, for the cuboid shown alongside:

Area of face ABFE = area of face DCGH = \(l \times b\) sq. units,

Area of face ABCD = area of face EFGH = \(b \times h\) sq. units,

and area of face BCGF = area of face AEHD = \(h \times l\) sq. units.

And so, surface area of cuboid = \(2 \times\) area of ABFE \(+ 2 \times\) area of ABCD \(+ 2 \times\) area of BCGF = \(2(l \times b) + 2(b \times h) + 2(h \times l) = 2(l \times b + b \times h + h \times l)\)

A cube has six faces, all of them equal in area. Each face of a cube is a square. Therefore, area of each face of a cube = \(a^2\) square units. Surface area of a cube = \(6a^2\) sq. units (where a is one side of the cube).

Example 1

The length, breadth and height of a cuboid are 15 cm, 8 cm and 6 cm respectively. Find: (i) its volume (ii) its surface area.

Solution

Since \(l = 15 \text{ cm}, b = 8 \text{ cm}\) and \(h = 6 \text{ cm}\)

Volume of the cuboid = \(l \times b \times h = 15 \text{ cm} \times 8 \text{ cm} \times 6 \text{ cm} = 720 \text{ cm}^3\)

Surface area of the cuboid = \(2(l \times b + b \times h + h \times l) = 2(15 \times 8 + 8 \times 6 + 6 \times 15) \text{ cm}^2 = 2(120 + 48 + 90) \text{ cm}^2 = 516 \text{ cm}^2\)

Example 2

The volume of a cuboid is 240 cm\(^3\). If its length is 8 cm and height 5 cm, find its breadth.

Solution

The breadth of a cuboid, \(b = \frac{V}{l \times h} = \frac{240}{8 \times 5} \text{ cm} = 6 \text{ cm}\)

Example 3

One side of a cube is 8 cm. Find: (i) its volume (ii) its surface area.

Solution

Since each side of the cube = 8 cm, i.e. \(a = 8 \text{ cm}\),

Its volume = \(a^3 = 8^3 \text{ cm}^3 = 512 \text{ cm}^3\)

Its surface area = \(6a^2 = 6 \times (8)^2 \text{ cm}^3 = 384 \text{ cm}^2\)

Example 4

The surface area of a cube is 96 cm\(^2\). Find: (i) the length of one of its sides (edges) (ii) its volume.

Solution

\(6 \text{(side)}^2 = 96\)

\(\Rightarrow \text{(side)}^2 = \frac{96}{6} = 16\) and \(\text{side} = 4 \text{ cm}\)

Volume = \(\text{(side)}^3 = (4)^3 \text{ cm}^3 = 64 \text{ cm}^3\)

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