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Applied Math Mensuration of Solid Chapter 15

Mensuration of Solid

15.1 Solid:

It is a body occupying a portion of three-dimensional space and therefore bounded by a closed surface which may be curved (e.g., sphere), curved and planer (e.g., cylinder) or planer (e.g., cube or prism).

15.2 Mensuration of Prisms:

Prism:

A solid bounded by congruent parallel bases or ends and the side faces (called the lateral faces) are the parallelograms, formed by joining the corresponding vertices of the bases. It is called a right prism if the lateral are rectangles. Otherwise an oblique prism. A common side of the two lateral faces is called a lateral edge. Prism are named according to the shape of ends. A prism with a square base, a rectangular base, a hexagonal base, and a parallelogram base, is called a square prism, a rectangular prism, a hexagonal prism and a parallelepiped respectively.

Altitude:

The altitude of a prism is the vertical distance from the centre of the top to the base of the prism.

Fig. 15.1

Axis: The axis of a prism is the distance between the centre of the top to the centre of the base. In a right prism the altitude, the axis and the latera edge are the same lengths. In the Fig. 1 the lateral faces are OAEC, BDGF, OBCF dan ADEG. The bases are OABD and ECFG. LM = h is the axis of the prism, where L and M are the centres of the bases.

15.3 Surface Area of a Prims:

Lateral surface area of a prism is the sum of areas of the lateral faces. From Fig. 1. Lateral surface area = Area of (OAEC + ADEG + BDGF + OBCF) = (OA)h + (AD)h + (DB)h + (OB)h 323

Applied Math Mensuration of Solid = (OA + AD + DB + BO)h

= Perimter of the base x height of the prism (1) Total surface area = Lateral surface area + Area of the bases (2) Note: that, Total surface, surface area, total surface area and the surface of any figure represent the same meanings.

15.4 Volume of a Prism:

A sold occupies an amount of space called its volume. Certain solids have an internal volume or cubic capacity. (The term capacity, when not associated with cubic, is usually reserved for the volume of liquids or materials which pour, and special sets of units e.g.; gallon, litre, are used). The volume of solid is measured as the total number of unit cubes that it contains. If the solid is a Prism the volume can be computed directly from the formula,

V = l . b . h

Where, l, b and h denoted the length, breadth and height of the prism respectively. Also l . b denotes the area A of the base of the prism. Then

Volume of the Prims = Ah

= Area of the base x height of the prism (4)

Fig. 15.2

l = 6 units b = 3 units and h = 5 units

The total number of unit cubes

= l b h = 6 x 3 x 5 = 90

So volume of prism = 90 cubic unit

Weight of solid = volume of solid x density of solid (density means weight of unit volume)

15.5 Types of Prism:

1. Rectangular Prism:

If the base of prism is a rectangle, it is called a rectangular prism. Consider a rectangular prism with length a, breadth b and height c (Fig. 15.3) 324

Applied Math Mensuration of Solid Fig. 15.3

(i) Volume of rectangular prism = abc cu. Unit (ii) Lateral surface area = area of four lateral faces = 2 ac + 2bc sq. unit = (2a + 2b)c = Perimetr of base x height (iii) Total surface area = Area of six faces = 2ab + 2ac + 2ca = 2 (ab + bc + ca) sq. unit (iv) Length of the diagonal OG In the light triangle ODG, by Pythagorean theorem,

OG2 = OD2 + DG2

= OD

2 + C2 (I)

Also in the right triangle OAD,

OD2 = OA2 + AD2

= a

2 + b2

Put OD

2 in equation (1)

(the line joining the opposite corners of the rectangular prism is called its diagonal).

OG2 = a2 + b2 + c2

Or |OG| = 2 2 2a b c

2. Cube: A cube is a right prism will all sides equals. Let a be the

side of the cube Fig. 4

Fig. 15.4

325

Applied Math Mensuration of Solid (i) Volume of the cube = a . a. a

= a

3 cu . unit

(ii) Lateral surface area = Area of four lateral faces = 2a . a + 2a . a = 4a2 sq. unit (iii) Total surface area = Area of six faces = 6a2 sq. unit (iv) The length of the diagonal |OG| = 222aaa= a 3

Example 1:

Find the volume, total surface, diagonal and weight of rectangular block of wood 7.5 cm long, 8.7 cm wide and 12 cm deep, 1 cu. cm = 0.7 gm.

Solution:

Let a = 7.5 cm, b = 8.7 cm, and c = 12cm

Then (i) Volume = abc = 7.5 x 8.7 x 12 = 783.00 cu. cm. (ii) Total surface = 2 (ab + bc + ca) = 2 (65.25 + 104.4 + 90) = 519.3 sq. cm. (iii) diagonal = 2 2 2a b c = 2 2 27.5 8.7 12 = 275.94 16.6 cm (iv) Weight = Volume x density = 783 x 0.7 = 548.1 gms

Example 2:

The side of a triangular prism are 25, 51 and 52 cm. and height is

60 cm. Find the side of a cube of equivalent volume.

Solution:

Volume of a prism = Area of base x height

= S(s a)(s b)(s c) x h

When a = 25cm, b = 51cm, c = 52cm, h = 60cm

S = a + b + c 25 51 526422 S a = 6425 = 39, S b = 6451 = 13, S c = 6452 = 12 Volume of prism = 64(39)(13)(12) x 60 37440 cu. cm

Volume of a cube of side l cm = l3 cu. cm.

l3 = 37440 l = 13(37440) = 33.45 cu. cm. 326

Applied Math Mensuration of Solid Example 3:

The volume of the cube is 95 cu. cm. Find the surface area and the edge of the cube.

Solution:

Volume of cube = 95 cu. cm.

Let a be the side of cube, then

Volume = a

3 a

3 = 95

a = 13(95) = 4.56 cm

Surface area = 6a

2 = 6(4.56)2 = 124.92 sq. cm

Example 4:

Find the number of bricks used in a wall 100 ft long, 10 ft high and one and half brick in thickness. The size of each is 9x 142
x 3

Solution:

a = Length of wall = 100 ft = 1200 inches b = Breadth of wall = 992
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