Note on Solids

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Solid means having three dimensions (length, breadth and thickness), as a geometrical body or figure. Or it may also be defined as relating to bodies or figures of three dimensions.

Prism and their surface area and volume

Prisms are the solid object that has two opposite faces congruent and parallel.

 

The above figures are different solid figures which have congruent opposite face.

Each congruent face of a prism is called its cross-section. There are infinite numbers of imaginary surfaces that cut the prism perpendicular to its height or length with congruent surfaces.

Total surface area of prism

Let's take a cartoon box. While unfolding the cartoon box, there we can find many faces of the boxes. Measure all the surfaces and find the area of each surface. If the length is “l” breadth is “b” and height is ‘h’, what will be the total surface area?

In a box, altogether there are 6 surfaces. Among them three pairs are congruent. Thus, total surface area (A) = 2(l \(\times\) b) + 2(b \(\times\)h) + 2(l \(\times\) h) square units

= 2(lb + bh + lh) square units.

In the case of irregular shape, area of surface are calculated separately and added to find out the total surface area.

Lateral surface area of prism

The sum of areas of four lateral surfaces of the prism is called the lateral surface area of the prism.

Lateral surface area (S) = 2(l \(\times\) h) + 2(b \(\times\) h)

= 2lh + 2bh

= h \(\times\) 2(l+b)

= h \(\times\) p

\(\therefore\) lateral surface area (S) = height \(\times\) perimeter of base

Total surface area of the prism (TSA) = Lateral surface area + 2 \(\times\) area of cross section

i.e. TSA =LSA + 2A

Volume of prism

let's take a cuboidal prism.

For a cuboid For a cube

Volume (V) = l \(\times\) b \(\times\) h

V = A \(\times\) H

Where, A = l \(\times\) b 

Volume (V) = l \(\times\) l \(\times\) l

(V) = l2 \(\times\) l

(V) = A \(\times\) l

Thus, Volume = Area of cross-section × height

S.N

Solid Figures

Area of base or cross section

Lateral Surface Area

Total surface Area

Volume

1. Cuboid A = l \(\times\) b 2h(l + B) 2(lb + bh + lh) V = l\(\times\) b\(\times\) h
2. Cube A = l2 4l2 or 4A 6l2or 6A V = l3
3. Prism A = base area or A = area of cross section h \(\times\) p p\(\times\) h + 2A V = A\(\times\) h

  • Lateral Surface Area (LSA) = height \(\times\) perimeter of base
  • Total Surface Area of the prism (TSA) = Lateral surface area + 2 \(\times\) area of cross section
  • Volume of cuboid (V) = l \(\times\) b \(\times\) h

                                        V = A \(\times\) H

                            Where, A = l \(\times\) b

  • Volume of cube(V) = l \(\times\) l \(\times\) l

                               (V) = l2 \(\times\) l

                               (V) = A \(\times\) l

.

Very Short Questions

Solution:

Here, length (l) = 8cm

Total surface area (TSA) = 6l2

= 6× (8cm)2

= 6× 64cm2

= 384cm2

Again,

Volume (V) = l3

= (8cm)3

= 512cm3

Hence, TSA = 384cm2 and volume = 512cm3

Solution:

Here,

Lateral Surface Area (LSA) = perimeter× height

= (1cm +cm + 4cm + 1cm + 8cm + 6cm + 8cm) × 6cm

= 34cm× 6cm

= 204cm2

Also,

Volume (V) = Area of cross section× height

= (8cm× 6cm - 4cm× 3cm)× 6cm

= (48cm2 - 12cm2)× 6cm

= 36cm2× 6cm

= 216cm3

Solution:

Here,

Length (l) = 10cm

Breadth (b) = 5cm

Heigth (h) = 8cm

Now,

LSA = 2h(l + b)

= 2× 8cm (10cm + 5cm)

= 16cm× 15cm

= 240cm2

Again,

TSA = 2(lb + bh + lh)

= 2 (10cm× 5cm + 5cm× 8cm +10cm× 8cm)

= 2 (50cm2 + 40cm2 + 80cm2)

= 2× 170cm2

= 340cm2

Also,

Volume (V) = l× b× h

= 10cm× 5cm× 8cm

= 400cm3

Solution:

Here,

Length of box (l) = 30cm

Breadth of box (b) = 20cm

Height of box (h) = 10cm

Length of soap (l1) = 3cm

Breadth of soap (b1) = 2cm

Height of soap (h1) = 2cm

Now,

Volume of box (V) = l× b× h

= 30cm× 20cm× 10cm

= 6000cm3

Volume of soap (V1) =l1× b1× h1

= 3cm × 2cm ×2cm

= 12cm3

∴ Number of soap that can be fitted on the box (N) = \(\frac{V}{V_1}\)

= \(\frac{600cm^3}{12cm^3}\)

= 50

Hence, 50 soap can be fitted on the box.

Solution:

Here,

Length (l) = 16m

Breadth (b) = 15m

Heigth (h) = 13m

Area of 4 walls (A) = 2h(l + b)

= 2× 13m (16m + 15m)

= 26m× 31m

= 806m2

Cost of painting wall (C) = Rs 120m2

Total Cost (T) = ?

∴ Total Cost (T) = C× A

= Rs 120 ×806m2

= Rs 96,720

LSA TSA
The lateral surface area of a three-dimensional object is the surface area of the object minus the area of its bases. The total surface area of a three- dimensional object is the surface area of the object adding all of its areas.
For example, dice. We found that the surface area of a six-sided dice.Since the dice has two bases, we subtractthe area of the two bases. For example,dice. We found that the surface area of a six-sided dice. We add all the areas of a dice.

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