NCERT Solutions Chapter 13 - Surface Areas and Volumes, Class 9, Maths PDF Download

Exercise 13.1

1. A plastic box 1.5 m long, 1.25 m wide and 65 cm deep is to be made. It is opened at the top. Ignoring the thickness of the plastic sheet, determine:
 (i) The area of the sheet required for making the box.
 (ii) The cost of sheet for it, if a sheet measuring 1m² costs Rs 20.

Answer
Length of plastic box = 1.5 m
Width of plastic box = 1.25 m
Depth of plastic box = 1.25 m
(i) The area of sheet required to make the box is equal to the surface area of the box excluding the top.
Surface area of the box = Lateral surface area + Area of the base
= 2(l+b)×h + (l×b)
= 2[(1.5 + 1.25)×1.25] + (1.5 × 1.25) m² 
= (3.575 + 1.875) m² 
= 5.45 m² 
The sheet required required to make the box is 5.45 m² 
(ii) Cost of 1 m² of sheet = Rs 20
∴ Cost of 5.45 m² of sheet = Rs (20 × 5.45) = Rs 109

2. The length, breadth and height of a room are 5m, 4m and 3m respectively. Find the cost of white washing the walls of the room and the ceiling at the rate of Rs.7.50 per m².

 Answer
length of the room = 5m
breadth of the room = 4m
height of the room = 3m
Area of four walls including the ceiling
= 2(l+b)×h + (l×b)
= 2(5+4)×3 + (5×4) m² 
= (54 + 20) m² 
= 74 m² Cost of white washing = Rs.7.50 per m²
Total cost = Rs. (74×7.50) = Rs. 555

3. The floor of a rectangular hall has a perimeter 250 m. If the cost of painting the four walls at the rate of Rs.10 per m² is Rs.15000, find the height of the hall.
 [Hint : Area of the four walls = Lateral surface area.]

Answer
Perimeter of rectangular hall = 2(l + b) = 250 m
Total cost of painting = Rs.15000
Rate per m² = Rs.10
Area of four walls = 2(l + b) h m² = (250×h) m²
A/q,
(250×h)×10 = Rs.15000
⇒ 2500×h = Rs.15000 
⇒ h = 15000/2500 m
⇒ h = 6 m 
Thus the height of the hall is 6 m.

4. The paint in a certain container is sufficient to paint an area equal to 9.375 m². How many bricks of dimensions 22.5 cm×10 cm×7.5 cm can be painted out of this container?

Answer
Volume of paint = 9.375 m² = 93750 cm²
Dimension of brick = 22.5 cm×10 cm×7.5 cm
Total surface area of a brick = 2(lb + bh + lh) cm² 
= 2(22.5×10 + 10×7.5 + 22.5×7.5) cm²
= 2(225 + 75 + 168.75) cm²
= 2×468.75 cm² = 937.5 cm²
Number of bricks can be painted = 93750/937.5 = 100

5. A cubical box has each edge 10 cm and another cuboidal box is 12.5 cm long, 10 cm wide and 8 cm high.
 (i) Which box has the greater lateral surface area and by how much?
 (ii) Which box has the smaller total surface area and by how much?

Answer
(i) Lateral surface area of cubical box of edge 10cm = 4×10² cm² = 400 cm²
Lateral surface area of cuboid box = 2(l+b)×h
= 2×(12.5+10)×8 cm²
= 2×22.5×8 cm² = 360 cm²
Thus, lateral surface area of the cubical box is greater by (400 – 360) cm² = 40 cm²

(ii) Total surface area of cubical box of edge 10 cm =6×102cm2=600cm2
Total surface area of cuboidal box = 2(lb + bh + lh)
= 2(12.5×10 + 10×8 + 8×12.5) cm²
= 2(125+80+100) cm²
= (2×305) cm² = 610 cm²
Thus, total surface area of cubical box is smaller by 10 cm²

6. A small indoor greenhouse (herbarium) is made entirely of glass panes (including base) held together with tape. It is 30 cm long, 25 cm wide and 25 cm high.
 (i) What is the area of the glass?
 (ii) How much of tape is needed for all the 12 edges?

Answer
(i) Dimensions of greenhouse:
l = 30 cm, b = 25 cm, h = 25 cm
Total surface area of green house = 2(lb + bh + lh)
= 2(30×25 + 25×25 + 25×30) cm²
= 2(750 + 625 + 750) cm²
= 4250 cm²
(ii) Length of the tape needed = 4(l + b + h)
= 4(30 + 25 + 25) cm
= 4×80 cm = 320 cm

7. Shanti Sweets Stall was placing an order for making cardboard boxes for packing their sweets. Two sizes of boxes were required. The bigger of dimensions 25 cm × 20 cm × 5 cm and the smaller of dimensions 15 cm × 12 cm × 5 cm. For all the overlaps, 5% of the total surface area is required extra. If the cost of the cardboard is Rs.4 for 1000 cm² , find the cost of cardboard required for supplying 250 boxes of each kind.

Answer
Dimension of bigger box = 25 cm × 20 cm × 5 cm
Total surface area of bigger box = 2(lb + bh + lh)
= 2(25×20 + 20×5 + 25×5) cm²
= 2(500 + 100 + 125) cm²
= 1450 cm²
Dimension of smaller box = 15 cm × 12 cm × 5 cm
Total surface area of smaller box = 2(lb + bh + lh)
= 2(15×12 + 12×5 + 15×5) cm²
 = 2(180 + 60 + 75) cm²
= 630 cm²
Total surface area of 250 boxes of each type = 250(1450 + 630) cm²
 = 250×2080 cm² = 520000 cm²
Extra area required = 5/100(1450 + 630) × 250 cm² = 26000 cm²

Total Cardboard required = 520000 + 26000 cm² = 546000 cm²
Total cost of cardboard sheet = Rs. (546000 × 4)/1000 = Rs. 2184

8. Parveen wanted to make a temporary shelter for her car, by making a box-like structure with tarpaulin that covers all the four sides and the top of the car (with the front face as a flap which can be rolled up). Assuming that the stitching margins are very small, and therefore negligible, how much tarpaulin would be required to make the shelter of height 2.5 m, with base dimensions 4m × 3m?

Answer
Dimensions of the box- like structure = 4m × 3m × 2.5
Tarpaulin only required for all the four sides and top.
Thus, Tarpaulin required = 2(l+b)×h + lb
     = [2(4+3)×2.5 + 4×3] m²
     = (35×12) m²
     = 47 m²

Page No: 216

Exercise 13.2

1. The curved surface area of a right circular cylinder of height 14 cm is 88 cm². Find the diameter of the base of the cylinder.

 Answer
Let r be the radius of the base and h = 14 cm be the height of the cylinder.
Curved surface area of cylinder = 2πrh = 88 cm²
⇒ 2 × 22/7 × r × 14 = 88
⇒ r = 88/ (2 × 22/7 × 14)
⇒ r = 1 cm
Thus, the diameter of the base = 2r = 2×1 = 2cm

2. It is required to make a closed cylindrical tank of height 1 m and base diameter 140 cm from a metal sheet. How many square metres of the sheet are required for the same?

Answer
Let r be the radius of the base and h be the height of the cylinder.
Base diameter = 140 cm and Height (h) = 1m
Radius of base (r) = 140/2 = 70 cm = 0.7 m
Metal sheet required to make a closed cylindrical tank = 2πr(h + r)
= (2 × 22/7 × 0.7) (1 + 0.7) m²
= (2 × 22 × 0.1 × 1.7) m²
=7.48 m²

3. A metal pipe is 77 cm long. The inner diameter of a cross section is 4 cm, the outer diameter being 4.4 cm (see Fig. 13.11). Find its
 (i) inner curved surface area,
 (ii) outer curved surface area,
 (iii) total surface area.

Answer

Let R be external radius and r be the internal radius h be the length of the pipe.

R  = 4.4/2 cm = 2.2 cm

r = 4/2 cm = 2 cm
h = 77 cm
(i) Inner curved surface = 2πrh cm²
       = 2 × 22/7 × 2 × 77cm²
       = 968 cm²

(ii) Outer curved surface = 2πRh cm²

        = 2 × 22/7 × 2.2 × 77 cm²
        = 1064.8 cm²

(iii) Total surface area of a pipe = Inner curved surface area + outer curved surface area + areas of two bases

= 2πrh + 2πRh + 2π(R^{2 -} r²)
= [968 + 1064.8 + (2 × 22/7) (4.84 - 4)] cm²
= (2032.8 + 44/7 × 0.84) cm²
= (2032.8 + 5.28) cm² = 2038.08 cm²

4. The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions to move once over to level a playground. Find the area of the playground in m².

Answer
Length of the roller (h) = 120 cm = 1.2 m
Radius of the cylinder = 84/2 cm = 42 cm = 0.42 m
Total no. of revolutions = 500
Distance covered by roller in one revolution = Curved surface area = 2πrh
                 = (2 × 22/7 × 0.42 × 1.2) m²
                 = 3.168 m²
Area of the playground = (500 × 3.168) m² = 1584 m²

5. A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of painting the curved surface of the pillar at the rate of Rs.12.50 per m².

Answer
Radius of the pillar (r) = 50/2 cm = 25 cm = 0.25 m
Height of the pillar (h) = 3.5 m.
Rate of painting = Rs.12.50 per m²
Curved surface = 2πrh
                         = (2 × 22/7 × 0.25 × 3.5) m²
                              =5.5 m²
Total cost of painting = (5.5 × 12.5) = Rs.68.75

6. Curved surface area of a right circular cylinder is 4.4 m². If the radius of the base of the cylinder is 0.7 m, find its height.

 Answer
Let r be the radius of the base and h be the height of the cylinder.
Curved surface area = 2πrh = 4.4 m²
⇒ 2 × 22/7 × 0.7 × h = 4.4 
⇒ h = 4.4/(2 × 22/7 × 0.7) = 1m
⇒ h = 1m

7. The inner diameter of a circular well is 3.5 m. It is 10 m deep. Find
 (i) its inner curved surface area,
 (ii) the cost of plastering this curved surface at the rate of Rs.40 per m².

 Answer
Radius of circular well (r) = 3.5/2 m = 1.75 m
Depth of the well (h) = 10 m
Rate of plastering = Rs.40 per m²
(i) Curved surface = 2πrh
                              = (2 × 22/7 × 1.75 × 10) m²
                              = 110 m²

(ii) Cost of plastering = Rs.(110 × 40) = Rs.4400

8. In a hot water heating system, there is a cylindrical pipe of length 28 m and diameter 5 cm. Find the total radiating surface in the system.

 Answer
Radius of the pipe (r) = 5/2 cm = 2.5 cm = 0.025 m
Length of the pipe (h) = 28/2 m = 14 m
Total radiating surface = Curved surface area of the pipe = 2πrh
     = (2 × 22/7 × 0.025 × 28) m²
       = 4.4 m²
9. Find
 (i) the lateral or curved surface area of a closed cylindrical petrol storage tank that is 4.2 m in diameter and 4.5 m high.
 (ii) how much steel was actually used, if 1/12 of the steel actually used was wasted in making the tank.

 Answer
(i) Radius of the tank (r) = 4.2/2 m = 2.1 m
Height of the tank (h) = 4.5 m
Curved surface area = 2πrh m²
 = (2 × 22/7 × 2.1 × 4.5) m²
 = 59.4 m²

(ii) Total surface area of the tank = 2πr(r + h) m²
= [2 × 22/7 × 2.1 (2.1 + 4.5)] m² = 87.12 m²
Let x be the actual steel used in making tank.
∴ (1 - 1/12) × x = 87.12
⇒ x = 87.12 × 12/11
⇒ x = 95.04 m²

10. In Fig. 13.12, you see the frame of a lampshade. It is to be covered with a decorative cloth. The frame has a base diameter of 20 cm and height of 30 cm. A margin of 2.5 cm is to be given for folding it over the top and bottom of the frame. Find how much cloth is required for covering the lampshade.

Answer
Radius of the frame (r) = 20/2 cm = 10 cm 
Height of the frame (h) = 30 cm + 2×2.5 cm = 35 cm
2.5 cm of margin will be added both side in the height.
Cloth required for covering the lampshade = curved surface area = 2πrh
    = (2 × 22/7 × 10 × 35)cm²
                  = 2200 cm²

11. The students of a Vidyalaya were asked to participate in a competition for making and decorating penholders in the shape of a cylinder with a base, using cardboard. Each penholder was to be of radius 3 cm and height 10.5 cm. The Vidyalaya was to supply the competitors with cardboard. If there were 35 competitors, how much cardboard was required to be bought for the competition?

 Answer
Radius of the penholder (r) = 3cm
Height of the penholder (h) = 10.5cm
Cardboard required by 1 competitor = CSA of one penholder + area of the base
= 2πrh + πr²
= [(2 × 22/7 × 3 × 10.5) + 22/7 × 3²] cm²
= (198 + 198/7) cm²
= 1584/7 cm²
Cardboard required for 35 competitors = (35 × 1584/7) cm²
 = 7920 cm²

Exercise 13.2

1. Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area.

 Answer
Radius (r) = 10.5/2 cm = 5.25 cm
Slant height (l) = 10 cm
Curved surface area of the cone = (πrl) cm²
=(22/7 × 5.25 × 10) cm²
=165 cm²

2. Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.

Answer
Radius (r) = 24/2 m = 12 m
Slant height (l) = 21 m
Total surface area of the cone = πr (l + r) m²
  = 22/7 × 12 × (21 + 12) m²
  = (22/7 × 12 × 33) m²
  = 1244.57 m²

3. Curved surface area of a cone is 308 cm² and its slant height is 14 cm. Find
 (i) radius of the base and
 (ii) total surface area of the cone.

 Answer
(i) Curved surface of a cone = 308 cm²
Slant height (l) = 14cm
Let r be the radius of the base
∴ πrℓ = 308
⇒ 22/7 × r × 14 = 308
⇒ r =308/(22/7 × 14) = 7 cm

(ii) TSA of the cone = πr(l + r) cm²
 = 22/7 × 7 ×(14 + 7) cm²
 = (22 × 21) cm²
 = 462 cm²

4. A conical tent is 10 m high and the radius of its base is 24 m. Find
 (i) slant height of the tent.
 (ii) cost of the canvas required to make the tent, if the cost of 1 m² canvas is Rs.70

 Answer
Radius of the base (r) = 24 m
Height of the conical tent (h) = 10 m
Let l be the slant height of the cone.
∴ l² = h² + r²
⇒ l = √h² + r² 
⇒ l = √10² + 24² 
⇒ l = √100 + 576 
⇒ l = 26 m
(ii) Canvas required to make the conical tent = Curved surface of the cone
Cost of 1 m² canvas = Rs.70
∴ πrl = (22/7 × 24 × 26) m² = 13728/7 m²
∴ Cost of canvas = Rs. 13728/7 × 70 = Rs.137280

5. What length of tarpaulin 3m wide will be required to make conical tent of height 8m and base radius 6m? Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately 20cm (Use π = 3.14).

 Answer
Radius of the base (r) = 6 m
Height of the conical tent (h) = 8 m
Let l be the slant height of the cone.
∴ l = √h² + r² 
⇒ l = √8² + 6² 
⇒ l = √100 
⇒ l = 10 m
CSA of conical tent = πrl  = (3.14 × 6 × 10) m² = 188.4 m²
Breadth of tarpaulin = 3 m 
Let length of tarpaulin sheet required be x.
20 cm will be wasted in cutting.
So, the length will be (x - 0.2 m)
Breadth of tarpaulin = 3 m
Area of sheet = CSA of tent
[(x - 0.2 m) × 3] m = 188.4 m²
⇒ x - 0.2 m = 62.8 m
⇒ x = 63 m

6. The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of Rs.210 per 100 m².

Answer
Radius (r) = 14/2 m = 7 m 
Slant height tomb (l) = 25 m
Curved surface area = πrl m²
=(227×25×7) m²
=550  m²
Rate of white- washing = Rs.210 per 100 m²
Total cost of white-washing the tomb = Rs.(550 × 210/100) = Rs.1155

7. A joker’s cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps.

Answer
Radius of the cone (r) = 7 cm
Height of the cone (h) = 24 cm
Let l be the slant height 
∴ l = √h² + r² 
⇒ l = √24² + 7² 
⇒ l = √625 
⇒ l = 25 m
Sheet required for one cap = Curved surface of the cone
  = πrl cm²
  = (22/7 × 7 × 25) cm²
  = 550 cm²
Sheet required for 10 caps = 550 × 10 cm² = 5500 cm²

8. A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is Rs.12 per m² , what will be the cost of painting all these cones? (Use π = 3.14 and take √1.04 = 1.02)

Answer
Radius of the cone (r) = 40/2 cm = 20 cm = 0.2 m
Height of the cone (h) = 1 m
Let l  be the slant height of a cone. 
∴ l = √h² + r² 
⇒ l = √1² + 0.2² 
⇒ l = √1.04 
⇒ l = 1.02 m
Rate of painting = Rs.12 per m² 
Curved surface of 1 cone = πrl m²
= (3.14 × 0.2 × 1.02) m²
= 0.64056 m²
Curved surface of such 50 cones = (50 × 0.64056) m²
= 32.028 m²
Cost of painting all these cones = Rs.(32.028 × 12)
     = Rs.384.34