Surface Areas and Volumes NCERT Solution, study material, CBSE Notes
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  • Surface Areas and Volumes
NCERT Solution: Surface Areas and Volumes

If the lateral surface of a cylinder is 94.2 cm2 and its height is 5 cm, then find (i) radius of its base (ii) its volume. (Use π = 3.14)

Given; curved surface area of cylinder = 94.2 sq cm, h = 5 cm
CSA of cylinder = 2 π rh
Or, 94.2 = 2 X (3.14) X r X 5
Or, r = 94.2/(2 X 3.14 X 5) = 3 cm

Now, volume of cylinder = π r2 h
= 3.14 X 32 X 5 = 141.30 cubic cm

It costs Rs 2200 to paint the inner curved surface of a cylindrical vessel 10 m deep. If the cost of painting is at the rate of Rs 20 per m2, find
(i) inner curved surface area of the vessel,
(ii) radius of the base,
(iii) capacity of the vessel.

Cost = Rs. 2200, rate = Rs. 20 per sq m, h = 10 m
Curved surface area = Cost/Rate
= 2200/20 = 110 sq cm

CSA of cylinder = 2 π rh
Or, 110 = 2 X (22/7) X r X 10
Or, r = (110 X 7)/(2 X 22 X 10) = 1.75 cm

Now, volume of cylinder = π r2 h
= (22/7) X (1.75)2 X 10 = 96.25 cubic m

The capacity of a closed cylindrical vessel of height 1 m is 15.4 litres. How many square metres of metal sheet would be needed to make it?

Capacity = 15.4 litre = 15400 cubic cm, h = 1 m = 100 cm
Volume of cylinder = π r2 h
Or, 15400 = (22/7) X r2 X 100
Or, r2 = (15400 X 7)/(22 X 100) = 49
Or, r = 7 cm

Now, total surface area of cylinder = 2 π r(r + h)
= 2 X (22/7) X 7 (7 + 10)
= 2 X 22 X 17 = 748 sq cm

A lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior. The diameter of the pencil is 7 mm and the diameter of the graphite is 1 mm. If the length of the pencil is 14 cm, find the volume of the wood and that of the graphite.

Radius of pencil = 3.5 mm, h = 14 cm = 140 mm
Volume of pencil = π r2h
= (22/7) X (3.5)2 X 140 = 5390 cubic mm

Radius of lead = 0.5 mm, h = 140 mm
Volume of lead = π r2 h
= (22/7) X (0.5)2 X 140 = 110 cubic mm
Hence, volume of wood = 5390 – 110 = 5280 cubic mm

A patient in a hospital is given soup daily in a cylindrical bowl of diameter 7 cm. If the bowl is filled with soup to a height of 4 cm, how much soup the hospital has to prepare daily to serve 250 patients?

Given; r = 3.5 cm, h = 4 cm
Volume of cylinder = π r2 h
= (22/7) X (3.5)2 X 4 = 154 cubic cm
Volume of 250 bowls = 250 X 154 = 38500 cubic cm = 38.5 litre

Find the volume of the right circular cone with

(i) radius 6 cm, height 7 cm
(ii) radius 3.5 cm, height 12 cm

(i) radius 6 cm, height 7 cm

Given; r = 6 cm, h = 7 cm
Volume of cone = 1/3 π r2h
= (1/3) X (22/7) X 62 X 7 = 264 cubic cm

(ii) radius 3.5 cm, height 12 cm

 Given; r = 3.5 cm, h = 12 cm
Volume of cone = 1/3 π r2 h
= (1/3) X (22/7) X (3.5)2 X 12 = 154 cubic cm

Find the capacity in litres of a conical vessel with

(i) radius 7 cm, slant height 25 cm
(ii) height 12 cm, slant height 13 cm

(i) radius 7 cm, slant height 25 cm

Answer: Given; r = 7 cm, l = 25 cm
Here; h2 = l2 - r2
= 252 - 72
= 625 – 49 = 576
Or, h = 24 cm

Volume of cone = 1/3 π r2 h
= (1/3) X (22/7) X 72 X 24 = 1232 cubic cm
= 1.232 litre

(ii) height 12 cm, slant height 13 cm

Answer: Given; h = 12 cm, l = 13 cm
Here, r2 = l2 - h2
= 132 - 122
= 169 – 144 = 25
Or, r = 5 cm

Volume of cone = 1/3 π r2 h
= (1/3) X (3.14) X 52 X 12 = 314 cubic cm
= 0.314 litre

The height of a cone is 15 cm. If its volume is 1570 cm3, find the radius of the base. (Use π = 3.14)

Given; volume = 1570 cubic cm, h = 15 cm, r = ?
Volume of cone = 1/3 π r2h
Or, 1570 = (1/3) X (3.14) X r2 X 15
Or, r2 = (1570 X 3)/(3.14 X 15) = 100
Or, r = 10 cm

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