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NCERT Solutions · Class 10 · Chapter 12: Surface Areas and Volumes

NCERT Solutions for Class 10 Maths Chapter 12 Exercise 12.1

Exercise 12.1: Surface area of combined solids. Add the areas of the faces you can see and leave out any face that is hidden where two solids join. A hemisphere's curved surface is \(2\pi r^2\); a cone's curved surface is \(\pi r l\) with \(l = \sqrt{r^2 + h^2}\); a cylinder's is \(2\pi r h\). Unless the question gives another value, take \(\pi = \tfrac{22}{7}\), as the NCERT exercises do (no calculator: CBSE is a no-calculator exam).

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Exercise 12.1 questions and solutions

Exercise 12.1, Question 1

Two cubes of volume \(64\ \text{cm}^3\) are joined end to end. Find the surface area of the cuboid formed.
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  1. Each edge is 4 cm, so the cuboid is \(8 \times 4 \times 4\).
  2. \[2(8 \times 4 + 4 \times 4 + 8 \times 4) = 2(32 + 16 + 32)\]
Answer: \(160\ \text{cm}^2\)

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Exercise 12.1, Question 2

A vessel is a hollow hemisphere topped by a hollow cylinder; diameter 14 cm, total height 13 cm. Find its inner surface area.
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  1. \(r = 7\); cylinder height \(13 - 7 = 6\).
  2. Hemisphere \(2\pi r^2 = 308\); cylinder \(2\pi rh = 264\).
Answer: \(572\ \text{cm}^2\)

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Exercise 12.1, Question 3

A toy is a cone of radius 3.5 cm on a hemisphere of the same radius; total height 15.5 cm. Find its total surface area.
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  1. Cone height \(15.5 - 3.5 = 12\), slant \(\sqrt{3.5^2 + 12^2} = 12.5\).
  2. Cone \[\begin{aligned}\pi rl &= \dfrac{22}{7} \times 3.5 \times 12.5 \\ &= 137.5\end{aligned}\]; hemisphere \(2\pi r^2 = 77\).
Answer: \(214.5\ \text{cm}^2\)

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Exercise 12.1, Question 4

A 7 cm cube is topped by a hemisphere. What is the largest possible diameter of the hemisphere? Find the surface area of the solid.
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  1. The largest diameter is the edge, 7 cm (\(r = 3.5\)).
  2. Surface \[\begin{aligned}= 6 \times 49 - \pi r^2 + 2\pi r^2 &= 294 + \pi r^2 \\ &= 294 + 38.5\end{aligned}\]
Answer: Diameter 7 cm; \(332.5\ \text{cm}^2\)

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Exercise 12.1, Question 5

A hemispherical hollow is cut from one face of a cube of edge \(l\), with the diameter of the hemisphere equal to \(l\). Find the surface area of the remaining solid.
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  1. Remove a circle of area \(\pi\left(\tfrac l2\right)^2\) from one face and add the hemisphere's curved surface \(2\pi\left(\tfrac l2\right)^2\).
  2. \[6l^2 + \dfrac{\pi l^2}{4} = \dfrac{l^2}{4}(24 + \pi)\]
Answer: \(\dfrac{l^2}{4}(\pi + 24)\)

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Exercise 12.1, Question 6

A capsule is a cylinder with a hemisphere at each end; total length 14 mm, diameter 5 mm. Find its surface area.
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  1. \(r = 2.5\); cylinder length \(14 - 5 = 9\).
  2. \[\begin{aligned}2\pi rh + 4\pi r^2 &= 45\pi + 25\pi \\ &= 70\pi \\ &= 220\end{aligned}\]
Answer: \(220\ \text{mm}^2\)

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Exercise 12.1, Question 7

A tent is a cylinder (height 2.1 m, diameter 4 m) with a conical top of slant height 2.8 m. Find the canvas area and its cost at ₹500 per m² (the base is not covered).
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  1. Cylinder \(2\pi(2)(2.1) = 8.4\pi\); cone \(\pi(2)(2.8) = 5.6\pi\).
  2. Total \(14\pi = 44\ \text{m}^2\); cost \(44 \times 500\).
Answer: \(44\ \text{m}^2\); ₹22 000

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Exercise 12.1, Question 8

A cone of the same height and diameter is hollowed out of a solid cylinder (height 2.4 cm, diameter 1.4 cm). Find the total surface area of the remaining solid, to the nearest cm².
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  1. \(r = 0.7\), \(h = 2.4\), slant \(\sqrt{0.49 + 5.76} = 2.5\).
  2. Curved cylinder \(2\pi rh = 3.36\pi\), top \(\pi r^2 = 0.49\pi\), cone inside \(\pi rl = 1.75\pi\).
  3. Total \(5.6\pi = 17.6\).
Answer: \(18\ \text{cm}^2\) (17.6 cm²)

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Exercise 12.1, Question 9

A hemisphere is scooped from each end of a solid cylinder of height 10 cm and radius 3.5 cm. Find the total surface area.
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  1. Curved cylinder \(2\pi(3.5)(10) = 70\pi\); two hemispherical hollows \(2 \times 2\pi(3.5)^2 = 49\pi\).
  2. Total \(119\pi = 374\).
Answer: \(374\ \text{cm}^2\)

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Done the NCERT exercises? The board paper asks more

Surface Areas and Volumes has 37 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.

Surface Areas and Volumes: notes and questionsCreate a free accountTeachers: set it as homework

Surface Areas and Volumes in our sample papers: Sample paper 1 (questions 14, 31) · Sample paper 2 (questions 15, 16, 34) · Sample paper 3 (questions 15, 20, 35) · Sample paper 4 (questions 15, 16, 25, 36) · Sample paper 5 (questions 15, 16, 34).

Also useful: free MCQs and case studies for Surface Areas and Volumes · Class 10 formula sheet · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan

Textbook: NCERT Mathematics Class 10 (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.