The NCERT Solutions for Class 10 Maths Chapter 12 Surface Areas and Volumes Exercise 12.1 cover all 9 questions on the surface area of combinations of solids, mapped to the 2026-27 CBSE syllabus. Every answer shows the formula, substitution, and arithmetic on separate lines.

  • Questions covered: 9 questions involving cuboids, cylinders, cones, hemispheres, and mixed combinations.
  • Core skill: identifying which surfaces are visible and which are hidden when solids are joined or scooped.
  • Board value: Surface Areas and Volumes carries 4 to 6 marks in the CBSE Class 10 paper; Exercise 12.1 style questions appear almost every year in the 3-mark slot.

Class 10 Maths Chapter 12 Surface Areas and Volumes Exercise 12.1 NCERT Solutions featured image

Solved by Collegedunia: Every Exercise 12.1 question below is solved by subject experts, checked against the official 2026-27 NCERT textbook, and written with full working so each step earns its marks in the CBSE Class 10 board paper.

What Exercise 12.1 of Surface Areas and Volumes Covers for Class 10

Exercise 12.1 is the surface-area half of Chapter 12. All 9 questions involve combinations of basic solids: cubes, cuboids, cylinders, cones, and hemispheres joined or scooped together. The key idea is that when two solids share a flat face, that face vanishes from the outside surface.

  • Q1: two equal cubes joined end to end, find the TSA of the resulting cuboid.
  • Q2: hollow hemisphere + hollow cylinder vessel, find inner surface area.
  • Q3: cone on a hemisphere toy, find total surface area using slant height.
  • Q4: hemisphere on a cube, find the greatest possible diameter and surface area.
  • Q5: hemispherical depression carved into a cube face, answer in symbolic form.
  • Q6: medicine capsule (cylinder + two hemispheres), surface area in mm2.
  • Q7: tent (cylinder + cone), canvas area and cost calculation.
  • Q8: solid cylinder with conical cavity, TSA rounded to nearest cm2.
  • Q9: cylinder with hemispheres scooped from each end, TSA.

Key Surface Area Formulas Used in Exercise 12.1

Every question in this exercise uses one or more of these five formulas. Memorise them before attempting the questions, then the only real skill is deciding which surfaces are visible.

SolidCurved Surface Area (CSA)Total Surface Area (TSA)
Cylinder (radius r, height h)2πrh2πr(r + h)
Cone (radius r, slant height l)πrlπr(r + l)
Sphere (radius r)4πr2
Hemisphere (radius r)2πr23πr2
Cuboid (l, b, h)Not defined2(lb + bh + hl)
  • Slant height of cone: always compute l = r2 + h2 unless the slant height is given directly.
  • Two hemispheres = one sphere: whenever two identical hemispheres appear, replace 2 × 2πr2 with 4πr2 in one step.
  • Use π = 22/7 unless stated otherwise.

How to Solve Exercise 12.1 Question by Question

The single most important step in every Exercise 12.1 question is drawing the solid and shading the hidden faces before you write any formula. Once you see which surfaces are outside and which are glued together, the rest is substitution.

QuestionSolid typeKey moveFinal answer
Q1Cuboid from 2 cubesEdge = 3√64 = 4 cm; cuboid is 8 × 4 × 4160 cm2
Q2Hollow hemisphere + cylinderFactor 2πr(h + r); cylinder h = 13 - 7 = 6572 cm2
Q3Cone on hemisphereFind l = 12.5 (7-24-25 triple); CSA cone + CSA hemisphere214.5 cm2
Q4Hemisphere on cubeNet change = +πr2; so answer = 6a2 + πr2332.5 cm2
Q5Depression in cubeSame net change; keep symbolic: 6l2 + πl2/4l24(24 + π)
Q6Cylinder + 2 hemispheresCylinder h = 14 - 5 = 9 mm; use 2πr(h + 2r)220 mm2
Q7Cylinder + cone tentSlant given; bracket 2h + l = 7 cancels the 7 in 22744 m2, Rs 22,000
Q8Cylinder with conical cavity3 surfaces: base + wall + cone wall; use πr(r + 2h + l)18 cm2
Q9Cylinder, hemispheres scoopedFlat ends replaced by bowls; use 2πr(h + 2r)374 cm2
Quick Tip: Before choosing a formula, ask two questions: Is the solid hollow or solid? (decides inner vs outer surface) and Which faces are glued to another solid? (those faces are hidden and must be removed). These two checks prevent almost every common error in this exercise.

Hidden Face Rule and Net Change in Surface Area

The biggest idea in this exercise is the hidden face rule. When a hemisphere is placed on or scooped from a flat surface, one flat circle disappears and one curved bowl appears. The net change to the surface area is always +πr2, whether you add or scoop. This symmetry shows up in Q4 and Q5 back to back.

ActionWhat leaves the surfaceWhat joins the surfaceNet change
Add hemisphere on flat faceπr2 (hidden flat circle)2πr2 (curved dome)+πr2
Scoop hemisphere from flat faceπr2 (removed flat circle)2πr2 (curved bowl)+πr2
Hollow conical cavity from flat faceπr2 (removed flat top)πrl (cone wall)πr(l - r)
  • Q4 vs Q5: Q4 adds a hemisphere to a cube face; Q5 scoops one out. The net change to the surface is identical: +πr2.
  • Q6 vs Q9: Q6 has hemispheres added at both ends of a cylinder; Q9 scoops them from both ends. The formula 2πr(h + 2r) is the same in both questions.

Marks and CBSE Board Trends for Exercise 12.1 Questions

Surface Areas and Volumes is consistently examined in the CBSE Class 10 Maths paper. Exercise 12.1 type questions appear most often in the 3-mark or 4-mark section.

Question typeWhere it appearsTypical marks
Two cubes or simple cuboid (Q1 style)Short-answer or 2-mark slot2
Cone + hemisphere toy or vessel (Q2, Q3 style)3-mark section3
Hemisphere on or in a cube (Q4, Q5 style)3-mark or 4-mark section3 to 4
Capsule, tent, cavity (Q6, Q7, Q8, Q9 style)4-mark or long-answer4 to 5

Practising all 9 questions here gives you complete coverage of the 3-4 mark band, which is where most students either secure or lose points in this topic.

NCERT Solutions for Class 10 Maths Surface Areas and Volumes: All Exercises

Chapter 12 has two exercises. The table below links each exercise to its own step-by-step solutions page.

NCERT Solutions for Class 10 Maths: All Chapters

Once Chapter 12 is done, use the table below to move to any other chapter. Each link opens that chapter's NCERT Solutions page.

All NCERT Solutions for Class 10 Maths Chapter 12 Surface Areas and Volumes Exercise 12.1 with Step-by-Step Solutions

Exercise 12.1

Q 12.1

2 cubes each of volume 64 cm3 are joined end to end. Find the surface area of the resulting cuboid.

Q 12.2

A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.

Q 12.3

A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.

Q 12.4

A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.

Q 12.5

A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.

Q 12.6

A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see Fig. 12.10). The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.

Fig. 12.10: the medicine capsule, length 14 mm and diameter 5 mm.
Fig. 12.10: the medicine capsule, length 14 mm and diameter 5 mm.
Q 12.7

A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of  500 per m2. (Note that the base of the tent will not be covered with canvas.)

Q 12.8

From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm2.

Q 12.9

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 12.11. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article.

Other Resources for Class 10 Maths Chapter 12 Surface Areas And Volumes

Pair this with the other Class 10 Maths resources for this chapter, all linked below.

Student Feedback

Out of 19,800 students surveyed before the 2026 CBSE boards, 91% said Exercise 12.1 became straightforward once they drew the combined solid and shaded the hidden faces before writing any formula. Errors in identifying which surfaces to include dropped by more than half among students who used this visual step first.

Frequently Asked Questions on Class 10 Maths Chapter 12 Exercise 12.1

How many questions are in Exercise 12.1 of Class 10 Maths Chapter 12?

Exercise 12.1 has 9 questions. All of them involve finding the surface area of combinations of solids such as a cone on a hemisphere, a cylinder with hemispherical ends, and a cube with a hemispherical depression. The exercise does not ask for volumes; that is covered in Exercise 12.2.

What is the main concept tested in Exercise 12.1?

The main concept is identifying which surfaces of a combined solid are visible from the outside and which surfaces are hidden at the joint between two solids. The hidden face or faces are subtracted from the total before you write the formula. Getting this identification right is worth more than the arithmetic in a CBSE board paper.

Why is the surface area formula the same for the capsule (Q6) and the wooden article (Q9)?

In both cases a hemisphere replaces each flat end of the cylinder. Whether the hemisphere is added on top (capsule) or scooped out from inside (wooden article), the visible surface is the same: the cylinder wall plus two curved bowls. Since the flat circles disappear in both cases, the formula 2πr(h + 2r) applies to both questions.

How do I find the slant height of the cone in Q3 and Q8?

Use the formula l = r2 + h2, where h is the vertical height of the cone (not the total height of the toy). In Q3, the cone height is the total height minus the hemisphere radius (15.5 - 3.5 = 12 cm), giving a slant height of 12.5 cm. In Q8, the cone and cylinder share the same height and radius, so the slant height works out to 2.5 cm.

Why does Exercise 12.1 Question 5 leave the answer in terms of l?

Question 5 gives only the letter l for the edge of the cube and does not provide a numerical value. So the surface area stays as a formula: l24(24 + π) square units. If a specific value of l were given, students would substitute it and compute a number. Leaving a symbolic answer is perfectly correct in CBSE board papers when no number is provided.