Need NCERT solutions for class 10 maths chapter 12 (Surface Areas and Volumes)? This page routes you to the right exercise and the method it wants — the step-by-step solutions live one level below.
Chapter 12 has 17 questions across two exercises. Exercise 12.1 (nine questions) tests the surface area of combined solids. Exercise 12.2 (eight questions) tests volume, capacity, mass and water displacement.
One idea drives the whole chapter: for surface area you add only the exposed surfaces, because joining hides faces; for volume you add the parts directly. The chapter opens with Fig. 12.1 — the four solids you know from Class IX: the cuboid, cone, cylinder and sphere — and the truck-container of Fig. 12.2, a cylinder capped by two hemispheres.
Exercise 12.1 and 12.2 — what each question tests
Both exercises build one basic solid on another. The routing below tells you which question is yours and what method it wants; the table renders the same 17 rows as links to the full solutions.
Exercise 12.1 — nine surface area questions
Every question here places one solid on another and asks for the surface area of the result. The method never changes: list the surfaces that stay exposed, add their curved (or flat) areas, and leave out the faces the join hid.
- Two cubes joined end to end — the joined faces disappear, so the resulting cuboid’s surface area comes from its outer faces alone.
- Hollow hemisphere on a hollow cylinder — add the two inner curved surfaces that share one radius; the hidden ring between them does not count.
- Cone on a hemisphere of the same radius — the cone’s height is the total height minus the radius; find its slant height by Pythagoras, then add the two curved surfaces.
- Cube topped by a hemisphere — the greatest diameter the hemisphere can have is the cube’s edge; subtract the covered circle and add the hemisphere’s curved surface.
- Hemispherical depression cut from a cube face — the cube loses the covered circle and gains the depression’s inner curved surface.
- Medicine capsule, cylinder plus two hemispheres (Fig. 12.10) — the two hemispheres together form one full sphere, so the surface area is the cylinder’s curved part plus the sphere’s surface.
- Tent, cylinder with a conical top — the open base is not covered by canvas; find the canvas area, then its cost at ₹500 per square metre.
- Solid cylinder with a conical cavity hollowed out — the remaining surface shows the outer cylinder wall, the inner cone wall and one exposed ring.
- Hemisphere scooped from each end of a cylinder (Fig. 12.11) — the two scoops form a sphere; add the cylinder’s curved wall to the sphere’s full surface.
Exercise 12.2 — eight volume questions
Every question here deals with volume — how much a combined solid holds, how much material it contains, or how much water it displaces. The volumes of the parts are simply added, or subtracted when material is removed.
- Cone on a hemisphere, both radii 1 cm — the volume is the straight sum of the two parts; the answer stays in terms of \( \pi \).
- Rachel’s model, a cylinder with two cones at its ends — the air inside is the sum of all three volumes.
- Gulab jamun, a cylinder with two hemispherical ends (Fig. 12.15) — find the capsule volume first, then take 30% of it and multiply across 45 sweets.
- Pen stand, a cuboid with four conical depressions — subtract the four cones from the cuboid to get the wood remaining.
- Lead shots dropped into a brim-full cone of water — the water that flows out is one quarter of the cone’s volume; each shot is a sphere; divide to get the count.
- Solid iron pole made of two cylinders — find the total volume first, then mass equals volume times 8 grams per cubic centimetre.
- A cone and hemisphere sunk in a full cylinder of water — the water left equals the cylinder’s volume minus the solid’s volume.
- Spherical vessel with a cylindrical neck — add the sphere’s and neck’s volumes and compare with the child’s 345 cubic centimetres to check her answer.
The three methods behind the whole chapter
Chapter 12 reuses the Class IX formulas for the five basic solids. You are expected to know them cold:
| Solid | Curved surface area | Total surface area | Volume |
|---|---|---|---|
| Cuboid | — | \( 2(lb + bh + hl) \) | \( lbh \) |
| Cone | \( \pi rl \) | \( \pi r(l + r) \) | \( \frac{1}{3}\pi r^2h \) |
| Cylinder | \( 2\pi rh \) | \( 2\pi rh + 2\pi r^2 \) | \( \pi r^2h \) |
| Sphere | \( 4\pi r^2 \) | \( 4\pi r^2 \) | \( \frac{4}{3}\pi r^3 \) |
| Hemisphere | \( 2\pi r^2 \) | \( 3\pi r^2 \) | \( \frac{2}{3}\pi r^3 \) |
Revise the derivations in our Class 10 Maths notes, then apply the three methods below. You can verify every formula and figure against the official NCERT textbook PDF for Chapter 12.
Rule 1 — surface areas are never simply added
The full total surface areas of the two parts are never added, because the join hides a face (or a base) that both parts once counted. You add only the exposed curved surfaces. Worked examples 1 and 2, and the bird-bath of example 4 (pp. 163-166), all run on this rule.
Rule 2 — volumes are simply added
Volume is different: joining two solids loses no material, so the volume of the combination is the straight sum of the parts’ volumes. Examples 5, 6 and 7 (pp. 167-169) each add, or subtract, the volumes of the constituents.
The slant-height route
When a cone surmounts another solid and only the total height is given, the cone’s height is the total height minus the radius of the solid below it. Then the slant height is \( l = \sqrt{r^2 + h^2} \), which feeds the cone’s curved surface area \( \pi r l \). Example 1 (p. 164) is the model.
Common mistakes in Chapter 12
- Adding the full total surface areas of both solids. This counts the hidden joint twice. Only exposed surfaces count (Exercise 12.1 Q1, Q9).
- Forgetting to subtract the covered base circle when one solid sits on a face of another — the base of the top solid is no longer visible (Exercise 12.1 Q4, Q5, Q8; example 2).
- Using the whole height as the cone’s height when a cone stands on a hemisphere. Subtract the radius first (Exercise 12.1 Q3; example 1).
- Reading a diameter as a radius. The hemisphere’s diameter equals the cube edge in Q4, and Q8 gives the cavity’s diameter — halve before using any formula.
- Adding volumes where the solid removes material. The pen stand of Exercise 12.2 Q4 must subtract its four cones from the cuboid.
- Mixing π values. Take \( \pi = \frac{22}{7} \) unless a question states otherwise (both exercises do state it, except Exercise 12.2 Q6, which says use 3.14). In that same question, mass equals volume times density (8 g per cm³).
How to pick the right exercise
One glance is enough:
- The question asks for surface area, canvas, paint or colour → Exercise 12.1 (nine questions).
- The question asks for volume, capacity, syrup, mass or water displaced → Exercise 12.2 (eight questions).
Each row of the exercise directory above links to that exercise’s step-by-step solutions. The chapter summary (p. 171) states the two skills the board examines: finding the surface area of a combined solid and finding its volume. Chapter 11 covered areas related to circles, and Chapter 13 moves on to statistics — this chapter sits between them in the Class 10 syllabus.
| Exercise | What you get |
|---|---|
| Exercise 12.1 | Solved |
| Exercise 12.2 | Solved |
Frequently asked questions
Which exercise in Chapter 12 has surface area of combined solids?
Exercise 12.1, which has nine questions on surface area.
Which exercise covers volume of combined solids?
Exercise 12.2, which has eight questions on volume, capacity, mass and water displacement.
How many questions are in the two exercises of Chapter 12?
17 total — 9 in Exercise 12.1 and 8 in Exercise 12.2.
Why is the surface area of a combined solid not the sum of the two full surface areas?
Because the faces where the solids join are hidden and must be excluded; you add only the exposed surfaces.
Do I take π as 22/7 in these questions?
Yes — unless a question states otherwise. Both exercises open with “Unless stated otherwise, take \( \pi = \frac{22}{7} \)”.
What Class IX formulas do I need for Chapter 12?
The curved and total surface areas and volumes of the cuboid, cone, cylinder, sphere and hemisphere — all listed in the table above.
Reference: NCERT textbooks (CBSE).