Chapter 12: Surface Areas and Volumes
Unique Previous Year Questions extracted from the CBSE Class 10 Basic Mathematics 2025 question paper sets. Redundant questions have been removed while retaining the major question types and variations.
I
Multiple Choice Questions
1. Conceptual Solid
1 Mark
For which of the following solids is the lateral/curved surface area
and total surface area the same?
Set 430/1/1
2. Ratio Analysis – Cone and Cylinder
1 Mark
A cone and a cylinder have the same base radius and volume.
The ratio of their heights
\[
\text{height of cone}:\text{height of cylinder}
\]
is:
Set 430/2/1
3. Combination Formula
1 Mark
A cone of height \(h\) and radius \(r\) is surmounted on a solid
cylinder of the same dimensions. The total surface area of the
entire solid will be:
Set 430/2/2
4. Volume Ratio
1 Mark
A cone and cylinder have the same height and same radius.
The volumes of the cone and cylinder are in the ratio:
Set 430/3/1
5. Fitting Solids
1 Mark
The largest possible cone is just fitted inside a hollow cube of
edge \(25\) cm. The radius of the base of the cone is:
Set 430/5/1
6. Figure-Based – Empty Space
1 Mark
The volume of air in a hollow cylinder is \(450\text{ cm}^3\).
A cone of the same height and radius as that of the cylinder is
kept inside it. The volume of empty space in the cylinder is:
Set 430/6/1
7. Joined Solids
1 Mark
Two identical cones are joined as shown in the figure, with their
bases touching. If the radius of the base is \(4\) cm and the slant
height of each cone is \(6\) cm, then the height of the solid is:
Set 430/6/3
II
Assertion – Reason Question
8. Hemisphere Carved from Cylinder
1 Mark
Assertion (A):
When a hemisphere of same radius \(r\) is carved out from one side
of a solid wooden cylinder, the total surface area of the remaining
solid is increased by \(2\pi r^2\).
Reason (R):
Curved surface area of a hemisphere is \(2\pi r^2\).
III
Short Answer Type Questions
9. Scooping Problem
3 Marks
A hemispherical depression is scooped out from the top face of a
wooden cubical block of side \(14\) cm. If the diameter of the
hemisphere is equal to the side of the cube, find the total surface
area of the remaining solid block.
\[
\text{Use }\pi=\frac{22}{7}
\]
Set 430/2/2
IV
Long Answer Type Questions
10. Combination Volume
5 Marks
A toy is in the form of a cone surmounted on a hemisphere.
The cone and hemisphere have the same radii. The height of the
conical part of the toy is equal to the diameter of its base.
If the radius of the conical part is \(5\) cm, find the volume
of the toy.
Set 430/1/1
11. Combination Surface Area
5 Marks
A cubical block is surmounted by a hemisphere of radius \(3.5\) cm.
What is the smallest possible length of the edge of the cube so that
the hemisphere can totally lie on the cube? Find the total surface
area of the solid so formed.
Set 430/1/1
12. Figure-Based – Wooden Article
5 Marks
A wooden article was made by scooping out a hemisphere
(of same diameter) from one end of a solid cylinder.
If the height of the cylinder is \(10\) cm and the diameter of
the cylinder is \(14\) cm, find the total surface area of the
remaining wooden article.
Set 430/2/1
13. Vessel Capacity
5 Marks
A spherical glass vessel has a cylindrical neck which is \(7\) cm
long and \(2\) cm in diameter. The diameter of the spherical part
is \(14\) cm. Find the capacity of the entire glass vessel.
\[
\text{Use }\pi=\frac{22}{7}
\]
Set 430/2/3
14. Figure-Based – Perfume Bottle
5 Marks
A perfume bottle is in the form of a cylinder but the bottom has
a hemispherical raised portion to reduce capacity. The inner
diameter is \(5\) cm and height is \(10\) cm. Find the capacity
of the bottle in mL.
\[
\text{Use }\pi=3.14,\qquad 1\text{ cm}^3=1\text{ mL}
\]
Set 430/3/1
15. Composite Cylinder
5 Marks
From a solid wooden cylinder of height \(10\) cm and radius \(14\) cm,
a cylinder of radius \(7\) cm and height \(5\) cm is scooped out to
form a cavity inside. Find the total surface area of the remaining
solid.
Set 430/3/2
16. Figure-Based – Wooden Beads
5 Marks
A necklace is made of wooden beads. Each bead is a sphere of
diameter \(4.2\) mm. A cylinder is hollowed out from each bead.
If the radius of the cylinder is \(1\) mm, find the volume of
wood left in each bead.
Set 430/3/3
17. Advanced Scooping
5 Marks
From each end of a solid cylinder of height \(20\) cm and base
radius \(7\) cm, a cone of base radius \(2.1\) cm and height
\(5\) cm is scooped out. Find the volume of the remaining solid.
Set 430/5/2
V
Case Study Based Questions
18. Data-Based – Ball Pool
4 Marks
Suhana bought \(600\) new balls of diameter \(7\) cm to fill a pool.
The cuboidal box containing the balls has dimensions
\(42\text{ cm}\times91\text{ cm}\times50\text{ cm}\).
(i) Find the volume of one ball. [1 Mark]
(ii) \(10\) balls are painted. Determine the area of
painted surface. [1 Mark]
(iii) Find the volume of empty space in the box.
OR Determine how many balls are in the lowermost
layer if they cover the base edge to edge.
[2 Marks]
19. Data-Based – Bowl in Box
4 Marks
A hemispherical bowl is packed in a cuboidal box. The bowl just fits
in the box. Its inner radius is \(10\) cm and outer radius is
\(10.5\) cm.
(i) Find the dimensions of the cuboidal box.
[1 Mark]
(ii) Find the total outer surface area of the box.
[1 Mark]
(iii) Find the difference between the capacity of the
bowl and volume of the box (use \(\pi=3.14\)).
OR Find the inner area of the bowl to be painted.
[2 Marks]