CHAPTER 12 : SURFACE AREAS AND VOLUMES
CBSE Class 10 Mathematics — PYQs 2026 | Basic + Standard
BASIC QUESTION PAPER 2026
SECTION A : MULTIPLE CHOICE QUESTIONS (1 Mark Each)
1.
For which of the following solids is the lateral/curved surface
area and total surface area the same?
(A) Cube
(B) Cuboid
(C) Hemisphere
(D) Sphere
[Sets: 430/1/1, 430/1/2, 430/1/3]
2.
A cone and a cylinder have the same base radius and volume.
The ratio of their heights, i.e. height of cone : height of
cylinder, is:
(A) \(1:1\)
(B) \(3:1\)
(C) \(1:3\)
(D) \(3:2\)
[Set: 430/2/1]
3.
A cone and cylinder have the same height and same radius.
The volumes of the cone and the cylinder are in the ratio:
(A) \(1:1\)
(B) \(1:3\)
(C) \(3:1\)
(D) \(1:2\)
[Sets: 430/3/1, 430/3/2, 430/3/3]
4.
A cone of height \(h\) and radius \(r\) is surmounted on a
hemisphere of the same radius. The total surface area of the
entire solid will be:
(A) \(\pi rh+\pi r^2\)
(B) \(\pi r\sqrt{h^2+r^2}+\pi r^2\)
(C) \(\pi r\sqrt{h^2+r^2}+2\pi r^2\)
(D) \(\pi r\sqrt{h^2+r^2}+3\pi r^2\)
[Set: 430/2/3]
5.
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:
(A) \(2\pi rh+\pi r\sqrt{h^2+r^2}\)
(B) \(2\pi rh+\pi r^2+\pi r\sqrt{h^2+r^2}\)
(C) \(2\pi rh+2\pi r^2+\pi r\sqrt{h^2+r^2}\)
(D) \(2\pi rh+\pi r\sqrt{h^2+r^2}-\pi r^2\)
[Set: 430/2/2]
SECTION B : LONG ANSWER QUESTIONS (5 Marks Each)
1.
Type A: Volume of Combined Toy
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.
[Sets: 430/1/1, 430/1/2, 430/1/3]
2.
Type B: Cube Surmounted by Hemisphere
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.
[Sets: 430/1/1, 430/1/2, 430/1/3]
3.
Type C: Scooping Out a Hemisphere from a Cylinder
A wooden article was made by scooping out a hemisphere
of the 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.
Figure: A hemisphere removed from the top of a solid cylinder.
[Set: 430/2/1]
4.
Type D: Hemispherical Depression in a Cube
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.
Use \(\pi=\dfrac{22}{7}\).
[Set: 430/2/2]
5.
Type E: Cylinder with Cylindrical Cavity
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. Find the total surface area
of the remaining solid.
[Set: 430/3/2]
6.
Type F: Spherical Vessel with Cylindrical Neck
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.
Use \(\pi=\dfrac{22}{7}\).
[Set: 430/2/3]
7.
Type G: Perfume Bottle
A perfume bottle is in the form of a cylinder but the bottom
of the bottle has a hemispherical raised portion
to reduce the capacity. The inner diameter of the bottle is
\(5\) cm and the height is \(10\) cm. Find the capacity of the
perfume bottle in mL.
Use \(\pi=3.14\) and
\(1\text{ cm}^3=1\text{ mL}\).
Figure: A cylindrical bottle with an inverted hemisphere at
the base.
[Set: 430/3/1]
8.
Type H: Wooden Bead with a Cylindrical Hole
A necklace is made up of wooden beads. Each bead is in the
form of 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.
Figure: A sphere with a narrow cylindrical hole removed
through the centre.
[Set: 430/3/3]
STANDARD QUESTION PAPER 2026
SECTION A : MULTIPLE CHOICE QUESTIONS (1 Mark Each)
1.
Type A: Ice-Cream Cone and Spherical Scoops
An ice-cream cone of radius \(r\) and height \(h\) is
completely filled by two spherical scoops. If the radius of
each spherical scoop is \(\dfrac{r}{2}\), then \(h:2r\) equals:
(A) \(1:8\)
(B) \(1:2\)
(C) \(1:1\)
(D) \(2:1\)
[Sets: 30/4/1 Q16, 30/4/2 Q2, 30/4/3 Q2]
2.
Type B: Balls Packed in a Cylindrical Jar
Three tennis balls are just packed in a cylindrical jar.
If the radius of each ball is \(r\), the volume of air inside
the jar is:
(A) \(2\pi r^3\)
(B) \(3\pi r^3\)
(C) \(5\pi r^3\)
(D) \(4\pi r^3\)
Figure: Three circular tennis balls packed inside a cylindrical jar.
[Sets: 30/4/1 Q13, 30/4/2 Q5, 30/4/3 Q6]
3.
Type C: Conical Cavity in a Hemisphere
A conical cavity of maximum volume is carved out from a
wooden solid hemisphere of radius \(10\) cm. The curved
surface area of the cavity carved out is:
Use \(\pi=3.14\).
(A) \(314\sqrt{2}\text{ cm}^2\)
(B) \(314\text{ cm}^2\)
(C) \(\dfrac{3140}{3}\text{ cm}^2\)
(D) \(3140\sqrt{2}\text{ cm}^2\)
Figure: A hemisphere with an inverted conical cavity.
[Sets: 30/5/1 Q7, 30/5/2 Q13, 30/5/3 Q1]
4.
Type D: Steel Used in a Hemispherical Bowl
A hemispherical bowl is made of steel of thickness \(1\) cm.
The outer radius of the bowl is \(6\) cm. The volume of steel
used (in cm³) is:
(A) \(182\pi\)
(B) \(\dfrac{182}{3}\pi\)
(C) \(682\pi\)
(D) \(\dfrac{364}{3}\pi\)
[Sets: 30/2/1 Q16, 30/2/2 Q11, 30/2/3 Q13]
5.
Type E: Cone Carved from a Cube
A cone of maximum size is carved out from a solid cube of
edge length \(l\). The volume of the cone is:
(A) \(\dfrac{\pi l^3}{12}\)
(B) \(\dfrac{\pi l^3}{3}\)
(C) \(l^3\left(1-\dfrac{\pi}{3}\right)\)
(D) \(\dfrac{\pi l^3}{8}\)
Figure: A cube with a maximum-size cone inscribed in it.
[Sets: 30/3/1 Q7, 30/3/2 Q15, 30/3/3 Q10]
6.
Type F: Total Surface Area of a Solid Hemisphere
The total surface area of a solid hemisphere of diameter
\(2d\) is:
(A) \(3\pi d^2\)
(B) \(2\pi d^2\)
(C) \(\dfrac{1}{2}\pi d^2\)
(D) \(\dfrac{3}{4}\pi d^2\)
[Sets: 30/1/1 Q16, 30/1/2 Q17]
7.
Type G: Radius of a Sphere
The radius of a sphere (in cm) whose volume is
\(36\pi\text{ cm}^3\) is:
(A) \(3\)
(B) \(3\sqrt{3}\)
(C) \(3^{2/3}\)
(D) \(3^{1/3}\)
[Set: 30/1/3 Q14]
8.
Type H: Hemispherical Camping Tent
A camping tent in hemispherical shape of radius \(1.4\) m
has a door opening of area \(0.50\text{ m}^2\).
The outer surface area of the tent is:
(A) \(11.78\text{ m}^2\)
(B) \(12.32\text{ m}^2\)
(C) \(11.82\text{ m}^2\)
(D) \(12.86\text{ m}^2\)
[Sets: 30/5/1 Q12, 30/5/2 Q16, 30/5/3 Q4]
SECTION B : ASSERTION–REASON QUESTIONS (1 Mark Each)
1.
Assertion (A):
The surface area of the cuboid formed by joining two cubes
of sides \(4\) cm each, end-to-end, is \(160\text{ cm}^2\).
Reason (R):
The surface area of a cuboid of dimensions
\(l\times b\times h\) is
\(2(lb+bh+hl)\).
[Sets: 30/2/1 Q19, 30/2/2 Q20, 30/2/3 Q19]
SECTION C : VERY SHORT ANSWER QUESTIONS (2 Marks Each)
1.
Type A: Toy Made of Cone and Hemisphere
A toy is made by surmounting a cone on a hemisphere of
radius \(7\) cm. The total height of the toy is \(31\) cm.
Find the total surface area of the toy.
[Sets: 30/3/3 Q21, 30/5/3 Q21]
Note: The same question appears with the above set references
as a variation in the source material.
2.
Type B: Thermocol Balls in a Spherical Balloon
\(1000\) small thermocol balls of radius \(0.5\) cm are kept
in a spherical balloon of radius \(20\) cm. Find the volume
of air in the balloon.
[Set: 30/3/2 Q25]
SECTION D : SHORT ANSWER QUESTIONS (3 Marks Each)
1.
Type A: Cylinder with Hemispherical Ends
A solid is in the form of a cylinder with hemispherical ends.
The total height of the solid is \(20\) cm and the diameter of
the cylinder is \(7\) cm. Find the total volume of the solid.
Use \(\pi=\dfrac{22}{7}\).
[Sets: 30/1/1 Q30, 30/1/2 Q29]
2.
Type B: Plant Protection Shed
To protect plants, a shed is made with a cuboidal lower part
and a semi-cylindrical top. If the cuboid has dimensions
\(14\text{ m}\times25\text{ m}\times16\text{ m}\), find the
area of the cloth required.
Figure: Shed consisting of a cuboidal lower portion and a
semi-cylindrical top.
[Sets: 30/4/1 Q29(a), 30/4/2 Q27(a), 30/4/3 Q26(a)]
3.
Type C: Hollow Hemisphere and Cone
The internal and external radii of a hollow hemisphere are
\(5\sqrt{2}\) cm and \(10\) cm respectively. A cone of height
\(5\sqrt{7}\) cm and radius \(5\sqrt{2}\) cm is surmounted
on it. Find the total surface area in terms of \(\pi\).
Use \(\sqrt{2}=1.4\).
Figure: Composite solid consisting of a hollow hemisphere
and a cone.
[Sets: 30/4/1 Q29(b), 30/4/2 Q27(b), 30/4/3 Q26(b)]
4.
Type D: Cylinder and Cone — Ratio of Radius to Height
A right circular cylinder and a right circular cone have
equal bases and equal heights. If their curved surface areas
are in the ratio \(8:5\), find the ratio between the radius
of their bases and their height.
[Set: 30/1/3 Q28]
SECTION E : CASE STUDY QUESTIONS (4 Marks Each)
Case Study 1: The Wall Mounted Lamp
A wall-mounted lamp consists of a cuboid of dimensions
\(24\text{ cm}\times12\text{ cm}\times17\text{ cm}\).
The bulb is a sphere of diameter \(7\) cm.
(i)
Find the surface area of the bulb.
(1 Mark)
(ii)
What could be the maximum diameter of the bulb if at least
\(1\) cm space is left from each side?
(1 Mark)
(iii) (a)
Find the area of the fabric used if there is a fold of
\(2\) cm on the top and bottom edges.
(2 Marks)
OR
(iii) (b)
Find the space available (empty volume) inside the lamp.
(2 Marks)
Figure: Wall-mounted lamp showing the cuboid and spherical bulb.
[Sets: 30/5/1 Q37, 30/5/2 Q38, 30/5/3 Q38]
Case Study 2: The Leafy Ball Fountain
The diameter of the spherical ball is \(21\) cm.
The cylindrical pool has an outer diameter of \(50\) cm and
an inner diameter of \(40\) cm. The height of the solid base
is \(14\) cm. The height of water filled is \(7\) cm.
(i)
Determine the total height of the fountain.
(1 Mark)
(ii)
Find the volume of the ball.
(1 Mark)
(iii) (a)
If one-third of the ball is submerged in water, find the
volume of water filled in the pool.
(2 Marks)
OR
(iii) (b)
Find the sum of the outer curved surface area of the
cylindrical part and surface area of the ball.
(2 Marks)
Figure: Spherical ball placed in a cylindrical pool.
[Sets: 30/3/1 Q38, 30/3/2 Q37, 30/3/3 Q36]
Case Study 3: The Fair Visit — Rubik's Cube & Ice-Cream
(i)
Find the length of the diagonal of a Rubik's cube if each
edge measures \(6\) cm.
(1 Mark)
(ii)
Find the volume of a Rubik's cube if the edge length is
\(7\) cm.
(1 Mark)
(iii) (a)
What is the curved surface area of the hemisphere
(ice-cream) if the base radius is \(7\) cm?
(2 Marks)
OR
(iii) (b)
If two cubes of edges \(4\) cm are joined end-to-end,
find the surface area of the resulting cuboid.
(2 Marks)
[Sets: 30/2/1 Q38, 30/2/2 Q37, 30/2/3 Q36]