# NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Written by Team Trustudies
Updated at 2021-05-07

## NCERT solutions for class 10 Maths Chapter 13 Surface Areas And Volume Exercise - 13.1

Q1 ) 2 cubes each of volume 64 cm3 are joined end to end. Find the surface area of the resulting cuboid.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Let the length of each edge of the cube be a cm.

Then, Volume cm3

cm

When two cubes of equal volumes (i.e., equal edges) are joined end to end, we get a cuboid such that its,

l = Length cm

b = Breadth = 4 cm

and h = Height = 4 cm

$?$ Surface area of the cuboid = 2(lb + bh + hl)

cm2

Q2 ) 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.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Here r, the radius of hemisphere = 7 cm ,

h , the height of cylinder = (13 - 7) = 6 cm

Clearly, radius of the base of cylindrical part is also r cm

Surface area of the vessel = Curved surface area of the cylindrical part + Curved surface area of hemispherical part

cm2

Q3 ) 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.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

We have VO = 15.5 cm, OA = OO' = 3.5 cm

Let r be the radius of the base of cone and h be the height of conical part of the toy.

Then r = OA = 3.5 cm

h = VO = VO' - OO = (15.5 - 3.5) cm = 12 cm

Also radius of the hemisphere = OA = r = 3.5 cm

l = Slant height, is given by :
$?l=\sqrt{{r}^{2}+{h}^{2}}$

cm

Total surface area of the toy = Curved surface area of cone + Curved surface area of hemisphere

cm2

Q4 ) A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter of the hemisphere can have? Find the surface area of the solid?

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

The greatest diameter that a hemisphere can have = 7 cm .

Surface area of the solid after surmounted hemisphere

cm2

Q5 ) A hemispherical depression is cut of one face of a cubical wooden block such the diameter l of the hemisphere is equal to the edge of the cube. Determine th surface area of the remaining solid.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Edge of the cube =$l$

Diameter of the hemisphere = $l$

$?$ Radius of the hemisphere $=\frac{l}{2}$

$?$ Area of the remaining solid after cutting out the hemispherical depression.

Q6 ) A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see figure). The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Let r cm be the radius and h cm be the height of the cylinder. Then.

mm

mm

Also the radius of hemisphere cm

Now, surface area of the capsule = Curved surface of cylinder + Surface area of two hemispheres

mm2

Q7 ) 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 Rs 500 per m2 . (Note that the base of the tent will not covered with canvas).

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

We have,

Height of the cylinderical part = 2.1 m

Diameter of the cylinderal part = 4 m

Radius of the cylinerical part = 2 m

Slant height of the conical part = 2.8 m

Thus, Total canvas used = Curved surface area of cylinder + Curved surface area of cone

m2

Now, cost of 1 m2 the canvas for the tent = Rs 500

So, cost of 44 m2 the canvas for the tent = Rs 44 × 500 = Rs 22000

Q8 ) 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.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Height of the cylinder cm

Height of the cone cm

Slant height of the cone

cm

Surface area of the remaining solid = Curved surface area of cylinder + Curved surface of the cone + Area of upper circular base of cylinder

cm2

Q9 ) A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in figure. If the height of the cylinder is 10 cm,and its base is of radius 3.5 m, find the total surface area of the article.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Surface area of the article when it is ready = Curved surface area of cylinder + 2 × Curved surface area of hemisphere.

where r = 3.5 m and h = 10 cm

cm2

## NCERT solutions for class 10 Maths Chapter 13 Surface Areas And Volume Exercise - 13.2

Q1 ) A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of $?$.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Volume of the solid = Volume of the cone + Volume of the hemisphere

[$?$ h = r and R = r]

cm3 [$?$ r = 1 cm]

Q2 ) Rachel an engineering student was asked to make a model shaped like a cylinder with two cones attached at its two ends by using thin aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm,find the volume of air contained in the model the Rachel made. (Assume the outer and inner dimensions of the model be nearly the same).

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Volume of the air contained in the model = Volume of the cylindrical portion of the model + Volume of its two conical ends.

where cm , cm and cm

cm3

Q3 ) A gulab jamun, contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemsipherical ends with length 5 cm and diameter is 2.8 cm (see figure).

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Volume of the gulab jamun = Volume of the cylindrical portion + Volume of the hemispherical ends

where r = 1.4 cm, h = 2.2 cm

cm3

Volume of 45 gulab jamuns

cm3

Quantity of syrup in gulab jamuns = 30% of their volume

cm3

Q4 ) A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 m and the depth is 1.4 cm. Find the volume of wood in the entire stand (see figure).

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Volume of wood in the entire stand = Volume of the cuboid - 4 × Volume of a depression (i.e., cone)

cm3

Q5 ) A vessel in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one- fourth of the water flows out. Find the number of lead shots dropped in the vessel.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Height of the conical vessel, h = 8 cm.

Its radius r = 5 cm

Volume of cone = Volume of water in cone

cm3

Volume of water flows out = Volume of lead shots

of the volume of water in the cone

cm3

Volume of one spherical lead shot

cm3

$?$ Number of lead shots dropped into the vessel

Q6 ) A solid iron pole consists of a cylindrical height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm3 of iron has approximately 8 g mass. (Use )

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Volume of the solid iron pole = Volume of the cylindrical portion + Volume of the other cylindrical portion

cm3

$?$ The mass of the pole
grams

kg

Q7 ) A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water circular such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Volume of the cylinder

cm3

Volume of the solid = Volume of cone + Volume of hemisphere

cm3

Volume of water left in the cylinder = Volume of the cylinder - Volume of the solid

m3

Q8 ) A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in diameter ; the diameter of the spherical part is 8.5 cm. By measuring the amount of water it holds , a child finds its volume to be 345 cm3. Check whether she is correct, taking the above as the inside measurements and

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Volume of spherical vessel = Volume of sphere + Volume of cylinder

cm3

$?$ She is incorrect. The correct answer is 346.51 cm3

## NCERT solutions for class 10 Maths Chapter 13 Surface Areas And Volume Exercise - 13.3

Q1 ) A metallic sphere of radius 4.2 cm is melted and recast into the shape of a cylinder of radius 6 cm. Find the height of the cylinder.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Volume of the sphere cm3

If h is the height of a cylinder of radius 6 cm. Then its volume cm3

$?$ The volume of metal in the form of sphere and cylinder remains the same , we have

cm

Q2 ) Metallic spheres of radii 6 cm, 8 cm and 10 cm, respectively, are melted to form a single solid sphere. Find the radius of the resulting sphere.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Sum of the volumes of 3 gives spheres.

cm3

Let R be the radius of the new spheres whose volume is the sum of the volumes of 3 given spheres.

cm

$?$ The radius of the resulting sphere is 12 cm.

Q3 ) A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a platform 22 m by 14 m. Find the height of the platform.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Let $h$ be the required height of the platform.

The shape of the platform will be like the shape of a cuboid of dimensions with a hole in the shape of cylinder of radius $3.5m$ and depth $h$.

The volume of the platform will be equal to the volume of the earth dug out from the well.

Now, the volume of the earth = Volume of the cylindrical well

m3

Also, the volume of the platform m3

But volume of the platform = Volume of the well

m

$?$ Height of the platform = 2.5 m

Q4 ) A well of diameter 3 m is dug 14 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Let h be the required height of the embankment.

The shape of the embankment will be like the shape of a cylinder of internal radius 1.5 m and external radius m

The volume of the embankment will be equal to the volume of earth dug out from the well.

Now, the volume of the earth = Volume of the cylindrical well

m3

Also, the volume of the embankment

m3

m

Hence, the required height of the embankment = 1.125 m

Q5 ) A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice cream. The ice cream is to be filled into cones of height 12 cm and diameter 6 cm, having hemispherical shape on the top. Find the number of such cones which can be filled with ice cream.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

Volume of the cylinder

cm3

Volume of a cone having hemispherical shape on the top

cm3

Let the number of cone that can be filled with ice cream be n.

Then,

$?$ 10 cones can be filled with ice cream.

Q6 ) How many silver coins, 1.75 cm in diameter and of thickness 2 mm, must be melted to form a cuboid of dimensions 5.5 cm × 10 cm × 3.5 cm?

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume

The shape of the coin will be like the shape of a cylinder of radius
cm and of height 2 mm = 0.2 cm

Its volume

cm3

Volume of the cuboid cm3

Number of coins required to form the cuboid

$?$ 400 coins must be melted to form a cuboid.

Q7 ) A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. The bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap.

NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volume