A cuboid is 12 cm long, 9 cm wide and 5 cm high. Calculate the total surface area of the cuboid.
Length = l = 12 cm
Breadth = b = 9 cm
Height = h = 5 cm
Total surface area of cuboid = 2 × (lb + bh + lh)
= 2 × (12×9 + 9×5 + 12×5)
= 2 × (108 + 45 + 60)
= 2 × 213
= 426 cm2
Therefore, total surface area of the cuboid is 426 cm2
Three cubes with edges 8 cm, 6 cm and 1 cm respectively are melted together and formed into a single cube. Find the total surface area of the new cube.
To find the total surface area of the new cube formed we first need to find its edge
Volume of cube = (edge)3
Volume of cube with edge 8 cm = 83 = 512 cm3
Volume of cube with edge 6 cm = 63 = 216 cm3
Volume of cube with edge 1 cm = 13 = 1 cm3
Let the edge of new cube formed be ‘a’
Volume of new cube = a3
As the three cubes are melted and the new cube is formed
Volume of new cube = volume of all the three cubes
⇒ a3 = 512 + 216 + 1
⇒ a3 = 729
⇒ a = 9
Surface area of cube = 6 × (edge)2
⇒ surface area of new cube = 6 × a2
= 6 × 92
= 6 × 81
= 486 cm2
Therefore, surface area of new cube is 486 cm2
The dimensions of a box are 50 cm × 36 cm × 25 cm. How many square cm of cloth is needed to make a cover for the box?
The box is like the shape of a cuboid
Length = l = 50 cm
Breadth = b = 36 cm
Height = h = 25 cm
Total surface area of cuboid = 2 × (lb + bh + lh)
⇒ total surface area of box = 2 × (50×36 + 36×25 + 50×25)
= 2 × (1800 + 900 + 1250)
= 2 × 3950
= 7900 cm2
Amount of cloth required to make cover = total surface area of box
Therefore, 7900 cm2 cloth will be required to make cover for the box
The area of each face of a cube is 100 cm2. If the cube is cut into two equal parts by a plane parallel to the base then find the total surface area of each equal part.
A cube has 6 surfaces
We will get two cuboids after cutting
The cutting plane will pass through 4 surfaces halving their areas
So, each part will be having 4 surfaces with area 50 cm2
each part will have remaining 2 surfaces with area 100 cm2
total surface area of each part = 100 + 100 + 50 + 50 + 50 + 50
= 400 cm2
Therefore, total surface area of each part is 400 cm2
An open box is made of wood 3 cm thick. Its external length, breadth and height are 146 cm, 116 cm and 83 cm respectively. Find the cost of painting the inner surface at Rs 2 per 1000 sq. cm.
the box is as shown in figure
The wood is 3 cm thick
And given length, breadth and height are external which means including the wood thickness
So, to find the inner surface area we need to subtract the wooden thickness from external length, breadth and height
Thus length, breadth and height of inner surface area are as follows
In the length and breadth, we need to subtract 3 from both the sides which means we have to subtract 6 but in the height as the box is open from one end so we need to subtract only 3cm base thickness as seen in the figure
Length = 146 – 6 = 140 cm
Breadth = 116 – 6 = 110 cm
Height = 83 – 3 = 80 cm
The shape of the box is a cuboid and it is open box, therefore, we should subtract the top rectangular surface area from the total inner surface area
Total surface area of cuboid = 2 × (lb + bh + lh) – lb
⇒ total surface area of box = 2 × (140×110 + 110×80 + 140×80) – 140×110
= 2 × (15400 + 8800 + 11200) – 15400
= 2 × 35400 – 15400
= 70800 – 15400
= 55400 cm2
Cost of painting 1000 cm2 is Rs 2
Thus, cost of painting 55400 cm2 =
= 55.4 × 2
= 110.8 Rs
Therefore, cost of painting the inner surface = 110.8 Rs
The sum of length, breadth and height of a cuboid is 19 cm. If length of the diagonal is 11 cm then calculate the total surface area of the cuboid.
Let the length, breadth and height be l, b and h respectively
Given:
l + b + h = 19 cm
length of diagonal = 11 cm
length of diagonal of a cuboid =
⇒ = 11 cm
Squaring both sides we get
⇒ l2 + b2 + h2 = 121
Using identity (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac
⇒ (l + b + h)2 = l2 + b2 + h2 + 2lb + 2bh + 2hl
Substituting given values
⇒ 192 = 121 + 2 × (lb + bh + hl)
⇒ 361 = 121 + 2 × (lb + bh + hl)
⇒ 2 × (lb + bh + hl) = 361 – 121
⇒ 2 × (lb + bh + hl) = 240 cm2
Therefore, total surface area of cuboid is 240 cm2
A room whose floor is a square of side 6 m contains 180 cubic metre of air. Find the height of the room.
Length of room= l = 6 m
Breadth of room = b = 6 m
Let height of room be h
Room contains 180 cubic metre of air which means volume of room is 180 m3
Room is in the shape of a cuboid
Volume of cuboid = lbh
⇒ 180 = 6 × 6 × h
⇒ 180 = 36h
⇒ h =
⇒ h = 5 m
Height of room is 5 m
Find the number of bricks, each measuring 22 cm × 10 cm × 7 cm required to construct a wall 44 m long, 1.5 m high and 85 cm thick.
Length of brick = 22 cm
Breadth of brick = 10 cm
Height of brick = 7 cm
Brick is in the shape of a cuboid
Volume of cuboid = length × breadth × height
⇒ volume of 1 brick = 22 × 10 × 7
= 1540 cm3
Let n bricks be required to make the wall
Length of wall = 44 m = 4400 cm
Breadth of wall = 85 cm
Height of wall = 1.5m = 150 cm
Volume of wall = 4400 × 85 × 150
= 56100000 cm3
The wall is completely made from bricks therefore
Volume of n bricks = volume of wall
⇒ n × volume of one brick = 56100000
⇒ n × 1540 = 56100000
⇒ n =
⇒ n =
Dividing numerator and denominator by 22, we get
⇒ n =
⇒ n = 36428.57 ~ 36429
Number of brick required = 36429
Find the length of the longest rod that can be placed in a room 10 m long, 8 m wide and 6 m high.
The longest edge of a cuboid is the body diagonal
So the length of longest rod will be same as the length of body diagonal
Length of room = l = 10 m
Breadth of room = b = 8 m
Height of room = h = 6 m
Length of body diagonal =
⇒ length of rod =
=
=
Therefore, length of longest rod = 10√2 m
The volume of a cube is 512 cubic m. Find its side.
Volume = 512 m3
Let a be the side of cube
Volume of cube = a3
⇒ a3 = 512
⇒ a =
⇒ a = 8 m
Find the number of bricks, each measuring 20 cm × 10 cm × 7.5 cm required to construct a wall 5 m long, 30 cm wide and 3 m high.
Length of brick = 20 cm
Breadth of brick = 10 cm
Height of brick = 7.5 cm
Brick is in the shape of a cuboid
Volume of cuboid = length × breadth × height
⇒ volume of 1 brick = 20 × 10 × 7.5
= 1500 cm3
Let n bricks be required to make the wall
Length of wall = 5 m = 500 cm
Breadth of wall = 30 cm
Height of wall = 3 m = 300 cm
Volume of wall = 500 × 30 × 300
= 4500000 cm3
The wall is completely made from bricks therefore
Volume of n bricks = volume of wall
⇒ n × volume of one brick = 56100000
⇒ n × 1500 = 4500000
⇒ n =
⇒ n =
⇒ n = 3000
Number of brick required = 3000
The dimensions of a cuboid are in the ratio 5 : 3 : 2. If total surface area of the cuboid is 558 cm2, then find the length of edges of the cuboid.
Let
Length of cuboid = l = 5x
Breadth of cuboid = b = 3x
Height of cuboid = h = 2x
Such that the ratio is 5:3:2 as mentioned in question where x is any positive number
Total surface area of cuboid = 558 cm2
Total surface area of cuboid = 2 × (lb + bh + hl)
⇒ 558 = 2 × [(5x)(3x) + (3x)(2x) + (2x)(5x)]
⇒ = 15x2 + 6x2 + 10x2
⇒ 279 = 31x2
⇒ x2 =
⇒ x2 = 9
⇒ x = ± 3
x is length therefore x cannot be negative therefore x = 3
therefore,
length of cuboid = 5x = 5 × 3 = 15 cm
breadth of cuboid = 3x = 3 × 3 = 9 cm
height of cuboid = 2x = 2 × 3 = 6 cm
length of edges of cuboid are 15 cm, 9 cm and 6 cm
The diameter of a cylinder is 14 cm and its height is 15 cm. Calculate the total surface area and volume of the cylinder.
Diameter of cylinder = d = 14 cm
⇒ radius of cylinder = r = = = 7 cm
Height of cylinder = h = 15 cm
Total surface area will be the sum of curved surface area and the area of two circles flat surfaces
Total surface area of cylinder = 2πrh + 2πr2
= 2 × × 7 × 15 + 2 × × 72
= 30 × 22 + 2 × 22 × 7
= 660 + 308
= 968 cm2
Volume of cylinder = πr2h
= × 72 × 15
= 22 × 7 × 15
= 22 × 105
= 2310 cm3
Therefore, total surface area of cylinder is 968 cm2 and volume is 2310 cm3
The height of a right circular cylinder is 7 cm and radius of its base is 3 cm. Find the curved surface area, total surface area and volume of the cylinder.
Height of cylinder = h = 7 cm
Radius of base of cylinder = r = 3 cm
Curved surface area of cylinder = 2πrh
= 2 × × 3 × 7
= 6 × 22
= 132 cm2
Total surface area will be the sum of the curved surface area and the area of two circles flat surfaces
Total surface area of cylinder = 132 + 2πr2
= 132 + 2 × 3.14 × 32
= 132 + 6.28 × 9
= 132 + 56.52
= 188.52 cm2
Volume of cylinder = πr2h
= × 32 × 7
= 9 × 22
= 198 cm3
Therefore, curved surface area of cylinder is 132 cm2, total surface area is 188.52 cm2 and volume is 198 cm3
The area of circular base of a cylinder is 154 cm2 and its height is 21 cm. Find the volume and curved surface area of the cylinder.
Area of circular base = 154 cm2
Height = h = 21 cm
Let r be radius of circular base
Area = πr2
⇒
⇒ = r2
⇒ r2 = 7 × 7 = 49
⇒ r = ± 7
Radius cannot be negative hence r = 7
Volume of cylinder = πr2h
= × 72 × 21
= 7 × 21 × 22
= 3234 cm3
Curved surface area of cylinder = 2πrh
= 2 × × 7 × 21
= 14 × 22 × 3
= 924 cm2
Therefore, volume of cylinder is 3234 cm3 and curved surface area is 924 cm2
The radii of two right circular cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 4. Calculate the ratio of their curved surface areas and the ratio of their volumes.
Let the radii of two circles r1 and r2
⇒ =
Let the heights of two cylinders h1 and h2
⇒ =
Curved surface area of cylinder = 2πrh
Ratio of their surface areas =
= ×
= ×
=
Volume of cylinder = πr2h
Ratio of their volumes =
= ×
= ×
= ×
=
Therefore, ratio of their curved surface areas is 5:6 and the ratio of their volumes is 5:9
A solid cylinder has total surface area of 462 m2. Its curved surface area is one–third of its total surface area. Find the volume of the cylinder.
Total surface area of cylinder = 462 m2
Curved surface area of cylinder = × Total surface area of cylinder
= × 462
= 154 m2
Curved surface area of cylinder = 154 m2
We know
Total surface area of cylinder = curved surface area + area of circles at top and bottom of cylinder
⇒ 462 = 154 + area of circles at top and bottom of cylinder
⇒ area of circles at top and bottom of cylinder = 462 – 154 = 308 m2
Let r be the radius of to and bottom circle of cylinder
area of circles at top and bottom of cylinder = 2πr2
⇒ 2πr2 = 308
⇒
⇒ r2 =
⇒ r2 = 7 × 7
⇒ r = ± 7
r is radius and radius cannot be negative hence r = 7 m
now to find volume we need to find one more parameter about the cylinder which is the height
let us assume the height to be h
curved surface area of cylinder = 2πrh
⇒ 2πrh = 154
⇒ 2 × × 7 × h = 154
⇒ h =
⇒ h = m
volume of cylinder = πr2h
= × 72 ×
= 11 × 7 × 7
= 539 m3
Therefore, volume of cylinder is 539 m3
The lateral curved surface area of a cylinder is 660 cm2 and its height is 15 cm. Find the volume of the cylinder.
Height = 15 cm
lateral curved surface area of a cylinder = 660 sq. cm
let r be the radius
lateral curved surface area of a cylinder = 2πrh
⇒ 2πrh = 660
⇒ 2 × × r × 15 = 660
⇒ 30 × r =
⇒ 30 × r = 30 × 7
⇒ r = 7 cm
Volume of cylinder = πr2h
= × 72 × 15
= 22 × 7 × 15
= 2310 cm3
Therefore, volume is 2310 cm3
The volume of a cylinder is 30π cm3 and the area of it base is 6π cm2. Find the height of the cylinder.
Let the radius of base of cylinder be r
Area of base = 6π cm2
Area of base = πr2
⇒ πr2 = 6π
⇒ r2 = 6
Volume of cylinder = 30π cm3
Let h be the height of cylinder
Volume of cylinder = πr2h
⇒ πr2h = 30π
⇒ 6h = 30
⇒ h = 5 cm
Height of cylinder is 5 cm
The volume and curved surface area of a cylinder are 1650 cm3 and 660 cm2 respectively. Find the radius and the height of the cylinder.
Let r be the radius of base and h be the height of cylinder
Curved surface area of cylinder = 660 cm2
Curved surface area = 2πrh
⇒ 2πrh = 660
⇒ 2 × × r × h = 660
⇒ r × h =
⇒ r × h =
⇒ r × h = 15 × 7
⇒ r × h = 105
⇒ rh = 105
Volume of cylinder = 1650 cm3
Volume of cylinder = πr2h
⇒ πr2h = 1650
⇒ × r × r × h = 1650
Using r × h = 105 we get
⇒ × r × 105 = 1650
⇒ 22 × 15 × r = 1650
⇒ r =
⇒ r =
⇒ r = 5 cm
Substitute r = 5 in r × h = 105 to get h
⇒ 5 × h = 105
⇒ h =
⇒ h = 21 cm
Therefore, radius is 5 cm and height is 21 cm
The height and radius of a cylinder are 7.5 cm and 3.5 cm respectively. Find the ratio between the total surface area and curved surface area of the cylinder
Let r be the radius and h be the height of cylinder
h = 7.5 cm = cm
r = 3.5 cm = cm
total surface area = 2πrh + 2πr2
curved surface area = 2πrh
ratio of total surface area to curved surface area =
⇒ ratio =
⇒ ratio =
⇒ ratio = 1 +
⇒ ratio = 1 +
⇒ ratio = 1 +
⇒ ratio =
⇒ ratio =
Therefore, ratio between the total surface area and curved surface area of the cylinder is 22:15
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.
The volume of earth removed by digging for cylindrical well is the same volume used to make platform which is cuboidal in shape
Therefore, volume of well dug = volume of platform formed
Parameters of well
Diameter = d = 7 m
⇒ radius = r = = m
The well is 20 m deep so we can say that its height is 20 m
Height = h = 20 m
Volume of cylinder = πr2h
⇒ volume of well dug = × × 20
= × × 20
= 22 × 7 × 5
= 770 m3
Parameters for platform
Length = l = 22 m
Breadth = b = 14 m
Let height be h
Volume of cuboid = l × b × h
⇒ Volume of platform = 22 × 14 × h
= 308h m3
As volume of well dug = volume of platform formed
⇒ 770 = 308h
⇒ h =
Divide numerator and denominator by 7 we get
⇒ h =
Divide numerator and denominator by 22 we get
⇒ h = = 2.5
Therefore, height of platform is 2.5 m
30800 cm3 of water can be filled in a cylindrical container. The inner radius of the container is 14 cm. Find the inner curved surface area of the container.
Volume of water filled in cylinder = volume of cylindrical container
Radius = r = 14 cm
Let Height = h
Volume of cylindrical container = 30800
Volume of cylinder = πr2h
⇒ πr2h = 30800
⇒ × 142 × h = 30800
⇒ × 14× 14 × h = 30800
⇒ 28 × 22 × h = 30800
⇒ h =
⇒ h =
Divide numerator and denominator by 14
⇒ h = = 50 cm
Curved surface area of cylinder = 2πrh
⇒ inner curved surface area = 2 × × 14 × 50
= 100 × 22 × 2
= 4400 cm2
Therefore, inner curved surface area of the container is 4400 cm2
A hollow cylinder, open at both ends has a thickness of 2 cm. If the inner diameter is 14 cm and its height is 26 cm then calculate the total surface of the hollow cylinder.
Cross section of cylinder i.e. top/bottom of cylinder will look like as shown
Inner diameter of cylinder = 14 cm
⇒ Inner radius = ri = = 7 cm
As the thickness is 2 cm it can be seen from figure that external radius = inner radius + 2
⇒ external radius = re = 7 + 2 = 9 cm
Height of cylinder = h = 26 cm
Curved surface area of cylinder = 2πrh
⇒ external curved surface area of cylinder = 2πreh
⇒ inner curved surface area of cylinder = 2πrih
As seen the figure the area between the external circle and inner circle will also be counted in total surface area
That area is given by subtracting inner circle area from external circle area
Let us call that area as ring area
⇒ ring area = πre2 – πri2
= π × (re2 – ri2)
Using identity a2 – b2 = (a + b)(a – b)
⇒ ring area = π × (re+ ri) × (re– ri)
We have two such ring areas at the top and bottom of cylinder
So total ring area = 2 × π × (re+ ri) × (re– ri)
Now,
Total surface area of hollow cylinder = external curved surface area of cylinder + inner curved surface area of cylinder + total ring area
⇒ Total surface area of hollow cylinder = 2πreh + 2πrih + 2 × π × (re+ ri) × (re– ri)
⇒ Total surface area of hollow cylinder = 2πh(re + ri) + 2π(re+ ri)(re– ri)
⇒ Total surface area of hollow cylinder = 2π(re + ri)[h + re– ri]
Substituting values
⇒ Total surface area of hollow cylinder = 2 × × (9 + 7)[26 + 9– 7]
= 2 × × 16 × 28
= 32 × 22 × 4
= 2816
Therefore, total surface of the hollow cylinder is 2816 cm2
A hollow cylinder, open at both ends has a height of 20 cm. The inner and outer diameters are 26 cm and 30 cm respectively. Find the volume of this hollow cylinder.
Height of cylinder = h = 20 cm
Inner diameter = 26 cm
⇒ Inner radius = ri = = 13 cm
Inner diameter = 30 cm
⇒ Inner radius = ro = = 15 cm
Volume of hollow cylinder = π(ri2 – ro2)h
⇒ volume of hollow cylinder = × (152 – 132) × 20
= × (225 – 169) × 20
= × 56 × 20
= 22 × 8 × 20
= 3520 cm3
Therefore, volume of cylinder is 3520 cm3
The height of a cone is 28 cm and radius of its base is 21 cm. Find its curved surface area, total surface area and its volume.
Height = h = 28 cm
Radius of base = r = 21 cm
Curved surface area of cone = πrl
Where l =
⇒ l =
⇒ l =
⇒ l =
⇒ l = 35 cm
Therefore,
Curved surface area = × 21 × 35
= 22 × 21 × 5
= 2310 cm2
Total surface area of cone = curved surface area + area of base
= 2310 + πr2
= 2310 + × 21 × 21
= 2310 + 22 × 3 × 21
= 2310 + 1386
= 3696 cm2
Volume of cone = πr2h
= × × 21 × 21 × 28
= 22 × 21 × 28
= 12936 cm3
Therefore, curved surface area is 2310 cm2, total surface area is 3696 cm2 and volume is 12936 cm3
The volume of a right circular cone is 1232 cm3 and its height is 24 cm. Find the slant height of the cone.
Height of cone = h = 24 cm
Volume of cone = 1232 cm3
Let r be the radius of base of cone
Volume of cone = πr2h
⇒ × × r2 × 24 = 1232
⇒ × r2 × 8 = 1232
⇒ × r2 =
⇒ × r2 = 154
⇒ r2 =
⇒ r2 = 7 × 7
⇒ r = 7
Slant height l =
⇒ l =
⇒ l =
⇒ l =
⇒ l = 25 cm
Therefore, slant height of cone = 25 cm
The diameter of the base of a cone is 14 m and its slant height is 25 m. Find the total surface area of the cone.
Diameter of base = 14 m
⇒ radius of base = r = = 7 m
Let h be the height of cone
Slant height = l = 25 m
Total surface area of cone = curved surface area of cone + base area of cone
⇒ total surface area = πrl + πr2
= πr(l + r)
= × 7 × (25 + 7)
= 22 × 32
= 704 m2
Therefore, total surface area of the cone is 704 m2
Radius of the base of a cone is 14 cm and its slant height is 50 cm. Find the curved surface area and the total surface area of the cone.
Radius of base of cone = r = 14 cm
Slant height = l = 50 cm
Curved surface area of cone = πrl
= × 14 × 50
= 22 × 2 × 50
= 2200 cm2
Total surface area of cone = curved surface area of cone + base area of cone
⇒ total surface area = 4400 + πr2
= 4400 + × 14 × 14
= 2200 + 22 × 2 × 14
= 2816 cm2
Therefore, curved surface area is 2200 cm2 and total surface area is 2816 cm2
The height of a right circular cone is 8 cm and the radius of its base is 6 cm. Find its volume.
Height of cone = 8 cm
Radius of base = r = 6 cm
Volume of cone = πr2h
= × × 6 × 6 × 8
= × 2 × 48
=
=
= 301.714 cm3
Therefore, volume is 301.714 cm3
The curved surface area of a cone is 1884.4 m2 and its slant height is 12 m. On the basis of this information, calculate the radius of the cone.
Curved surface area of cone = 1884.4 m2
Slant height = l = 12 m
Let r be the radius of base of cone
Curved surface area of cone = πrl
⇒ 1884.4 = × r × 12
⇒ 1884.4 = × r × 12
⇒ r =
⇒ r =
⇒ r = 49.96 ~ 50 m
Therefore, radius of base of cone is 50 m
The area of the base of a right circular cone is 154 cm2. If its slant height is 25 cm then find the height of the cone.
Area of base of cone = 154 cm2
Let r be the radius of cone
Area of base of cone = πr2
⇒ πr2 = 154
⇒ × r2 = 154
⇒ r2 =
⇒ r2 = 7 × 7
⇒ r = 7 cm
Slant height = 25 cm
Let h be the height of cone
Slant height =
⇒ 25 =
Squaring both sides we get
⇒ 625 = 72 + h2
⇒ 625 = 49 + h2
⇒ h2 = 625 – 49
⇒ h2 = 576
⇒ h = ± 24
Height cannot be negative therefore h = 24 cm
Height of cone is 24 cm
The diameters of two cones are equal and their slant height are in the ratio 5 : 4. If the curved surface area of the smaller cone is 400 cm2 then find the curved surface area of the bigger cone.
As the diameters of the cones are equal, their radius is also equal
Let the radius of both cones be r
Let the slant height of bigger cone be l1 and that of smaller cone be l2
l1:l2 = 5:4
⇒ =
Curved surface area of smaller cone = 400 cm2
Curved surface area of cone = πrl
Curved surface area of bigger cone = πrl1
Curved surface area of smaller cone = πrl2
Consider ratio of curved surface area of bigger cone to smaller cone
⇒ =
⇒ =
⇒ =
⇒ Curved surface area of bigger cone =
⇒ Curved surface area of bigger cone = 5 × 100
⇒ Curved surface area of bigger cone = 500 cm2
Therefore, curved surface area of bigger cone is 500 cm2
The slant height and radius of a cone are in the ratio 7 : 4. If its curved surface area is 792 cm2, find its radius.
Let the slant height of cone be l and radius of base of cone be r
Ratio of slant height to radius = 7:4
⇒ =
⇒ l = × r
Curved surface area of cone = 792 cm2
⇒ πrl = 792
put l = × r
⇒ × r × × r = 792
⇒ × r2 = 792
⇒ r2 =
⇒ r2 = 36 × 4
⇒ r2 = 144
⇒ r2 = √144
⇒ r = ± 12
r is radius and radius cannot be negative thus r = 12 cm
The circumference of the base of a 9 m high cone is 44 m. Find the volume of air present in it.
Height of cone = h = 9 m
Circumference of base of cone = 44 m
Let r be the radius of base of cone
Circumference of base of cone = 2πr
⇒ 2πr = 44
⇒ πr = 22
⇒ × r = 22
⇒ r =
⇒ r = 7 m
Volume of air present inside the cone = volume of cone
Volume of cone = πr2h
= × × 72 × 9
= 22 × 7 × 3
= 462 m3
Therefore, volume of air inside cone is 462 m3
A conical vessel whose internal radius is 10 cm and height 18 cm is full of water. The water is emptied into a cylindrical vessel of internal radius 5 cm. Find the height to which the water rises.
Radius of base of conical vessel = r = 10 cm
Height of conical vessel = h = 18 cm
Volume of water in conical vessel = volume of cone
Volume of cone = πr2h
⇒ Volume of water in conical vessel = × π × 102 × 18
= π × 100 × 6
= 600π
Now this volume of water is transferred in cylindrical vessel whose radius is 5 cm
Let h1 be the height up to which the water rises in the cylindrical vessel
Volume of water in cylindrical vessel = π(radius)2h
= π × 52 × h
= 25hπ
As water is transferred from conical vessel to cylindrical vessel volume of water is same
⇒ Volume of water in conical vessel = Volume of water in cylindrical vessel
⇒ 600π = 25hπ
⇒ 25h = 600
⇒ h =
⇒ h = 24 cm
Therefore, the height to which the water rises is 24 cm
Find the volume of the largest right circular cone that can be cut out of a cube whose edge is 14 cm.
Volume of cone = πr2h
For volume to be largest the radius of base of cone ‘r’ and height of cone ‘h’ should be maximum
In a cube whose edge is 14 cm
As it can be seen from the figure that the maximum height of cone can be 14 cm and also the maximum diameter can be 14 cm
Radius = r = = 7 cm
Height = h = 14 cm
⇒ Volume of cone = × × 72 × 14
= × 22 × 49 × 2
=
=
= 718.67 cm2
Therefore, volume of the largest right circular cone that can be cut out of the cube is 718.67 cm2
The base radius and the height of cone are 7 cm and 24 cm respectively. Find the slant height, curved surface area, total surface area and its volume.
Base radius of cone = r = 7 cm
Height of cone = h = 24 cm
Slant height = l =
⇒ l =
⇒ l =
⇒ l =
⇒ l = 25 cm
Curved surface area = πrl
= × 7 × 25
= 22 × 25
= 550 cm2
Total surface area of cone = curved surface area + area of base
= πrl + πr2
= πr(l + r)
= × 7 × (25 + 7)
= 22 × 32
= 704 cm2
Volume of cone = πr2h
= × × 72 × 24
= 22 × 7 × 8
= 1232 cm3
Therefore, slant height of cone is 25 cm, curved surface area is 550 cm2, total surface area is 704 cm2 and volume is 1232 cm3
The radius of a sector of a circle is 12 cm and its angle is 120°. Its straight edges are joined together to form a cone. Find the volume of this cone.
The sector is as shown in the figure with TU = 12 as radius and ∠VTU = 120°
Now edges VT and UT are joined to form the cone as shown and hence points U and V become the same
For the cone formed TU or TV is the slant height of cone
TU = l = 12 cm
To find volume of cone we need to find base radius of cone and height of cone
Radius of base of cone as seen in figure is OV or OU let us assume that to be rc
To find base radius of cone we can use the fact that the cone is made by folding the sector which means the curved surface area of cone will be same as area of sector thus we will find those areas and equate them
Area of sector =
r is the radius of sector which is TU = 12 cm and θ is the angle which is ∠VTU = 120°
⇒ area of sector =
=
=
= 48π
Curved surface area of cone = πrcl
= π × rc × 12
= 12πrc
Therefore,
⇒ 48π = 12πrc
⇒ rc = 4 cm
Now we have found the base radius of cone we need to find one more parameter which is the height to find volume
Let the height of the cone be h which is TO in the diagram
using the formula
l =
⇒ 12 =
Squaring both the sides
⇒ 144 = 16 + h2
⇒ h2 = 144 – 16
⇒ h2 = 128
⇒ h2 = 64 × 2
⇒ h = 8√2 cm
Volume of cone = πrc2h
= × × 42 × 8√2
= × × 16 × 8√2
=
=
= 134.095 × √2
= 134.095 × 1.414
= 189.61 cm3
Therefore, volume of cone is 189.61 cm3
Find the surface area and volume of a sphere whose radius is 1.4 cm.
Radius of sphere = r = 1.4 cm
Surface area of sphere = 4πr2
= 4 × × 1.4 × 1.4
= 5.6 × 22 × 0.2
= 24.64 cm2
Volume of sphere = πr3
= × × 1.4 × 1.4 × 1.4
= × 22 × 0.2 × 1.4 × 1.4
= × 8.624
= 11.49 ~ 11.5 cm3
Therefore, surface area of sphere is 24.64 cm2 and volume of sphere is 11.5 cm3
The surface area of a sphere is 616 cm2. Find its volume.
Surface area of sphere = 616 cm2
Let the radius of sphere be r
Surface area of sphere = 4πr2
⇒ 4πr2 = 616
⇒ πr2 =
⇒ πr2 = 154
⇒ × r2 = 154
⇒ r2 =
⇒ r2 = 7 × 7
⇒ r = 7 cm
Volume of sphere = πr3
= × × 73
= × 22 × 72
=
=
= 1437.33 cm3
Therefore, volume of sphere is 1437.33 cm3
The radius of a hemisphere is 4.5 cm. Find its surface area and its volume.
Hemisphere is half of a sphere
So the volume and area will be halved
But for total surface area there would be one more area which is the flat circle of a hemisphere
Radius of hemisphere = r = 4.5 cm =
Surface area of hemisphere = 3πr2
= 3 × ×
= 3 × ×
=
=
= 190.92 cm2
Volume of hemisphere = πr3
= × ×
=
=
= 190.92 cm3
Therefore, surface area is 190.92 cm2 and volume is 190.92 cm3
Volume of a sphere is 38808 cm3. Find its surface area.
Volume of sphere = 38808 cm3
Let r be the radius of sphere
Volume of sphere = πr3
⇒ πr3 = 38808
⇒ × × r3 = 38808
⇒ r3 =
⇒ r3 =
⇒ r3 = 441 × 21
⇒ r3 = 9261
⇒ r = 21 cm
Surface area of sphere = 4πr2
= 4 × × 21 × 21
= 4 × 22 × 3 × 21
= 5544 cm2
Therefore, surface area is 5544 cm2
A cylinder made out of lead has radius 4 cm and height 10 cm. It is melted to form lead shots of radius 2 cm. Find the number of lead shots.
Radius of each lead shot = rs = 2 cm
Radius of cylinder = rc = 4 cm
Height of cylinder = h = 10 cm
volume of sphere = πrs3
Volume of cylinder= πrc2h
The lead shots area made by melting the cylinder which means same melted volume is used to make the lead shots thus the volume remains the same
Let ‘n’ be the number of lead shots formed
i.e. volume of n lead shots = volume of cylinder
⇒ n × πrs3 = πrc2h
⇒ 4 × rs3 × n = 3 × rc2 × h
Substituting values
⇒ 4 × 23 × n = 3 × 42 × 10
⇒ 4 × 8 × n = 3 × 16 × 10
⇒ 4 × n = 3 × 2 × 10
⇒ 4 × n = 3 × 20
⇒ n = 3 × 5
⇒ n = 15
Number of lead shots made = 15
A hollow spherical shell is 2 cm thick. If its outer radius is 8 cm, then find the volume of the metal to make such a shell.
Outer radius = r1 = 8 cm
The shell is 2 cm thick therefore we must subtract 2 cm from outer radius to get the inner radius
Inner radius = r2 = 8 – 2 = 6 cm
Volume of metal required to make shell = volume of shell
Volume of shell we can get by subtracting the volume of hollow part of sphere from the complete volume of sphere with 8 cm radius
⇒ Volume of shell = πr13 – πr23
= π × (r13 – r23)
Substituting values
⇒ Volume of shell = × × (83 – 63)
= × × (512 – 216)
= × × 296
=
= 1240.38 cm3
Therefore, volume of the metal to make the shell is 1240.38 cm3
How many cones of radius 3 cm and height 6 cm can be made by melting a metallic sphere of radius 9 cm.
base radius of cone= rc = 3 cm
Height of cone = h = 6 cm
Radius of sphere = rs = 9 cm
volume of cone = πrc2h
Volume of sphere = πrs3
Let n be the number of cones made
As the sphere is melted and then the cones are made from the same amount of melted sphere therefore the volume remains same
i.e. volume of n cones made = volume of sphere
⇒ n × πrc2h = πrs3
⇒ nrc2h = 4rs3
Substituting values
⇒ n × 32 × 6 = 4 × 93
⇒ n × 6 = 4 × 9 × 9
⇒ n = 6 × 9
⇒ n = 54
Number of cones formed = 54
Eight spheres of equal radii are obtained by melting a metallic sphere of radius 10 cm. Find the surface area of each sphere thus obtained.
Let the radius of small spheres formed be r2
Radius of sphere which is melted = r1 = 10 cm
As the sphere is melted and then the small spheres are made from the same amount of melted sphere, therefore, the volume remains same
8 small spheres are made
Volume of sphere = πr3
⇒ 8 × πr23 = πr13
⇒ 8r23 = r13
Substituting values
⇒ 8 × r23 = 103
⇒ 8 × r23 = 1000
⇒ r23 =
⇒ r2 = = 5 cm
Surface area of sphere = 4πr2
⇒ surface area of small spheres made = 4πr22
= 4 × π × 52
= 4 × π × 25
= 100π cm2
Therefore, surface area of each sphere obtained is 100π cm2
If surface area of a sphere is 5544 cm2, then find the volume of the sphere.
Surface area of sphere = 5544 cm2
Let r be the radius of sphere
⇒ 4πr2 = 5544
⇒ πr2 = 1386
⇒ × r2 = 1386
⇒ r2 = 63 × 7
⇒ r2 = 441
⇒ r = ± 21
r is radius and radius cannot be negative thus r = 21
volume of sphere = πr3
= × × 21 × 21 × 21
= 4 × 22 × 3 × 7 × 21
= 38808 cm3
Therefore, volume of the sphere is 38808 cm3
How many spherical lead shots each 4.2 cm in diameter can be obtained from a rectangular solid of lead with dimensions 66 cm, 42 cm and 21 cm.
Diameter of spherical lead shot = 4.2 cm
⇒ radius of spherical lead shots = r = = 2.1 cm
Length of solid lead = l = 66 cm
Breadth = b = 42 cm
Height = h = 21 cm
Volume of sphere = πr3
Volume of rectangular lead = lbh
As the spherical lead shots are obtained from rectangular lead volume remains same
Let n spherical lead shots are made
⇒ n × πr3 = lbh
Substituting values
⇒ n × × × 2.13 = 66 × 42 × 21
⇒ n × × × 2.12 = 3 × 42 × 10
⇒ n × × × 2.1 = 3 × 20 × 10
⇒ n × × 0.3 = 600
⇒ n × 4 × 0.1 = 600
⇒ n × 4 × = 600
⇒ n × 4 = 6000
⇒ n = 1500
Therefore, 1500 spherical lead shots are formed
A sphere of diameter 6 cm is dropped in a right circular cylindrical vessel partly filled with water. The diameter of the cylindrical vessel is 12 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel?
Diameter of sphere = 6 cm
Radius of sphere = rs = = 3 cm
The diameter of cylindrical vessel = 12 cm
Radius of cylindrical vessel = rc = = 6 cm
The volume of sphere submerged in water is equal to the volume of water rise
Let h be the rise of water level
it is given that the sphere is submerged completely
thus, volume of sphere = volume of water rise
volume of sphere = πrs3
volume of cylinder = πrc2h
⇒ πrs3 = πrc2h
⇒ 4rs3 = 3rc2h
⇒ 4 × 33 = 3 × 62 × h
⇒ 4 × 9 × 3 = 3 × 36 × h
⇒ 4 × 3 = 3 × 4 × h
⇒ 3 = 3 × h
⇒ h = 1 cm
Therefore, rise in water level in the cylindrical vessel is 1 cm
A hemispherical bowl of internal radius 9 cm contains liquid. This liquid is to be filled in cylindrical bottles of diameter 3 cm and height 4 cm. How many bottles are required to empty the bowl?
Radius of hemispherical bowl = rh = 9 cm
Diameter of cylindrical bottle = 3 cm
Radius of cylindrical bottle = rc = cm
Height of cylindrical bottle = h = 4 cm
The volume of water is transferred from hemispherical bowl to cylindrical bottles
Suppose there are n bottles then volume of water filled in n bottles will be same as volume of water in hemispherical bowl
Volume of hemisphere = πrh3
Volume of cylinder = πrc2h
⇒ πrh3 = n × πrc2h
⇒ 2rh3 = 3nrc2h
Substitute values
⇒ 2 × 93 = 3n × × 4
⇒ 2 × 93 = 3n × 9
⇒ 2 × 92 = 3n
⇒ 162 = 3n
⇒ n = 54
Therefore, 54 bottles are required to empty the bowl
The diameter of a sphere is 0.7 cm. By putting 3000 such spheres in a tank full of water, the tank is emptied completely. Find the volume of water that overflows.
Total volume of sphere = 3000 × volume of one sphere
diameter of sphere = 0.7 cm
radius of sphere = r = = = cm
Volume of sphere = πr3
⇒ volume of one sphere = × × × ×
= × cm3
= × cm3
= cm3
3000 such spheres are submerged
Total volume of sphere = 3000 × volume of one sphere
= 3000 ×
= 539 cm2
The total volume of spheres submerged in tank is the total amount of water that overflows
Therefore, 539 cm2 volume of water overflows
The outer and inner diameter of a hollow hemi–spherical vessel are 43 cm and 42 cm. Find the cost of colouring the vessel at 7 paise per square cm.
The hemispherical bowl will look like as shown
To find the cost required to paint we first need to calculate its total surface area
Total surface area will include the area of outter curved surface the inner curved surface and the flat surface as shown
Diameter of outer surface = 43 cm
Radius of outer surface = ro = cm = 21.5 cm
Diameter of inner surface = 42 cm
Radius of inner surface = ri = cm = 21 cm
Curved surface area of hemisphere = 2πr2
⇒ outter curved surface area = 2πro2
= 2 × π ×
= π cm2
= 924.5π cm2
⇒ inner curved surface area = 2πri2
= 2 × π × 212
= 882π cm2
The flat area can be found by subtracting the inner circle area from outter circle area
⇒ flat area = πro2 – πri2
= π(21.5)2 – π(21)2
= 462.25π – 441π
= 21.25π cm2
Total surface area of hemispherical vessel = outter curved surface area + inner curved surface area + flat area
⇒ Total surface area of hemispherical vessel = 924.5π + 882π + 21.25π = 1827.75π cm2
Now cost of colouring 1 cm2 is 7 paise
So, cost of colouring 1827.75π cm2 = 7 × 1827.75 × π
= 7 × 1827.75 ×
= 1827.75 × 22
= 40210.5 paise
100 paise is 1 Rs
⇒ 40210.5 paise = = 402.105 Rs
Therefore, cost of colouring the vessel is 402.105 Rs
Total surface area of a cube is 486 cm2. The side of the cube is:
A. 6 cm
B. 8 cm
C. 9 cm
D. 7 cm
Let the side of cube be a
Total surface area of cube = 6a2
⇒ 6a2 = 486
⇒ a2 = 81
⇒ a = 9 cm
The length, breadth and height of a cuboid are 9 m, 2 m and 1 m respectively. The lateral surface area of the cuboid is:
A. 12 m2
B. 11 m2
C. 21 m2
D. 22 m2
Length of cuboid = l = 9 m
Breadth of cuboid = b = 2 m
Height of cuboid = h = 1 m
Lateral surface area of cuboid is surface area without the top and bottom surfaces
So here we won’t consider the area l × b
Lateral surface area = 2lh + 2bh
= 2h(l + b)
= 2 × 1 × (9 + 2)
= 22 m2
The diameter of a sphere is 6 cm. Its volume is—
A. 16π cm3
B. 20π cm3
C. 36π cm3
D. 30π cm3
Diameter of sphere = 6 cm
Radius of sphere = r = = 3 cm
Volume of sphere = πr3
= × π × 33
= 4 × 32 × π
= 36π cm3
The radius of base of a cylinder is 14 cm and its height is 10 cm. The curved surface area of the cylinder is—
A. 810 cm2
B. 880 cm2
C. 888 cm2
D. 890 cm2
Radius of cylinder = r = 14 cm
Height of cylinder = h = 10 cm
Curved surface area of cylinder = 2πrh
= 2 × × 14 × 10
= 2 × 22 × 2 × 10
= 880 cm2
The volume of a cone is 308 cm3. Its height is 6 cm. The radius of its base is—
A. 7 cm
B. 8 cm
C. 6 cm
D. None of these
Let r be the base radius of cone
Volume of cone = 308 cm3
Height = h = 6 cm
Volume of cone = πr2h
⇒ 308 = × × r2 × 6
⇒ = r2 × 2
⇒ 14 × 7 = r2 × 2
⇒ r2 = 7 × 7
⇒ r = 7 cm
The diameter of a solid metallic hemisphere is 42 cm. Find the cost of polishing its total surface at 20 paise per square cm.
Diameter of hemisphere = 42 cm
Radius of hemisphere = r = = 21 cm
Total surface area of hemisphere = 3πr2
= 3 × × 212
= 3 × 22 × 3 × 21
= 4158 cm2
Cost of polishing 1 sq. cm is 20 paise
⇒ cost of polishing 4158 cm2 = 4158 × 20
= 83160 paise
100 paise is 1 Rs
⇒ 83160 paise = = 831.6 Rs
Therefore, cost of polishing its total surface is 831.6 Rs
A cone, a hemisphere and a cylinder stand on equal bases and have the same height. Find the ratio of their volumes.
The cone, hemisphere and cylinder stand on equal bases which means their base radius is same
Let that base radius be r
It is given that height = r
Volume of cone = πr2h = πr3
Volume of hemisphere = πr3
Volume of cylinder = πr2h =πr3
Ratio of cone to hemisphere = =
Ratio of hemisphere to cylinder = =
⇒ ratio of volumes of all three solid = 1:2:3
The left part of a solid is in the shape of a cylinder while the right part is in the shape of cone. The diameter of cylinder is 14 cm and height is 40 cm while the diameter of the conical part is 14 cm and height is 12 cm. Find the volume of the solid.
The solid will look like as shown
Diameter of cylinder = 14 cm
Radius of cylinder = r = = 7 cm
Height of cylinder = h = 40 cm
Volume of cylinder = πr2h
= × 72 × 40
= 22 × 7 × 40
= 6160 cm3
Diameter of base of cone = 14 cm
Radius of base of cone = r1 = = 7 cm
Height of cone = 12 cm
Volume of cone = πr2h
= × × 72 × 12
= 22 × 7 × 4
= 616 cm3
Volume of solid = Volume of cylinder + Volume of cone
= 6160 + 616
= 6776 cm3
Therefore, volume of solid is 6776 cm3
Find the number of cones of radius 3 cm and height 6 cm that can be made by melting a metallic sphere of radius 9 cm.
Radius of metal sphere = rs = 9 cm
Radius of cone formed = rc = 3 cm
Height of cone = h = 6 cm
The sphere is melted and the volume of sphere is used to make cones
Volume of sphere = πrs3
Volume of cone = πrc2h
Let n cones are made then the volume of n cones is same as the volume of sphere
⇒ πrs3 = n × πrc2h
⇒ 4rs3 = nrc2h
Substitute values
⇒ 4 × 93 = n × 32 × 6
⇒ 4 × 92 = n × 6
⇒ 4 × 92 = 6n
⇒ 324 = 6n
⇒ n = 54
Therefore, 54 cones are made
A village, having a population of 4000, requires 150 litres of water per head per day. It has a tank measuring 20 m × 15 m × 6 m. For how many days will the water of this tank last?
Length of tank = l = 20 m
Breadth of tank = b = 15 m
Height of tank = h = 6 m
Volume of water in tank = volume of cuboidal tank
Volume of cuboid = lbh
⇒ volume of cuboidal tank = 20 × 15 × 6
= 1800 m3
1 m3 is 1000 litres
⇒ 1800 m3 = 1800 × 1000 = 1800000 litres
150 litres of water is required per head per day and the population of village is 4000
⇒ volume of water consumed by 4000 people per day = 4000 × 150 = 600000 litres
In one day village consumes 600000 litres of water
⇒ days require to consume 1800000 litres of water = = 3
Thus water of tank will last 3 days
Three solid sphere of iron whose radii are 6 cm, 8 cm and 10 cm respectively are melted into a single solid sphere. Find the radius of the solid sphere.
Three spheres are melted and one big sphere is made out of the melted spheres
Thus volume of sphere formed will be same as volume of 3 melted spheres
Let the radius of spheres be r1 = 6 cm, r2 = 8 cm and r3 = 10 cm
⇒ Volume of spheres melted = πr13 + πr23 + πr33
= π(r13 + r23 + r33)
Let r be the radius of sphere formed
⇒ Volume of sphere formed = πr3
As volumes are same
⇒ Volume of spheres melted = Volume of sphere formed
⇒ π(r13 + r23 + r33) = πr3
⇒ r13 + r23 + r33 = r3
Substituting values
⇒ 63 + 83 + 103 = r3
⇒ 216 + 512 + 1000 = r3
⇒ r3 = 1728
⇒ r = 12 cm
Therefore, radius of sphere made is 12 cm
A conical vessel of radius 10 cm and height 18 cm is filled with water. If the water is poured in a cylindrical vessel of radius 5 cm, find the height of water in the cylindrical vessel.
Base radius of conical vessel = r1 = 10 cm
Height of conical vessel = h1 =18 cm
Base radius of cylindrical vessel = r2 = 5 cm
Let h be the height of water in the cylindrical vessel
(note that we does not need to know the height of cylindrical vessel just assume that the height will be till the water level)
As the water is transferred from conical vessel to cylindrical vessel volume is unchanged
Volume of conical vessel = πr12h1
Volume of cylindrical vessel = πr22h
⇒ πr12h1 = πr22h
⇒ r12h1 = 3r22h
Substituting values
⇒ 102 × 18 = 3 × 52 × h
⇒ 100 × 6 = 25× h
⇒ h = 4 × 6
⇒ h = 24 cm
Therefore, height of water in the cylindrical vessel is 24 cm
From a wax cuboid of dimensions 11 cm × 3.5 cm × 2.5 cm, a candle of diameter 2.8 cm is made. Find the length of the candle.
Length of wax cuboid = l = 11 cm
Breadth of wax cuboid = b = 3.5 cm = cm
Height of wax cuboid = h = 2.5 cm = cm
Diameter of candle = 2.8 cm
Radius of candle = r = = 1.4 cm
Let the length of candle be l1
The candle is cylindrical so the length of candle will be its height
Candle is made from wax cuboid hence volume of cuboid wax is equal to volume of cylindrical candle candle
Volume of wax cuboid = lbh
Volume of cylindrical candle = πr2l1
⇒ lbh = πr2l1
⇒ 11 × × = × 1.42 × l1
⇒ × = 2 × 0.2 × 1.4 × l1
⇒ × = 0.56 × l1
⇒ × = 0.08 × l1
⇒ × = × l1
⇒ l1 =
⇒ l1 =
⇒ l1 =
⇒ l1 = 15.625 cm
Therefore, candle is 15.625 cm long
The diameter of a metallic sphere is 6 cm. It is melted section. If the length of the wire is 36 m, then find the radius of the wire.
Diameter of sphere = 6 cm
Radius of sphere = rs = = 3 cm
Length of wire = h = 36 m = 3600 cm
Let the radius of wire be rw
Wire is cylindrical
Wire is formed by melting the sphere thus volume of wire is same as volume of sphere
Volume of sphere = πrs3
Volume of cylindrical wire = πrw2h
⇒ πrs3 = πrw2h
⇒ × rs3 = rw2 × h
Substitute values
⇒ × 33 = rw2 × 3600
⇒ 4 × 32 = rw2 × 3600
⇒ 36 = rw2 × 3600
⇒ rw2 =
⇒ rw = 0.1 cm
Therefore, radius of wire is 0.1 cm