CHAPTER: 11
(AREA) & (VOLUME & SURFACE)
Area – Formulas:
FUNDAMENTAL CONCEPTS
Results on Triangles:
1. Sum of the angles of a triangle is 180°
2. The sum of any two sides of a triangle is greater than the third side.
3. Pythagoras Theorem: In a right-angled triangle, (Hypotenuse)2 = (Base)2 + (Height)2.
4. The line joining the mid-point of a side of a triangle to the opposite vertex is called the median.
5. The point where the three medians of a triangle meet, is called centroid. The centroid divided each of the
medians in the ratio 2: 1.
6. In an isosceles triangle, the altitude from the vertex bisects the base.
7. The median of a triangle divides it into two triangles of the same area.
8. The area of the triangle formed by joining the mid-points of the sides of a given triangle is one-fourth of the
area of the given triangle.
Results on Quadrilaterals:
1. The diagonals of a parallelogram bisect each other.
2. Each diagonal of a parallelogram divides it into triangles of the same area.
3. The diagonals of a rectangle are equal and bisect each other.
4. The diagonals of a square are equal and bisect each other at right angles
5. The diagonals of a rhombus are unequal and bisect each other at right angles.
6. A parallelogram and a rectangle on the same base and between the same parallels are equal in area.
7. Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area.
IMPORTANT FORMULAE
1. A. Area of a rectangle = (Length x Breadth).
𝐴𝑟𝑒𝑎 𝐴𝑟𝑒𝑎
∴ Length = (𝐵𝑟𝑒𝑎𝑑𝑡ℎ) and Breadth = (𝐿𝑒𝑛𝑔𝑡ℎ)
B. Perimeter of a rectangle = 2(Length + Breadth).
1
2. Area of a square = (side)2 = 2 (diagonal)2.
3. Area of 4 walls of a room = 2 (Length + Breadth) x Height.
1
4. A. Area of a triangle = 2 x Base x Height.
B. Area of a triangle = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)
1
where a, b, c are the sides of the triangle and s = 2 (a + b + c).
√3
C. Area of an equilateral triangle = 4 × (𝑠𝑖𝑑𝑒)2
𝑎
D. Radius of incircle of an equilateral triangle of side a = 2
√3
𝑎
E. Radius of circumcircle of an equilateral triangle of side a =
√3
∆
F. Radius of incircle of a triangle of area and semi-perimeter r = 𝑆
5. A. Area of parallelogram = (Base x Height).
1
B. Area of a rhombus = x (Product of diagonals).
2
1
C. Area of a trapezium = 2 x (sum of parallel sides) x distance between them
6. A. Area of a circle = 𝜋R2, where R is the radius.
B. Circumference of a circle = 2𝜋R.
2𝜋𝑅𝜃
C. Length of an arc = , where 𝜃 is the central angle.
360
1 𝜋𝑅2 𝜃
D. Area of a sector = 2 (arc × 𝑅) = 360
.
7. A. Circumference of a semi-circle = 𝜋R.
𝜋𝑅2
B. Area of semi-circle = 2
CLASS PRACTICE
1. The ratio between the length and the breadth of a rectangular park is 3: 2. If a man cycling along the boundary of the
park at the speed of 12 km/hr. completes one round in 8 minutes, then the area of the park (in sq. m) is:
a. 15360 b. 153600 c. 30720 d. 307200
2. An error 2% in excess is made while measuring the side of a square. The percentage of error in the calculated area of
the square is:
a. 2% b. 2.02% c. 4% d. 4.04%
3. The ratio between the perimeter and the breadth of a rectangle is 5: 1. If the area of the rectangle is 216 sq. cm, what
is the length of the rectangle?
a. 16 cm b. 18 cm c. 24 cm d. Data inadequate
4. The percentage increase in the area of a rectangle, if each of its sides is increased by 20% is:
a. 40% b. 42% c. 44% d. 46%
5. A rectangular park 60 m long and 40 m wide has two concrete crossroads running in the middle of the park and rest of
the park has been used as a lawn. If the area of the lawn is 2109 sq. m, then what is the width of the road?
a. 2.91 m b. 3 m c. 5.82 m d. None of these
1 1
6. The diagonal of the floor of a rectangular closet is 7 feet. The shorter side of the closet is 4 feet. What is the area of
2 2
the closet in square feet?
1 1
a. 5 b. 13 c. 27 d. 37
4 2
7. A towel, when bleached, was found to have lost 20% of its length and 10% of its breadth. The percentage of decrease
in area is:
a. 10% b. 10.08% c. 20% d. 28%
8. A man walked diagonally across a square lot. Approximately, what was the percent saved by not walking along the
edges?
a. 20 b. 24 c. 30 d. 33
9. The diagonal of a rectangle is √41 cm and its area are 20 sq. cm. The perimeter of the rectangle must be:
a. 9 cm b. 18 cm c. 20 cm d. 41 cm
10. What is the least number of squares tiles required to pave the floor of a room 15 m 17 cm long and 9 m 2 cm broad?
a. 814 b. 820 c. 840 d. 844
11. The difference between the length and breadth of a rectangle is 23 m. If its perimeter is 206 m, then its area is:
a. 1520 m2 b. 2420 m2 c. 2480 m2 d. 2520 m2
12. The length of a rectangle is halved, while its breadth is tripled. What is the percentage change in area?
a. 25% increase b. 50% increase c. 50% decrease d. 75% decrease
13. The length of a rectangular plot is 20 meters more than its breadth. If the cost of fencing the plot @ 26.50 per meter
is tk. 5300, what is the length of the plot in meters?
a. 40 b. 50 c. 120 d. None of these
Volume and Surface Area – Formulas:
1. CUBOID
Let length = l, breadth = b and height = h units. Then
A. Volume = (l x b x h) cubic units.
B. Surface area = 2(lb + bh + lh) sq. units.
C. Diagonal = √12 + 𝑏2 + ℎ2 units.
2. CUBE
Let each edge of a cube be of length a. Then,
A. Volume = a3 cubic units.
B. Surface area = 6a2 sq. units.
C. Diagonal = √3a units.
3. CYLINDER
Let radius of base = r and Height (or length) = h. Then,
A. Volume = (𝜋r2h) cubic units.
B. Curved surface area = (2𝜋rh) sq. units.
C. Total surface area = 2𝜋r (h + r) sq. units.
4. CONE
Let radius of base = r and Height = h. Then,
A. Slant height, l = √ℎ2 + 𝑟 2 units.
1
B. Volume = (3 𝜋𝑟 2 ℎ)cubic units.
C. Curved surface area = (𝜋rl) sq. units.
D. Total surface area = (𝜋rl + 𝜋r2) sq. units.
5. SPHERE
Let the radius of the sphere be r. Then,
4
A. Volume = (3 𝜋𝑟 3 )cubic units.
B. Surface area = (4𝜋r2) sq. units.
6. HEMISPHERE
Let the radius of a hemisphere be r. Then,
2
A. Volume = (3 𝜋𝑟 3 )cubic units.
B. Curved surface area = (2𝜋r2) sq. units.
C. Total surface area = (3𝜋r2) sq. units.
Note: 1 liter = 1000 cm3
CLASS PRACTICE
14. A right triangle with sides 3 cm, 4 cm and 5 cm is rotated the side of 3 cm to form a cone. The volume of the cone so
formed is:
a. 12 cm3 b. 15 cm3 c. 16 cm3 d. 20 cm3
15. In a shower, 5 cm of rain falls. The volume of water that falls on 1.5 hectares of ground is:
a. 75 cu. m b. 750 cu. m c. 7500 cu. m d. 75000 cu. m
16. A hall is 15 m long and 12 m broad. If the sum of the areas of the floor and the ceiling is equal to the sum of the areas
of four walls, the volume of the hall is:
a. 720 b. 900 c. 1200 d. 1800
17. A hollow iron pipe is 21 cm long and its external diameter is 8 cm. If the thickness of the pipe is 1 cm and iron weighs
8 g/cm3, then the weight of the pipe is:
a. 3.6 kg b. 3.696 kg c. 36 kg d. 36.9 kg
18. A boat having a length 3 m and breadth 2 m is floating on a lake. The boat sinks by 1 cm when a man gets on it. The
mass of the man is:
a. 12 kg b. 60 kg c. 72 kg d. 96 kg
19. The slant height of a right circular cone is 10 m and its height is 8 m. Find the area of its curved surface.
a. 30 𝑚2 b. 40 𝑚2 c. 60 𝑚2 d. 80 𝑚2
20. A cistern 6m long and 4 m wide contains water up to a depth of 1 m 25 cm. The total area of the wet surface is:
a. 49 𝑚2 b. 50 𝑚2 C. 53.5 𝑚2 d. 55 𝑚2
21. A metallic sheet is of rectangular shape with dimensions 48 m x 36 m. From each of its corners, a square is cut off so
as to make an open box. If the length of the square is 8 m, the volume of the box (in m3) is:
a. 4830 b. 5120 c. 6420 d. 8960
22. The curved surface area of a cylindrical pillar is 264 𝑚2 and its volume is 924 𝑚3 . Find the ratio of its diameter to its
height.
a. 3: 7 b. 7: 3 c. 6: 7 d. 7: 6
23. A cistern of capacity 8000 liters measures externally 3.3 m by 2.6 m by 1.1 m and its walls are 5 cm thick. The
thickness of the bottom is:
a. 90 cm b. 1 dm c. 1 m d. 1.1 cm
24. What is the total surface area of a right circular cone of height 14 cm and base radius 7 cm?
a. 344.35 c𝑚2 b. 462 c𝑚2 C. 498.35 c𝑚2 d. None of these
25. A large cube is formed from the material obtained by melting three smaller cubes of 3, 4 and 5 cm side. What is the
ratio of the total surface areas of the smaller cubes and the large cube?
a. 2: 1 b. 3: 2 c. 25: 18 d. 27: 20
TAKE HOME ASSIGNMENT
AREA
1. A tank is 25 m long, 12 m wide and 6 m deep. The cost of plastering its walls and bottom at 75 paise per sq. m, is:
a. tk. 456 b. tk. 458 c. tk. 558 d. tk. 568
2. A rectangular field is to be fenced on three sides leaving a side of 20 feet uncovered. If the area of the field is 680 sq.
feet, how many feet of fencing will be required?
a. 34 b. 40 c. 68 d. 88
3. Find the area of the triangle whose sides are 9cm, 10cm and 11cm respectively
a. 2242 cm² b. 25/3 cm² c. 2743 cm² d. 30/2 cm²
4. Find the area of the square whose diagonals is 32 cm long?
a. 4.98 c𝑚2 b. 5.12 cm³ c. 5.23 cm² d. 6.15 cm³
5. A Wheel makes 1000 rotation in covering of 44 km. Find the radius of the wheel?
a. 5.5 m b. 6m c. 7 m d. 7.5 m
6. The perimeter of a rectangle is 60 m. If its length is twice its breadth, then its area is
a. 185 m² b. 190 m² c. 200 m² d. 220 m²
7. The diagonal of a square is 4/2 cm. The diagonal of another square whose area is double that of the first
square is
a. 7 cm b. 8 cm c. 10 cm d. 11 cm
8. The area of a square plot is 5000 m². The length of its diagonal is
a. 75 m b. 90 m c. 100 m d. 120 m
9. If each side of a square is increased by 50%, the ratio of the area of the resulting square to the area of the
given square is
a. 8:5 b. 8:3 c. 9:4 d. 10:3
10. The sides of a triangle are in the ratio ½: 1/3: ¼, If the perimeter is 52 cm, then the length of the smallest
side is
a. 8 cm b. 10 cm c. 11 cm d. 12 cm
11. The area of a circle is increased by 22 c𝑚2 when its radius is increased by 1 cm. The original radius of the
circle is
a. 3 cm b. 4 cm c. 6 cm d. 7 cm
12. The length of a rectangular plot is twice its breadth. If the length of its diagonal is 5√5 m, the area of the
rectangle is
a. 38 m² b. 46 m² c. 50 m² d. 53 m²
13. Area of four walls of a room is 66 m² and its length and breadth are 7.5 m and 3.5 m respectively. The height
of the room is
a. 3 m b. 5 m c. 6 m d. 7 m
14. A Wheel makes 1000 revolutions in covering a distance of 88 km, The diameter of the wheel is
a. 20 m b. 22 m c. 25 m d. 28 m
15. The length of minute hand of a wall clock is 7 cm. The area swept by it in 15 minutes is
a. 23.5 cm² b. 32.5 cm² c. 38.5 cm² d. 47.5 cm²
16. The radius of a wheel is 0.21 m. How many times does it revolve during a journey of 0.792 km?
a. 550 b. 600 c. 650 d. 675
17. In a circle of radius 14 cm, an arc subtends an angle of 600 at the center. The length of the arc is
a. 12 1/3 cm b. 14 2/3 cm c. 15 1/3 cm d. 19 1/3 cm
18. The radius of the wheel of a vehicle is 70 cm. The wheel makes 100 revolutions in 50 seconds. The speed of
the vehicle is
a. 8.8 m/sec b. 9.2 m/sec c. 10.5 m/sec d. 11.2 m/sec
19. The area of a rhombus is 72 cm². One of its diagonals is thrice the other. The length of the shorter diagonal
is
a. 2√3 cm b. 3√3 cm c. 4√3 cm d. 5√3 cm
20. If a square and a rhombus stand on the same base, then the ratio of the areas of the square and rhombus is
a. 1 b. 2 c. 4 d. 5
21. A triangle has sides of length 3 cm, 4 cm and 5 cm. What is the area of the circle inscribed in the triangle?
a. 𝜋c𝑚2 b. r cm² c. 1 cm² d. ½ cm²
22. What is the diameter of a wheel that makes 113 revolutions to go 2 km 26 decameters?
a. 60/11 m b. 67/11 m c. 70/11 m d. 75/11 m
23. The area of a rectangle is 12 sq meters and its length is 3 times that of its breadth. What is the perimeter of
the rectangle?
a. 16 m b. 19 m c. 21 m d. 24 m
24. A road of uniform width runs round the inside of a rectangular field 38 m long and 32 m wide. If the road
occupies 600 m², the width of the road is
a. 4 m b. 5 m c. 6 m d. 8 m
25. The area of a triangle is x square cm. and its base is y cm. What is the height of the triangle?
a. x/y cm b. x/2y cm c. 2x/y cm d. 2𝑥 2 /y cm
26. The two parallel sides of a trapezium are 2 cm and 4 cm respectively and the perpendicular distance
between them is 3 cm. The area of the trapezium is
a. 6 cm² b. 7 cm c. 8 cm² d. 9 cm²
27. What is the area of an equilateral triangle inscribed in a circle of unit radius?
a. 2√2/3 sq unit b. 3√3/4 sq unit c. 3√2/4 sq unit d. 4√3/3 sq unit
28. If the ratio of the areas of two squares is 16: 1, then the ratio of their perimeter is
a. 2:1 b. 3:1 c. 4:1 d. 5:1
1 1
29. The diagonal of the floor of a rectangular closet is 72 feet. The shorter side of the closet is 42 feet. What is
the area of the closet in square feet?
1 1
a. 54 sq. ft. b. 132 sq. ft. c. 27 sq. ft. d. 37 sq. ft.
30. The diagonal of a rectangle is √41 cm and its area are 20 sq. cm. The perimeter of the rectangle must be:
a. 9 cm b. 18 cm c. 20 cm d. 41 cm
31. A man walked diagonally across a square lot. Approximately, what was the percent saved by not walking
along the edges?
a. 20% b. 24% c. 30% d. 33%
32. The difference between the length and breadth of a rectangle is 23 m. If its perimeter is 206 m, then its area
is:
a. 1520 m² b. 2420 m² c. 2480 m² d. 2520 m²
33. In ∆ PQR, side PQ = 32 cm and side PR = 25cm. What is the area of ∆ PQR?
a. 2 √154 sq.cm b. 3 √154 sq.cm c. 4 √308 sq.cm d. Cannot be determined
34. The following squares represent the monthly income of two families
If the monthly income of family A is Tk. 40000, the monthly income of family B is ?
a. Tk. 50000 b. Tk. 60000 c. Tk. 90000 d. Tk. 120000
35.The area of the largest circle, that can be drawn inside a rectangle with side 18 cm by 14 cm, is:
a. 49 cm² b. 154 cm² c. 378 cm² d. 1978 cm³
36. A diagonal of a rhombus is 6 cm. If its area is 24 cm² then the length of each side of the rhombus is:
a. 5 cm b. 6 cm c. 7 cm d. 8 cm
37.A horse is tied at the corner of a rectangular field whose length is 20 m and width is 16 m, with a rope whose
length is 14 m. Find the area which the horse can graze:
a.144 sq. m b. 154 sq. m c. 156 sq. m d. 164 sq. m
38. The adjoining figure contains three squares with areas of 100, 16 and 49 lying side by side as shown. By
how much should the area of the middle square be reduced in order that the total length PQ of the resulting
three squares is 19?
a. √2 b. 2 c. 4 d. 12
39. 66 cubic centimeters of silver is drawn into a wire 1 mm in diameter. The length of the wire in meters will be:
a. 84 b. 90 c. 168 d. 336
SURFACE
40. 50 men took a dip in a water tank 40 m long and 20 m broad on a religious day. If the average displacement of water
by a man is 4 m3, then the rise in the water level in the tank will be:
a. 20 cm b. 25 cm c. 35 cm d. 50 cm
41. How many bricks, each measuring 25 cm x 11.25 cm x 6 cm, will be needed to build a wall of 8 m x 6 m x 22.5 cm?
a. 5600 b. 6000 c. 6400 d. 7200
42.Find the volume and surface area of a cuboid 10m long, 8m broad and 4m high?
a. 256m² b. 284m² c. 304m² d.326m²
43. The surface area of a cube is 216m². Find its volume.
a. 188m² b. 204m³ c. 216m³ d. 226m³
44. Find the volume and the surface area of a sphere of diameter 14 cm
a. 116 cm² b. 332 cm² c. 498 cm² d. 616 cm²
45. The length of the longest pole that can be kept in a room 5m long, 4m broad and 2m height is
a. 3√5 m b. 4√5 m c. 5 √5 m d. 7√5 m
46. A rectangular water reservoir contains 54000 liters of water. If the length of reservoir is 9m and breadth is
4m, then the depth of the reservoir is
a. 1.3 m b. 1.5 m c. 2.2 m d. 2.3 m
47. What part of a ditch 48m long, 16.5m broad and 4m deep can be filled by the earth got by digging a
cylindrical tunnel of diameter 4m and length 56m 7
a. 2/7 b. 2/9 c. 3/5 d. 3/7
48.If the areas of three adjacent faces of a cuboid are x, y, z respectively, then the volume of the cuboid is
a. xyz b.(xyz)2
c. xy/z d. √ xyz
49. The diagonal of a cube measures 5√3 cm. Its volume is
a. 102 cm³ b. 105 cm³ c. 111 cm³ d.125 cm³
50. The area of the base of the rectangular tank is 6500 cm² and the volume of the water contained in it is 32.5
cubic meters. The depth of the water in the tank is
a. 42m b.50m c. 57m d. 62m
51. A covered wooden box has the inner measures as 110cm, 70cm and 30cm and the thickness of wood is
2.5cm. The volume of the wood is
a. 54982 cm³ b. 62547 cm c.69832 cm³ d. 70875 cm³
52. A hall is 15m long and 10m broad. If the sum of the areas of the floor and the ceiling is equal to the sum
of the areas of the 4 walls, the volume of the hall is
a. 815 m³ b. 875 m³ c. 900 m³ d. 950 m³
53. The size of a wooden block is (12cm × 10cm × 8cm). How many such blocks will be required to construct a solid
wooden cube of minimum size?
a. 1300 b. 1500 c. 1600 d. 1800
54. The height of a cylinder is 12cm and its diameter is 10cm. The volume of the cylinder is
a. 733.50 cm b. 856.21 cm³ c. 942.85 cm³ d. 993.88 cm³
55. The height of a cylinder is 14cm and its curved surface area is 264c𝑚2 . The volume of the cylinder is
a. 396 c𝑚3 b.451 c𝑚3 c. 482 c𝑚3 d.527 c𝑚3
56. A cylinder has a radius of 7 cm and the area of its curved surface is 176 cm². The volume of the cylinder is
a. 577𝑐𝑚3 b. 616𝑐𝑚3 c. 659𝑐𝑚3 d. 721𝑐𝑚3
57. The ratio of the radius of two cylinders is 2:3 and the ratio of their heights is 3.5. The ratio of their volumes will
be
a. 3:7 b. 3:17 c. 4:15 d. 4:17
58. The number of coins, 1.5 cm in diameter and 0.2 cm thick to be melted to form a right circular cylinder of height
10 cm and diameter 4.5 cm is
a. 375 b. 390 c. 420 d. 450
59. Find the area of the iron sheet required to prepare a cone 20 cm height with base radius 15 cm.
a. 1126.60 cm² b. 1365.25 cm²
c. 1750.35 cm² d. 1885.71 cm²
60. If the ratio of volumes of two Cones is 2:3 and the ratio of the radii of their bases is 1:2, then the ratio of
their heights will be
a. 5:4 b. 5:3 c. 7:3 d. 8:3
61. The volume of a hemisphere of radius 7 cm is
a. 712.5 cm³ b. 718.6 cm³ c. 723.6 cm³ d. 728.3 cm³
62. How many balls of diameter 0.4 cm each are formed a cuboid of measuring (4cm × 5cm × 6cm) ?
a. 3512 b. 3580 c. 3620 d. 3675
63. The volume of a sphere is 2145 11/21 c𝑚3 . Its diameter is
a. 16 cm b. 18 cm c. 21 cm d. 23 cm
64. 12 sphere of the same size are made by melting a solid cylinder of 16 cm diameter and 2 cm height. The
diameter of each sphere is
a. 4 cm b. 6 cm c. 7 cm d. 8 cm
65. The surface area of a cube is 600 c𝑚2 . The length of its diagonal is
10 10
a. cm b. cm c. 10 √3 cm d.10 √2 cm
√3 √2
66. The radius of hemisphere is 3 cm. The ratio of its volume to the total surface area is
a. 1: 3 b. 2: 1 C. 1: 1 d. 2: 3
67. The volume of a right circular cylinder is 392𝜋 cm³ and its height is 8 cm. Find the radius?
a. 6 cm b.7 cm c.8 cm d. 9 cm
68. If the height of right circular cylinder is 10 cm and radius of its base is 4 cm, find its total surface area?
a.350 cm³ b.351 cm³ c.352 cm² d. 353 cm³
69. length, Width and Height of a cuboid are in the ratio of 6:3:5. If the total surface area of cuboid is 1134
cm2 then find its Length, Width and Height
a. 18, 9, 15 b.17 10 15 c. 12, 15, 9 d. 11, 13, 16
70. If the surface area of two spheres is in the ratio 9:16, then the ratio of their volumes will be?
a.9:16 b.16:9 c. 27:64 d. 3:4
71. The volume of a cube is 512 cm³, its surface area is?
a. 64 cm² b.256 cm² c. 384 cm² d.512 cm²
72. A box is of 10 cm long, 8 cm broad and 5 cm high. What is the longest possible length of a pencil that
can be put in?
a. √150 cm b. √98 cm c. 3√21 cm d. 3√52 cm
73. If the volume and surface are of a sphere are numerically the same then its radius is?
a. 1 unit b. 2 unit c. 3 unit d. 4 unit
74. If the volume of the cube is 1331 m³, then the total surface area of the cube is (in m³):
a. 648 b. 484 c. 726 d. 216
75. The areas of two circles are in the ratio 1:2. Find the ratio of their radius
a. 1:2 b. 1:3 c. 1:√2 d. 1:4
