Mensuration & Geometry: Comprehensive Notes and 100 Practice Questions with Answers

 

Mensuration & Geometry: Comprehensive Notes and 100 Practice Questions with Answers

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1. Mensuration: Introduction and Key Concepts

Mensuration is the branch of mathematics that deals with measurement of geometric figures—calculating area, perimeter, volume, and surface area for 2D and 3D shapes1258. It is essential for understanding the size, shape, and space occupied by objects, with applications in engineering, architecture, and everyday life.

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2. 2D Geometry: Area and Perimeter Formulas

Shape	Area	Perimeter (Circumference)
Square	$a^2$	$4a$
Rectangle	$l\times b$	$2(l+b)$
Circle	$\pi r^2$	$2\pi r$
Triangle	$\frac{1}{2}\times \text{base}\times \text{height}$	$a+b+c$ (sum of sides)
Equilateral Triangle	$\frac{\sqrt{3}}{4}{a}^2$	$3a$
Isosceles Triangle	$\frac{1}{2}bh$	$2a+b$
Scalene Triangle	$\sqrt{s(s-a)(s-b)(s-c)}$, $s=\frac{a+b+c}{2}$	$a+b+c$
Parallelogram	$b\times h$	$2(a+b)$
Rhombus	$\frac{1}{2}{d}_1{d}_2$	$4a$
Trapezium	$\frac{1}{2}h(a+c)$	$a+b+c+d$

Key Points:

• Area is always in square units (e.g., cm², m²)236.

• Perimeter is always in linear units (e.g., cm, m).

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3. 3D Geometry: Volume and Surface Area Formulas

Shape	Volume	Surface Area (Total/Curved)
Cube	$a^3$	$6a^2$
Cuboid	$l\times b\times h$	$2(lb+bh+hl)$
Cylinder	$\pi r^{2}h$	Total: $2\pi r(r+h)$; Curved: $2\pi rh$
Cone	$\frac{1}{3}\pi r^{2}h$	Total: $\pi r(l+r)$; Curved: $\pi rl$, $l=\sqrt{r^2+h^2}$
Sphere	$\frac{4}{3}\pi r^3$	$4\pi r^2$
Hemisphere	$\frac{2}{3}\pi r^3$	Curved: $2\pi r^2$; Total: $3\pi r^2$

Key Points:

• Volume is always in cubic units (e.g., cm³, m³)12345.

• Surface area is in square units.

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4. Basic Geometric Theorems

• Pythagoras Theorem: In a right triangle, $a^2+b^2=c^2$ (where c is hypotenuse).

• Sum of Angles in Triangle: ${180}^{\circ }$.

• Sum of Angles in Quadrilateral: ${360}^{\circ }$.

• Circle Theorems:

  • Angle in a semicircle is ${90}^{\circ }$.

  • Tangent at any point is perpendicular to the radius at that point.

• Properties of Parallelogram: Opposite sides and angles are equal; diagonals bisect each other.

• Properties of Rhombus: All sides equal; diagonals bisect at right angles.

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5. Solved Examples

Example 1:
Area of a square with side 5 cm: $5^2=25$ cm²5.

Example 2:
Perimeter of a rectangle with length 10 cm, breadth 5 cm: $2(10+5)=30$ cm5.

Example 3:
Area of a right triangle with base 3 cm, height 4 cm: $\frac{1}{2}\times 3\times 4=6$ cm²5.

Example 4:
Volume of a cone, radius 7 cm, height 5 cm: $\frac{1}{3}\pi 7^2\times 5\approx 256.67$ cm³5.

Example 5:
Total surface area of a cylinder with radius 14 cm, height 10 cm: $2\pi rh+2\pi r^2=880+1232=2112$ cm²5.

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100 Practice Questions with Answers and Explanations

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2D Geometry: Area & Perimeter

1. Find the area and perimeter of a square with side 8 cm.
   Ans: Area = $64$ cm², Perimeter = $32$ cm.

2. Find the area and perimeter of a rectangle with length 12 cm, breadth 7 cm.
   Ans: Area = $84$ cm², Perimeter = $38$ cm.

3. Find the area and circumference of a circle with radius 5 cm.
   Ans: Area = $78.54$ cm², Circumference = $31.42$ cm (using $\pi =3.14$).

4. Find the area of a triangle with base 10 cm, height 6 cm.
   Ans: $\frac{1}{2}\times 10\times 6=30$ cm².

5. Find the perimeter of a triangle with sides 5 cm, 12 cm, 13 cm.
   Ans: $5+12+13=30$ cm.

6. Find the area of an equilateral triangle with side 6 cm.
   Ans: $\frac{\sqrt{3}}{4}\times 36=9\sqrt{3}\approx 15.59$ cm².

7. Find the perimeter of an equilateral triangle with side 9 cm.
   Ans: $3\times 9=27$ cm.

8. Find the area of a parallelogram with base 8 cm, height 5 cm.
   Ans: $8\times 5=40$ cm².

9. Find the perimeter of a parallelogram with sides 8 cm and 5 cm.
   Ans: $2\times (8+5)=26$ cm.

10. Find the area of a rhombus with diagonals 10 cm and 8 cm.
    Ans: $\frac{1}{2}\times 10\times 8=40$ cm².

11. Find the perimeter of a rhombus with side 7 cm.
    Ans: $4\times 7=28$ cm.

12. Find the area of a trapezium with parallel sides 12 cm, 8 cm, height 5 cm.
    Ans: $\frac{1}{2}\times 5\times (12+8)=50$ cm².

13. Find the perimeter of a trapezium with sides 8 cm, 10 cm, 6 cm, 12 cm.
    Ans: $8+10+6+12=36$ cm.

14. Find the area of a rectangle with length 15 m, breadth 9 m.
    Ans: $135$ m².

15. Find the circumference of a circle with diameter 14 cm.
    Ans: $\pi \times 14=43.96$ cm.

16. Find the area of a triangle with sides 7 cm, 8 cm, 9 cm (use Heron's formula).
    Ans: $s=12$, Area = $\sqrt{12\times 5\times 4\times 3}=\sqrt{720}\approx 26.83$ cm².

17. Find the area of a square with diagonal 10 cm.
    Ans: Side = $10\mathrm{/}\sqrt{2}=7.07$ cm, Area = $50$ cm².

18. Find the area of a circle with diameter 20 cm.
    Ans: Radius = 10 cm, Area = $314$ cm².

19. Find the perimeter of a rectangle with length 25 cm, breadth 15 cm.
    Ans: $2\times (25+15)=80$ cm.

20. Find the area of a parallelogram with base 14 cm, height 6 cm.
    Ans: $84$ cm².

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3D Geometry: Volume & Surface Area

21. Find the volume and surface area of a cube with side 5 cm.
    Ans: Volume = $125$ cm³, Surface area = $150$ cm².

22. Find the volume and total surface area of a cuboid with l = 10 cm, b = 6 cm, h = 4 cm.
    Ans: Volume = $240$ cm³, Surface area = $2(60+24+40)=248$ cm².

23. Find the volume and curved surface area of a cylinder with r = 7 cm, h = 10 cm.
    Ans: Volume = $1540$ cm³, Curved SA = $2\pi rh=440$ cm².

24. Find the total surface area of a cylinder with r = 7 cm, h = 10 cm.
    Ans: $2\pi r(r+h)=2\times 22\mathrm{/}7\times 7\times 17=748$ cm².

25. Find the volume and surface area of a sphere with r = 6 cm.
    Ans: Volume = $904.32$ cm³, Surface area = $452.16$ cm².

26. Find the volume and surface area of a hemisphere with r = 5 cm.
    Ans: Volume = $261.67$ cm³, Curved SA = $157.08$ cm², Total SA = $235.62$ cm².

27. Find the volume and total surface area of a cone with r = 3 cm, h = 4 cm.
    Ans: Volume = $37.7$ cm³, Slant height = 5 cm, Total SA = $\pi r(l+r)=75.4$ cm².

28. Find the volume of a cuboid with l = 8 cm, b = 5 cm, h = 3 cm.
    Ans: $120$ cm³.

29. Find the surface area of a cube with side 9 cm.
    Ans: $486$ cm².

30. Find the volume of a cylinder with diameter 10 cm, height 21 cm.
    Ans: r = 5 cm, Volume = $1650$ cm³.

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Basic Geometry Theorems and Properties

31. What is the sum of angles in a triangle?
    Ans: ${180}^{\circ }$.

32. What is the sum of angles in a quadrilateral?
    Ans: ${360}^{\circ }$.

33. State the Pythagoras theorem.
    Ans: $a^2+b^2=c^2$ in a right triangle.

34. If the hypotenuse of a right triangle is 10 cm and one side is 6 cm, find the other side.
    Ans: $\sqrt{{10}^2-6^2}=8$ cm.

35. If a parallelogram has base 10 cm and height 7 cm, what is its area?
    Ans: $70$ cm².

36. If the diagonals of a rhombus are 12 cm and 16 cm, what is its area?
    Ans: $96$ cm².

37. What is the area of a sector of a circle with radius 6 cm and angle 60°?
    Ans: $\frac{60}{360}\times \pi \times 36=18.85$ cm².

38. What is the length of the diagonal of a square with side 10 cm?
    Ans: $10\sqrt{2}\approx 14.14$ cm.

39. If the sides of a triangle are 5 cm, 12 cm, 13 cm, is it a right triangle?
    Ans: Yes, $5^2+{12}^2={13}^2$.

40. If the circumference of a circle is 44 cm, what is its radius?
    Ans: $r=44\mathrm{/}(2\pi )=7$ cm.

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Application & Mixed Problems

41. A rectangular field is 50 m long and 30 m wide. Find its area and perimeter.
    Ans: Area = $1500$ m², Perimeter = $160$ m.

42. A cylindrical tank has radius 3 m and height 7 m. Find its volume.
    Ans: $198$ m³.

43. A cube has volume 64 cm³. Find its side.
    Ans: $4$ cm.

44. A cone has base radius 4 cm and height 3 cm. Find its slant height.
    Ans: $5$ cm.

45. A circle has area 154 cm². Find its radius. ($\pi =22\mathrm{/}7$)
    Ans: $r=7$ cm.

46. A sphere has surface area 616 cm². Find its radius.
    Ans: $r=7$ cm.

47. A cuboid has length 8 cm, breadth 6 cm, height 2 cm. Find its total surface area.
    Ans: $2(48+12+16)=152$ cm².

48. A parallelogram has base 12 cm, height 5 cm. Find its area.
    Ans: $60$ cm².

49. A trapezium has parallel sides 10 cm, 6 cm, height 4 cm. Find its area.
    Ans: $32$ cm².

50. A right triangle has sides 9 cm, 12 cm, 15 cm. Find its area.
    Ans: $54$ cm².

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Challenging & Conceptual Questions

51. If the area of a square is 81 cm², find its perimeter.
    Ans: Side = 9 cm, Perimeter = 36 cm.

52. If the volume of a sphere is 113.04 cm³, find its radius ($\pi =3.14$).
    Ans: $r=3$ cm.

53. A rectangle and a square have equal areas. If rectangle is 8 cm × 2 cm, what is the side of the square?
    Ans: $\sqrt{16}=4$ cm.

54. If the area of a circle is 314 cm², find its circumference.
    Ans: $r=10$ cm, Circumference = $62.8$ cm.

55. If the surface area of a cube is 150 cm², find its volume.
    Ans: Side = 5 cm, Volume = 125 cm³.

56. A cuboid has length 10 cm, breadth 7 cm, height 4 cm. Find its diagonal.
    Ans: $\sqrt{{10}^2+7^2+4^2}=\sqrt{165}\approx 12.85$ cm.

57. A cylinder has volume 616 cm³, height 14 cm. Find its radius.
    Ans: $r=3.75$ cm.

58. If the area of an equilateral triangle is 16$\sqrt{3}$ cm², find its side.
    Ans: $a=8$ cm.

59. If the diagonal of a square is 10 cm, find its area.
    Ans: $50$ cm².

60. If the curved surface area of a cone is 132 cm², slant height 12 cm, find its radius.
    Ans: $r=3.5$ cm.

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Short Calculation and Reasoning Questions

61. What is the perimeter of a rectangle with sides 6 cm and 8 cm?
    Ans: $28$ cm.

62. Find the area of a square with diagonal 6 cm.
    Ans: $18$ cm².

63. Find the volume of a cube with side 6 cm.
    Ans: $216$ cm³.

64. Find the area of a circle with radius 3.5 cm.
    Ans: $38.5$ cm².

65. Find the total surface area of a cuboid with l = 2 cm, b = 3 cm, h = 4 cm.
    Ans: $52$ cm².

66. Find the volume of a cone with r = 2 cm, h = 6 cm.
    Ans: $25.12$ cm³.

67. Find the area of a parallelogram with base 10 cm, height 4 cm.
    Ans: $40$ cm².

68. Find the area of a trapezium with parallel sides 8 cm, 12 cm, height 5 cm.
    Ans: $50$ cm².

69. Find the surface area of a sphere with radius 2 cm.
    Ans: $50.24$ cm².

70. Find the perimeter of a rhombus with side 5 cm.
    Ans: $20$ cm.

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Application & Word Problems

71. A garden is 20 m long and 15 m wide. Find its area and fencing required.
    Ans: Area = 300 m², Perimeter = 70 m.

72. A cylindrical water tank has radius 2 m, height 3 m. Find its capacity in liters.
    Ans: Volume = $37.7$ m³ = $37,700$ liters.

73. A cube has surface area 96 cm². Find its side and volume.
    Ans: Side = 4 cm, Volume = 64 cm³.

74. A cone has base radius 3 cm, height 4 cm. Find its volume and slant height.
    Ans: Volume = $37.7$ cm³, Slant height = 5 cm.

75. A circle has circumference 31.4 cm. Find its area.
    Ans: $r=5$ cm, Area = $78.5$ cm².

76. A sphere has volume 268.08 cm³. Find its radius.
    Ans: $r=4$ cm.

77. A cuboid has dimensions 5 cm × 4 cm × 3 cm. Find its surface area.
    Ans: $94$ cm².

78. A parallelogram has base 9 cm, height 6 cm. Find its area.
    Ans: $54$ cm².

79. A trapezium has parallel sides 7 cm, 13 cm, height 4 cm. Find its area.
    Ans: $40$ cm².

80. A right triangle has sides 8 cm, 15 cm, 17 cm. Find its area.
    Ans: $60$ cm².

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Mixed Conceptual and Calculation Questions

81. If the area of a square is 49 cm², what is its side and perimeter?
    Ans: Side = 7 cm, Perimeter = 28 cm.

82. If the volume of a sphere is 904.32 cm³, what is its radius?
    Ans: $r=6$ cm.

83. A rectangle and a square have equal areas. If rectangle is 16 cm × 2 cm, what is the side of the square?
    Ans: $4\sqrt{2}$ cm.

84. If the area of a circle is 201.06 cm², what is its circumference?
    Ans: $r=8$ cm, Circumference = $50.24$ cm.

85. If the surface area of a cube is 384 cm², what is its volume?
    Ans: Side = 8 cm, Volume = 512 cm³.

86. A cuboid has length 7 cm, breadth 6 cm, height 5 cm. Find its diagonal.
    Ans: $\sqrt{7^2+6^2+5^2}=\sqrt{110}\approx 10.49$ cm.

87. A cylinder has volume 314 cm³, height 10 cm. Find its radius.
    Ans: $r=3.16$ cm.

88. If the area of an equilateral triangle is 16$\sqrt{3}$ cm², what is its side?
    Ans: 8 cm.

89. If the diagonal of a square is 14 cm, what is its area?
    Ans: $98$ cm².

90. If the curved surface area of a cone is 132 cm², slant height 11 cm, what is its radius?
    Ans: $r=12$ cm.

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Short Calculation Questions

91. Perimeter of a rectangle with sides 7 cm and 9 cm?
    Ans: 32 cm.

92. Area of a square with diagonal 8 cm?
    Ans: 32 cm².

93. Volume of a cube with side 8 cm?
    Ans: 512 cm³.

94. Area of a circle with radius 7 cm?
    Ans: 153.86 cm².

95. Total surface area of a cuboid with l = 3 cm, b = 4 cm, h = 5 cm?
    Ans: 94 cm².

96. Volume of a cone with r = 3 cm, h = 9 cm?
    Ans: 84.78 cm³.

97. Area of a parallelogram with base 12 cm, height 8 cm?
    Ans: 96 cm².

98. Area of a trapezium with parallel sides 5 cm, 15 cm, height 6 cm?
    Ans: 60 cm².

99. Surface area of a sphere with radius 7 cm?
    Ans: 615.44 cm².

100. Perimeter of a rhombus with side 9 cm?
     Ans: 36 cm.

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References:

• Adda247 Mensuration Formulas1

• BYJU’S Mensuration and Geometry26

• CareerPower Mensuration Formula3

• Cuemath Geometry Formulas4

• Mindspark Mensuration Examples5

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Practice these questions and review the formulas to master Mensuration and Geometry for all competitive exams!