Mensuration

Mensuration deals with measuring areas, volumes, surface areas, and perimeters of geometric figures.

2D Shapes - Area & Perimeter Formulas

  • Square: Area = a², Perimeter = 4a
  • Rectangle: Area = l × b, Perimeter = 2(l + b)
  • Triangle: Area = ½ × b × h
  • Equilateral Triangle: Area = (√3/4) × a²
  • Circle: Area = πr², Circumference = 2πr
  • Trapezium: Area = ½ × (a + b) × h

3D Shapes - Volume & Surface Area Formulas

  • Cuboid: Volume = l × b × h, TSA = 2(lb + bh + hl)
  • Cube: Volume = a³, TSA = 6a²
  • Cylinder: Volume = πr²h, CSA = 2πrh, TSA = 2πr(h + r)
  • Sphere: Volume = (4/3)πr³, Surface Area = 4πr²
  • Cone: Volume = (1/3)πr²h, TSA = πr(l + r)
  • Hemisphere: Volume = (2/3)πr³, TSA = 3πr²

Type 1: 2D Figures - Area & Perimeter

  1. Q1. SSC CGL 2021: Find the area of a square whose side is 14 cm.
    Solution: Area = a² = 14² = 196 cm²
  2. Q2. HSSC 2022: What is the perimeter of a rectangle with length 25 cm and breadth 15 cm?
    Solution: Perimeter = 2(l + b) = 2(25 + 15) = 80 cm
  3. Q3. RRB JE 2020: Find the area of an equilateral triangle with side 12 cm.
    Solution: Area = (√3/4) × a² = (1.732/4) × 144 ≈ 62.35 cm²
  4. Q4. SSC CHSL 2019: The base of a triangle is 10 cm and height is 15 cm. Find its area.
    Solution: Area = ½ × b × h = ½ × 10 × 15 = 75 cm²
  5. Q5. SSC MTS 2018: Find the circumference of a circle with radius 7 cm.
    Solution: C = 2πr = 2 × 3.14 × 7 = 43.96 cm

Type 2: Circle Related Questions

  1. Q6. SSC CGL 2019: The radius of a circle is doubled. How many times does the area increase?
    Solution: Area ∝ r², new area = 4 × original area. So, area increases 4 times.
  2. Q7. RRB NTPC 2021: Find the area of a circle with diameter 14 cm.
    Solution: Radius = 7 cm, Area = πr² = 3.14 × 7² = 153.86 cm²
  3. Q8. SSC GD 2020: Find the circumference of a semicircle with radius 10 cm.
    Solution: Circumference of semicircle = πr + 2r = 3.14 × 10 + 20 = 51.4 cm
  4. Q9. HSSC 2018: Find the area of the sector of a circle with radius 14 cm and angle 60°.
    Solution: Area of sector = (θ/360) × πr² = (60/360) × 3.14 × 14² ≈ 153.86 cm²
  5. Q10. SSC CGL 2017: Find the length of arc of a circle with radius 21 cm and angle 90°.
    Solution: Arc length = (θ/360) × 2πr = (90/360) × 2 × 3.14 × 21 = 33 cm

Type 3: Volume of 3D Shapes

  1. Q11. SSC CGL 2018: Find the volume of a cube with edge 5 cm.
    Solution: V = a³ = 5³ = 125 cm³
  2. Q12. RRB NTPC 2021: Calculate the volume of a cylinder with radius 7 cm and height 10 cm.
    Solution: V = πr²h = 3.14 × 49 × 10 = 1538.6 cm³
  3. Q13. SSC GD 2020: Find the volume of a cone with radius 3 cm and height 4 cm.
    Solution: V = (1/3)πr²h = (1/3) × 3.14 × 9 × 4 = 37.68 cm³
  4. Q14. SSC MTS 2019: Find the volume of a sphere with radius 7 cm.
    Solution: V = (4/3)πr³ = (4/3) × 3.14 × 343 = 1436.03 cm³
  5. Q15. HSSC 2022: Volume of hemisphere with radius 5 cm?
    Solution: V = (2/3)πr³ = (2/3) × 3.14 × 125 = 261.67 cm³

Type 4: Surface Area Questions

  1. Q16. SSC MTS 2019: Find the total surface area (TSA) of a cube with side 6 cm.
    Solution: TSA = 6a² = 6 × 36 = 216 cm²
  2. Q17. HSSC 2020: Find the curved surface area (CSA) of a cone with radius 5 cm and slant height 13 cm.
    Solution: CSA = πrl = 3.14 × 5 × 13 = 204.1 cm²
  3. Q18. SSC CHSL 2021: Find the surface area of a cylinder with radius 7 cm and height 10 cm.
    Solution: TSA = 2πr(h + r) = 2 × 3.14 × 7 × (10 + 7) = 747.6 cm²
  4. Q19. RRB JE 2019: Calculate the surface area of a sphere with radius 14 cm.
    Solution: Surface Area = 4πr² = 4 × 3.14 × 196 = 2461.76 cm²
  5. Q20. SSC GD 2022: Find the TSA of a cuboid with dimensions 5 m × 4 m × 3 m.
    Solution: TSA = 2(lb + bh + hl) = 2(20 + 12 + 15) = 2 × 47 = 94 m²

Type 5: Mixed Mensuration Problems

  1. Q21. SSC CGL 2021: A cone has radius 7 cm and height 24 cm. Find its volume.
    Solution: V = (1/3)πr²h = (1/3) × 3.14 × 49 × 24 = 1232.32 cm³
  2. Q22. SSC CHSL 2019: The length of a rectangle is twice its breadth. If the perimeter is 54 cm, find the area.
    Solution: Let breadth = x, length = 2x.
    Perimeter = 2(l + b) = 54 ⇒ 2(2x + x) = 54 ⇒ 6x = 54 ⇒ x = 9 cm.
    Area = l × b = 2x × x = 2 × 9 × 9 = 162 cm²
  3. Q23. RRB NTPC 2020: Find the radius of a circle whose area is 154 cm².
    Solution: Area = πr² ⇒ r² = 154/3.14 ≈ 49 ⇒ r = 7 cm.
  4. Q24. SSC MTS 2018: Find the height of a cylinder with volume 615.44 cm³ and radius 7 cm.
    Solution: V = πr²h ⇒ h = V/(πr²) = 615.44/(3.14 × 49) ≈ 4 cm.
  5. Q25. HSSC 2021: The base radius and height of a cone are 10 cm and 24 cm respectively. Find the slant height and surface area.
    Solution: Slant height, l = √(r² + h²) = √(100 + 576) = √676 = 26 cm.
    TSA = πr(l + r) = 3.14 × 10 × (26 + 10) = 3.14 × 10 × 36 = 1130.4 cm²

Additional Practice Questions

  1. Find the area of a trapezium with parallel sides 12 cm and 8 cm and height 7 cm.
  2. Calculate the volume of a cuboid with dimensions 10 cm, 6 cm, and 8 cm.
  3. A sphere has surface area 201.06 cm². Find its radius.
  4. Find the curved surface area of a cylinder with radius 14 cm and height 20 cm.
  5. A cube has surface area 384 cm². Find its edge length.

Answers for Additional Practice

  1. Area = ½ × (12 + 8) × 7 = 70 cm²
  2. Volume = 10 × 6 × 8 = 480 cm³
  3. Surface Area = 4πr² = 201.06 ⇒ r² = 201.06/(4 × 3.14) = 16 ⇒ r = 4 cm
  4. CSA = 2πrh = 2 × 3.14 × 14 × 20 = 1758.4 cm²
  5. TSA = 6a² = 384 ⇒ a² = 64 ⇒ a = 8 cm
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