Mesuration

Mensuration is the mathematics of measuring geometric figures-calculating their area, perimeter, and volume. This summary breaks down the key concepts, formulas, solved examples, and practice questions from the provided PDF, making it easy to understand and apply for exams.

1. What is Mensuration?

• Mensuration is the branch of mathematics that deals with the measurement of 2D (plane) and 3D (solid) shapes, including their area, perimeter, surface area, and volume¹².

2. Types of Shapes

• 2D shapes: Only length and breadth; measure area & perimeter.
• 3D shapes: Length, breadth, height/depth; measure volume & surface area¹².

3. Key Formulas Cheat Sheet

Triangles

• Perimeter: $ a + b + c $
• Area (General): $ \frac{1}{2} \times base \times height $
• Area (Heron’s Formula): $ \sqrt{s(s-a)(s-b)(s-c)} $, where $ s = \frac{a+b+c}{2} $
• Area (Equilateral): $ \frac{\sqrt{3}}{4}a^2 $
• Height (Equilateral): $ \frac{\sqrt{3}}{2}a $³

Quadrilaterals

• Square
  • Area: $ a^2 $
  • Perimeter: $ 4a $
  • Diagonal: $ a\sqrt{2} $
• Rectangle
  • Area: $ l \times b $
  • Perimeter: $ 2(l + b) $
  • Diagonal: $ \sqrt{l^2 + b^2} $³
• Parallelogram
  • Area: $ base \times height $
  • Perimeter: $ 2(a + b) $
• Rhombus
  • Area: $ \frac{d_1 \times d_2}{2} $
  • Perimeter: $ 4a $³
• Trapezium
  • Area: $ \frac{1}{2} \times (sum of parallel sides) \times height $
  • Perimeter: Sum of all sides

Circle

• Area: $ \pi r^2 $
• Circumference: $ 2\pi r $
• Diameter: $ 2r $³

3D Shapes

• Cube
  • Volume: $ a^3 $
  • Surface Area: $ 6a^2 $
  • Diagonal: $ a\sqrt{3} $
• Cuboid
  • Volume: $ l \times b \times h $
  • Surface Area: $ 2(lb + bh + hl) $
• Cylinder
  • Curved Surface Area: $ 2\pi rh $
  • Total Surface Area: $ 2\pi r(r + h) $
  • Volume: $ \pi r^2 h $
• Cone
  • Curved Surface Area: $ \pi r l $ (l = slant height)
  • Total Surface Area: $ \pi r(r + l) $
  • Volume: $ \frac{1}{3}\pi r^2 h $
• Sphere
  • Surface Area: $ 4\pi r^2 $
  • Volume: $ \frac{4}{3}\pi r^3 $³²

4. Solved Examples

Example 1: Triangle Area (Heron’s Formula)

Q: Sides are 12m, 13m, 11m. Find area and height with respect to side 12m.

• $ s = \frac{12+13+11}{2} = 18 $
• Area $ = \sqrt{18 \times 6 \times 5 \times 7} = 6\sqrt{105} $ m²
• Height $ = \frac{2 \times Area}{12} = \sqrt{105} $ m³

Example 2: Rectangle

Q: Area is 1352 m². Find perimeter and diagonal.

• Side $ = \sqrt{1352} = 26\sqrt{2} $ m
• Perimeter $ = 4 \times 26\sqrt{2} = 104\sqrt{2} $ m
• Diagonal $ = \sqrt{2} \times 26\sqrt{2} = 52 $ m³

Example 3: Volume of a Cylinder

Q: Radius is half the height, surface area is 616 cm². Find volume.

• Let radius = $ x $, height = $ 2x $
• $ 2\pi x(2x) + 2\pi x^2 = 616 $ ⇒ $ 6\pi x^2 = 616 $
• $ x^2 = \frac{616}{6\pi} $, find $ x $, then $ V = \pi x^2 \times 2x $³

5. Practice Questions

1. Three cubes with sides in ratio 3:4:5 are melted into one. If the diagonal of the new cube is $ 18\sqrt{3} $ cm, find the largest original edge.
2. Area of largest triangle inside a rectangle of length $ a $ and width $ b $?
3. Find perimeter and area of an isosceles triangle with equal sides 5 cm and height 4 cm.
4. A parallelogram of area $ A $ forms new parallelograms by joining midpoints of sides repeatedly. What is the sum of all areas?
5. A cylindrical container has radius half its height and inner surface area 616 cm². How much milk can it hold?
6. A rectangular sheet 22 cm × 12 cm is rolled into a cylinder along its length. Find the volume.
7. A hollow iron pipe is 21 cm long, outer diameter 8 cm, thickness 1 cm, iron weighs 8 g/cm³. Find the weight.
8. Water flows at 10 m/min from a 5 mm pipe. How long to fill a conical vessel with base diameter 30 cm and height 24 cm? (Answers and detailed solutions are in the PDF)³.

6. Tips & Tricks

• Always check units: Convert cm² to m², cm³ to liters, etc., as needed.
• For composite shapes: Break into known shapes, calculate separately, then add/subtract.
• Remember π ≈ 22/7 or 3.14 for calculations.
• For 3D shapes: Surface area = area covering the outside; Volume = space inside.

7. Quick Reference Table

8. Where to Practice More

• The PDF includes many solved and unsolved questions for practice³.
• For additional practice, use online resources like BYJU’S, Testbook, or Guidely for topic-wise quizzes and explanations⁴⁵².

Summary: Mensuration is about measuring shapes. Learn and memorize key formulas for area, perimeter, surface area, and volume. Practice with solved examples and attempt practice questions to master the topic for exams. Use the cheatsheet above for quick revision before tests!

[Sources: Mensuration.pdf, BYJU’S, Testbook, Guidely, NCERT]³¹⁴⁵⁶²