1. Basic Concepts in Mensuration

Measurement

Measurement involves assigning numbers to physical quantities such as length, area, or volume. In mensuration, we use units like centimeters, meters, and kilometers for length, square centimeters or meters for area, and cubic units for volume.

Perimeter

The perimeter is the total distance around a closed figure. For example, the perimeter of a rectangle is the sum of all its four sides.

Area

Area measures the surface covered by a shape. It is measured in square units, such as cm² or m².

Volume

Volume is the amount of space occupied by a solid. It is measured in cubic units, such as cm³ or m³.

2. Perimeter of Plane Figures

Perimeter of a Rectangle

If a rectangle has length l and breadth b, then
Perimeter = 2(l + b)

Perimeter of a Square

For a square with side a,
Perimeter = 4a

Perimeter of a Triangle

If a triangle has sides a, b, and c,
Perimeter = a + b + c

Perimeter of a Circle (Circumference)

For a circle with radius r,
Circumference = 2πr
(π is approximately 3.14)

Example

A garden is shaped like a rectangle of 20 m by 15 m. The fencing needed equals the perimeter: 2(20 + 15) = 70 meters.

3. Area of Plane Figures

Area of a Rectangle

Area = length × breadth = l × b

Area of a Square

Area = side × side = a²

Area of a Triangle

Area = ½ × base × height

Area of a Parallelogram

Area = base × height

Area of a Circle

Area = πr²

Area of a Trapezium

Area = ½ × (sum of parallel sides) × height

Example

Find the area of a triangle with base 8 cm and height 5 cm.
Area = ½ × 8 × 5 = 20 cm²

4. Surface Area and Volume of Solids

Cuboid

• Surface Area = 2(lb + bh + hl), where l = length, b = breadth, h = height
• Volume = l × b × h

Cube

• Surface Area = 6a², where a = length of a side
• Volume = a³

Cylinder

• Curved Surface Area (CSA) = 2πrh
• Total Surface Area = 2πr(r + h)
• Volume = πr²h

Sphere

• Surface Area = 4πr²
• Volume = (4/3)πr³

Hemisphere

• Curved Surface Area = 2πr²
• Total Surface Area = 3πr²
• Volume = (2/3)πr³

Cone

• Curved Surface Area = πrl, where l = slant height
• Total Surface Area = πr(r + l)
• Volume = (1/3)πr²h

Example

A cuboid has length 5 m, breadth 3 m, and height 2 m.
Surface area = 2(5×3 + 3×2 + 2×5) = 2(15 + 6 + 10) = 2(31) = 62 m²
Volume = 5 × 3 × 2 = 30 m³

5. Conversions and Units

• 1 meter = 100 centimeters
• 1 square meter = 10,000 square centimeters
• 1 cubic meter = 1,000,000 cubic centimeters
• 1 liter = 1,000 cubic centimeters

Always convert all measurements to the same units before calculating area or volume.

6. Applications of Mensuration

• Finding the area of fields, rooms, and playgrounds
• Calculating the amount of paint required for walls
• Estimating the capacity of containers and tanks
• Working out the cost of flooring or fencing
• Designing everyday objects like boxes, bottles, and domes

Example Problem

A water tank is a cylinder of radius 1.5 m and height 4 m. What is its capacity?
Volume = πr²h = 3.14 × (1.5)² × 4 = 3.14 × 2.25 × 4 = 28.26 m³

7. Mensuration in Daily Life

Mensuration is not just for exams. It helps in real-life tasks, such as planning construction projects, buying carpets, packaging gifts, and determining how much soil is needed for a garden bed. Understanding these concepts allows you to make smart choices and avoid unnecessary expenses.

• If you want to buy a rug for a rectangular room, you will need to calculate the area.
• To build a fence around a park, you must work out the perimeter.
• For filling a swimming pool, you must calculate the volume.

8. Important Formulas and Tips

• For a rectangle: Perimeter = 2(l + b), Area = l × b
• For a square: Perimeter = 4a, Area = a²
• For a circle: Circumference = 2πr, Area = πr²
• For a triangle: Area = ½ × base × height
• For a parallelogram: Area = base × height
• For a trapezium: Area = ½ × (sum of parallel sides) × height
• For a cuboid: Surface area = 2(lb + bh + hl), Volume = l × b × h
• For a cylinder: Surface area = 2πr(r + h), Volume = πr²h
• For a sphere: Surface area = 4πr², Volume = (4/3)πr³
• For a cone: Surface area = πr(r + l), Volume = (1/3)πr²h

Remember to always use the same units and check your calculations.

9. Common Errors and How to Avoid Them

• Using different units in the same problem. Always convert to the same unit first.
• Forgetting to square or cube measurements when working with area or volume.
• Mixing up perimeter with area, or area with volume.
• Using an incorrect value for π. Use 3.14 or 22/7, depending on instructions.
• Not labeling answers with the correct units (e.g., m, m², or m³).

10. Practice Problems

Basic

1. Find the perimeter of a rectangle with length 18 cm and breadth 12 cm.
2. What is the area of a square with side 9 m?
3. Calculate the circumference of a circle with radius 7 cm.
4. A triangle has a base of 10 cm and height of 6 cm. Find its area.

Intermediate

5. Find the surface area and volume of a cube with side 4 cm.
6. A cylinder has radius 5 cm and height 10 cm. Find its curved surface area.
7. What is the area of a trapezium with parallel sides 8 cm and 14 cm and height 5 cm?
8. Find the total surface area and volume of a cuboid with dimensions 10 m, 8 m, and 6 m.

Advanced

9. A solid metallic sphere has radius 3 cm. It is melted and recast into a cylinder of height 4 cm. Find the radius of the cylinder.
10. A cone has base radius 3.5 cm and height 12 cm. Calculate its volume.
11. A hemispherical bowl has radius 10 cm. Find its capacity in liters.
12. The floor of a room is 5 m long and 4 m wide. If you want to tile it with square tiles of side 20 cm, how many tiles will you need?

11. Solutions

1. Perimeter = 2(18 + 12) = 60 cm
2. Area = 9 × 9 = 81 m²
3. Circumference = 2 × 3.14 × 7 = 43.96 cm
4. Area = ½ × 10 × 6 = 30 cm²
5. Surface area = 6 × 4 × 4 = 96 cm²; Volume = 4 × 4 × 4 = 64 cm³
6. Curved surface area = 2 × 3.14 × 5 × 10 = 314 cm²
7. Area = ½ × (8 + 14) × 5 = ½ × 22 × 5 = 55 cm²
8. Total surface area = 2(10×8 + 8×6 + 6×10) = 2(80 + 48 + 60) = 2 × 188 = 376 m²; Volume = 10 × 8 × 6 = 480 m³
9. Volume of sphere = (4/3) × 3.14 × 3³ = (4/3) × 3.14 × 27 = 113.04 cm³; Let cylinder radius be r, volume = πr²h = 3.14 × r² × 4 = 113.04 ⇒ r² = 113.04 / (3.14 × 4) = 9 ⇒ r = 3 cm
10. Volume = (1/3) × 3.14 × (3.5)² × 12 = (1/3) × 3.14 × 12.25 × 12 = (1/3) × 3.14 × 147 = (1/3) × 461.58 = 153.86 cm³
11. Capacity = (2/3) × 3.14 × 10³ = (2/3) × 3.14 × 1000 = (2/3) × 3140 = 2093.33 cm³; 1 liter = 1000 cm³, so capacity = 2.093 liters
12. Area of floor = 5 × 4 = 20 m² = 20,000 cm²; Area of one tile = 20 × 20 = 400 cm²; Number of tiles = 20,000 / 400 = 50 tiles

Conclusion

Mensuration helps us measure and understand the world around us. From calculating the area of a field to estimating the volume of a container, these concepts are practical and relevant. By practicing the formulas and solving different types of problems, you will gain both confidence and skill in this essential branch of mathematics.