Mensuration Class 8 Notes — Formula Sheet & Chapter Revision

Understanding Mensuration: From 2D to 3D Shapes

Mensuration is the mathematical study of geometric measurements, including perimeter, area, and volume. In the early sections of this chapter, we focus on plane 2D figures. A plane figure is flat and bounded by lines or curves (e.g., trapeziums, rhombuses, and general quadrilaterals). For these shapes, we calculate the boundary distance (perimeter) and the surface enclosed (area).

As we transition to 3D space, we analyze solid shapes like cubes, cuboids, and cylinders. Solid shapes occupy space, which introduces two new concepts:

1. Surface Area: The sum of the areas of all external faces. This is divided into Lateral/Curved Surface Area (LSA/CSA) (excluding top and bottom faces) and Total Surface Area (TSA) (including all faces).
2. Volume: The total three-dimensional space enclosed inside a solid body.

Understanding the distinction between these terms and mastering unit conversion (e.g., changing cubic meters to liters) is crucial for securing perfect marks in your CBSE Class 8 exams.

Key Terms & Definitions

Perimeter

The continuous line forming the boundary of a closed geometric 2D shape.

Area

The amount of two-dimensional space enclosed inside a closed flat shape, measured in square units (e.g., m², cm²).

Lateral / Curved Surface Area

The total area of all the faces of a 3D solid excluding its top and bottom circular or flat bases.

Total Surface Area (TSA)

The sum of the areas of all the faces (including top and bottom bases) of a three-dimensional geometric shape.

Volume

The amount of three-dimensional space occupied by a solid object, measured in cubic units (e.g., m³, cm³).

Capacity

The volume of liquid or substance that a container can hold. While closely related to volume, capacity is typically measured in units like liters (L) or milliliters (mL).

Must-Remember Formulas (Formula Sheet)

• Area of a Trapezium: Area = 1/2 × (sum of parallel sides) × height = 1/2 × (a + b) × h
• Area of a Rhombus: Area = 1/2 × d1 × d2, where d1 and d2 are the lengths of the diagonals. (Also, diagonals of a rhombus are perpendicular bisectors of each other).
• Area of a General Quadrilateral: Area = 1/2 × diagonal × (h1 + h2), where h1 and h2 are the offsets drawn from opposite vertices to the diagonal.
• Total Surface Area (TSA) of a Cuboid: TSA = 2(lb + bh + hl)
• Lateral Surface Area (LSA) of a Cuboid: LSA = 2h(l + b) (Also known as the Area of Four Walls of a room).
• Total Surface Area of a Cube: TSA = 6a², where 'a' is the side length of the cube.
• Lateral Surface Area of a Cube: LSA = 4a²
• Curved Surface Area (CSA) of a Cylinder: CSA = 2πrh, where 'r' is the radius of the circular base and 'h' is the height.
• Total Surface Area (TSA) of a Cylinder: TSA = 2πr(r + h)
• Volume of 3D Solids: Volume of Cuboid = l × b × h | Volume of Cube = a³ | Volume of Cylinder = πr²h

3D Solids Quick Reference Comparison

Worked Revision Examples

• {"title":"Example 1: Finding Area of a Trapezium","description":"Find the area of a trapezium whose parallel sides are 12 cm and 8 cm, and the distance (height) between them is 6 cm.\n\nSolution:\n- Given: Parallel sides $a = 12$ cm, $b = 8$ cm, and height $h = 6$ cm.\n- Formula: Area = $1/2 \\times (a + b) \\times h$\n- Calculation: \n $\\text{Area} = \\frac{1}{2} \\times (12 + 8) \\times 6$\n $\\text{Area} = \\frac{1}{2} \\times 20 \\times 6 = 10 \\times 6 = 60\\text{ cm}^2$\n- Answer: The area of the trapezium is $60\\text{ cm}^2$."}
• {"title":"Example 2: Volume and Capacity of a Cylinder","description":"A cylindrical water tank has a radius of 1.4 m and a height of 2 m. Find its volume and capacity in Liters.\n\nSolution:\n- Given: $r = 1.4$ m, $h = 2$ m.\n- Volume Formula: $V = \\pi r^2 h$\n- Calculation:\n $V = \\frac{22}{7} \\times (1.4) \\times (1.4) \\times 2$\n $V = \\frac{22}{7} \\times 1.96 \\times 2 = 22 \\times 0.28 \\times 2 = 12.32\\text{ m}^3$\n- Unit Conversion: Since $1\\text{ m}^3 = 1000$ Liters:\n $\\text{Capacity} = 12.32 \\times 1000 = 12,320\\text{ Liters}$\n- Answer: The volume is $12.32\\text{ m}^3$ and the capacity is $12,320\\text{ Liters}$."}

Common Exam Traps & Unit Conversion Cues

• Watch the Units: This is where 90% of students lose marks. Never substitute values into a formula unless all dimensions are in the same unit (e.g., all in cm or all in m).
• Area vs. Volume Conversion: Remember that $1\text{ m}^2 = 10,000\text{ cm}^2$ (not 100!). Similarly, $1\text{ m}^3 = 1,000,000\text{ cm}^3$ (not 1,000!).
• Capacity Quick Rules: Keep these liquid volume rules memorized:
• $1\text{ cm}^3 = 1\text{ mL}$
• $1000\text{ cm}^3 = 1\text{ Liter}$
• $1\text{ m}^3 = 1000\text{ Liters}$
• Word Problem Trick: If an exam question asks for the 'cost of plastering/painting the walls of a room', you must find the Lateral Surface Area (LSA) of the cuboid, not the TSA, unless the ceiling or floor is explicitly included.

Quick Chapter Check

• Q1: If the diagonals of a rhombus are 10 cm and 24 cm, what is its area? A1: Area of a rhombus = 1/2 × d1 × d2. Here, Area = 1/2 × 10 × 24 = 120 cm².
• Q2: What is the formula for the 'Area of 4 walls' of a rectangular hall? A2: The area of 4 walls is equivalent to the Lateral Surface Area (LSA) of a cuboid, which is given by the formula: 2h(l + b).
• Q3: How many liters of water can a cylindrical container hold if its volume is 3.5 cubic meters? A3: Since 1 cubic meter (m³) = 1000 Liters, a container with a volume of 3.5 m³ can hold 3.5 × 1000 = 3500 Liters.
• Q4: If the side of a cube is doubled, how many times will its volume increase? A4: Volume of a cube = a³. If side becomes '2a', new volume = (2a)³ = 8a³. Thus, its volume will increase by 8 times.

Frequently Asked Questions