Mensuration (2D & 3D)

Introduction to Mensuration

The measuring of geometric figures, including length, area, volume, and perimeter of different 2D and 3D shapes, is the focus of the mathematical field of mensuration.

2D figures include: Squares, Rectangles, Triangles, Circles, Parallelograms, Trapeziums, Rhombuses, and more.
3D figures include: Cube, Cuboid, Sphere, Cylinder, Cone, Hemisphere, and so on.

Important Mensuration Terms and Definitions

Term	Abbreviation	Unit	Meaning
Area	A	cm² or m²	The amount of surface covered by a two-dimensional shape.
Perimeter	P	cm or m	The total length around the boundary of a closed figure.
Volume	V	cm³ or m³	The space that a three-dimensional object takes up.
Curved Surface Area	CSA	cm² or m²	The area of only the curved part of a 3D object (e.g., the side of a cylinder).
Lateral Surface Area	LSA	cm² or m²	The area of all the sides of a 3D object, excluding the top and bottom surfaces.
Total Surface Area	TSA	cm² or m²	The total area of all the surfaces (flat and curved) of a 3D shape.
Square Unit	–	cm² or m²	A unit used to measure area—represents a square with a side length of 1 unit.
Cubic Unit	–	cm³ or m³	A unit used to measure volume—represents a cube with a side length of 1 unit.

3. Mensuration Formulas

2D Figures

✅ Square

Perimeter = 4 × side
Area = side²

✅ Rectangle

Perimeter = 2 × (length + breadth)
Area = length × breadth

✅ Triangle

Perimeter = a + b + c
Area = (1/2) × base × height
Heron’s Formula:
Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c) / 2

✅ Isosceles Triangle

Area = (b / 4) × √(4a² − b²)
(Where a = equal side, b = base)

Note: Understanding the Formulas

The formula:

\text{Area} = \frac{b}{4} \times \sqrt{4a^2 - b^2}

is derived by first calculating the height (h) of the isosceles triangle using the Pythagorean theorem:

h = \sqrt{a^2 - \frac{b^2}{4}}

Then, applying the standard area formula for a triangle:

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

Substituting the expression for height:

\text{Area} = \frac{1}{2} \times b \times \sqrt{a^2 - \frac{b^2}{4}}

✅ Right-Angled Triangle

Area = (1/2) × base × height
Hypotenuse² = base² + height² (Pythagoras Theorem)

✅ Equilateral Triangle

Area = (√3 / 4) × side²

✅ Parallelogram

Area = base × height
Perimeter = 2 × (a + b)

✅ Trapezium

Area = (1/2) × (sum of parallel sides) × height

✅ Rhombus

Area = (1/2) × (diagonal₁ × diagonal₂)
Perimeter = 4 × side

✅ Circle

Circumference = 2πr
Area = πr² (π ≈ 3.1416 or 22/7)

3D Figures

✅ Cube

Volume = a³
Surface Area = 6a²
Diagonal = a√3

✅ Cuboid

Volume = l × b × h
Surface Area = 2(lb + bh + hl)
Diagonal = √(l² + b² + h²)

✅ Cylinder

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

✅ Cone

Volume = (1/3)πr²h
CSA = πrl, where l = √(r² + h²)
TSA = πr(l + r)

✅ Sphere

Volume = (4/3)πr³
Surface Area = 4πr²

✅ Hemisphere

Volume = (2/3)πr³
CSA = 2πr²
TSA = 3πr²

4. Key Concepts

Length: Measured in cm, m, or km
Area: Measured in cm², m², etc.
Volume: Measured in cm³, m³, etc.
Convert all units to the same system before applying formulas
Understand the difference between flat (2D) and solid (3D) figures
Diagonals, slant height, and height are critical for accurate calculations

5. Practice MCQs with Answers

Q1. What is the area of a circle with radius 7 cm?

A. 154 cm²
B. 44 cm²
C. 49 cm²
D. 22 cm²
✅ Answer: A
Explanation: Area = πr² = (22/7) × 7 × 7 = 154 cm²

Q2. Volume of a cube of side 5 cm?

A. 25 cm³
B. 125 cm³
C. 150 cm³
D. 100 cm³
✅ Answer: B
Explanation: Volume = a³ = 5³ = 125 cm³

Q3. What is the total surface area of a cuboid of dimensions 2 cm × 3 cm × 4 cm?

A. 52 cm²
B. 94 cm²
C. 88 cm²
D. 76 cm²
✅ Answer: C
Explanation: TSA = 2(lb + bh + hl) = 2(6 + 12 + 8) = 2 × 26 = 52 cm²

Q4. A cylinder has radius 3 cm and height 5 cm. Find its volume.

A. 141.3 cm³
B. 145 cm³
C. 180 cm³
D. 90 cm³
✅ Answer: A

Explanation: Volume = πr²h = 3.14 × 3 × 3 × 5 = 141.3 cm³

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