Chapter 6: Mensuration: Perimeter, Area & Volume
क्षेत्रमिति: परिमाप, क्षेत्रफल एवं आयतन की संकल्पना
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- Perimeter vs Area Disconnection: Two figures can have the exact same perimeter but completely different areas (e.g. a rectangle of 1×5 has perimeter 12, area 5; a rectangle of 3×3 has perimeter 12, area 9). Among all rectangles with a fixed perimeter, a SQUARE encloses the MAXIMUM area.
- Rectangle & Square Formulas: Perimeter of rectangle = 2(l + b); Area = l × b. Perimeter of square = 4s; Area = s².
- Triangle & Parallelogram: Area of triangle = 1/2 × base × height. Area of parallelogram = base × height.
- Circle Formulas: Circumference = 2πr = πd; Area = πr².
- 3D Volume: Volume of Cuboid = l × b × h; Total Surface Area = 2(lb + bh + hl). Volume of Cube = a³. Volume of Cylinder = πr²h.
- Trap: Does increasing the perimeter always increase the area? NO. Cutting a corner square out of a large sheet increases perimeter while reducing area.
- Trap: Which shape encloses the maximum area for a given fixed perimeter? **A Square** (or a Circle among all 2D shapes).
NCERT Core Facts: Perimeter, Area & 3D Spatial Geometry
Mensuration in CTET Paper 1 & 2 bridges numerical calculations with conceptual spatial reasoning:
1. Perimeter-Area Relationships (The Classic CTET Conceptual Trap)
- Perimeter: Total length of the outer boundary enclosing a 2D figure.
- Area: Total surface region enclosed within the boundary.
- Key Pedagogical Principle:
- Having the same perimeter does NOT guarantee the same area.
- Example:
- Shape A: Rectangle 6\text{ cm} × 2\text{ cm} \implies \text{Perimeter} = 16\text{ cm}, \text{Area} = 12\text{ cm}^2.
- Shape B: Square 4\text{ cm} × 4\text{ cm} \implies \text{Perimeter} = 16\text{ cm}, \text{Area} = 16\text{ cm}^2.
- Theorem: For any given fixed perimeter, the square encloses the largest area among all rectangles.
2. Standard Mensuration Formulae Reference
| Shape | Perimeter / Circumference | Surface Area | Volume |
|---|---|---|---|
| Rectangle | 2(l + b) | l × b | — |
| Square | 4s | s^2 | — |
| Triangle | a + b + c | (1)/(2) × \text{base} × \text{height} | — |
| Circle | 2π r | π r^2 | — |
| Cuboid | 4(l + b + h) (edges) | 2(lb + bh + hl) | l × b × h |
| Cylinder | — | 2π r h + 2π r^2 | π r^2 h |
Test Your Mastery: Mensuration: Perimeter, Area & Volume
Authentic CBSE exam questions tested directly from this NCERT chapter.
A wire in the shape of a rectangle of length 20 cm and breadth 10 cm is rebent to form a square. What will be the area enclosed by the square?
एक तार जो 20 सेमी लंबाई और 10 सेमी चौड़ाई वाले आयत के आकार में है, को मोड़कर एक वर्ग बनाया जाता है। वर्ग द्वारा घेरा गया क्षेत्रफल क्या होगा?
Which of the following statements regarding perimeter and area is conceptually CORRECT?
परिमाप और क्षेत्रफल के संदर्भ में निम्नलिखित में से कौन-सा कथन सत्य है?
Can two different shapes have the same perimeter but different areas?
Yes. For instance, a 1×7 rectangle and a 4×4 square both have a perimeter of 16 cm, but their areas are 7 cm² and 16 cm² respectively.
Which rectangular shape encloses the greatest area for a given fixed perimeter?
A square.