Mensuration

Perimeter, Area and Volume of Plane Figures and Solid Shapes

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Overview

Mensuration is the branch of mathematics dealing with measurement of geometric figures — their lengths, areas and volumes. For UPTET, this topic carries significant weight in the Mathematics section of both Paper I (Classes 1–5) and Paper II (Classes 6–8). Questions typically test your ability to apply formulas quickly and accurately rather than derive them.

At the primary level, expect questions on perimeter and area of basic 2D shapes (rectangles, squares, triangles, circles). At the upper-primary level, the scope expands to surface area and volume of 3D solids (cuboids, cylinders, cones, spheres). Many questions are word problems requiring you to identify the correct formula, substitute values carefully and handle unit conversions.

Mastery here demands two things: memorising the standard formulas and practising enough problems to recognise which formula applies in a given context. This is a high-scoring area if you are systematic.

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Key Concepts

• **Perimeter** is the total length of the boundary of a 2D figure. Think of it as the length of fence needed to enclose a plot.

• **Area** measures the surface enclosed by a 2D figure. It tells you how much paint is needed to cover a wall or how much carpet to buy for a floor.

• **Volume** measures the space occupied by a 3D object. It tells you the capacity of a container or the amount of material in a solid block.

• **Surface Area** is the total area of all outer faces of a 3D solid. Curved Surface Area (CSA) excludes the flat bases; Total Surface Area (TSA) includes them.

• **Units matter**: Perimeter is in linear units (cm, m), area in square units (cm², m²), volume in cubic units (cm³, m³). Converting between units (e.g., m to cm) requires squaring or cubing the conversion factor for area and volume respectively.

• **Composite figures** are shapes formed by combining basic shapes. Break them into simpler parts, compute separately, then add or subtract as needed.

• **Practical applications** include finding the cost of fencing (perimeter), tiling (area), or filling a tank (volume).

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Formulas / Key Facts

### 2D Figures — Perimeter (P) and Area (A)

| Figure | Perimeter | Area | |--------|-----------|------| | Rectangle (l × b) | P = 2(l + b) | A = l × b | | Square (side a) | P = 4a | A = a² | | Triangle (sides a, b, c; base b, height h) | P = a + b + c | A = ½ × b × h | | Right Triangle (legs p, q) | P = p + q + hypotenuse | A = ½ × p × q | | Equilateral Triangle (side a) | P = 3a | A = (√3/4) × a² | | Circle (radius r) | Circumference = 2πr | A = πr² | | Semicircle | P = πr + 2r | A = ½ πr² | | Parallelogram (base b, height h) | P = 2(a + b) | A = b × h | | Rhombus (diagonals d₁, d₂) | P = 4 × side | A = ½ × d₁ × d₂ | | Trapezium (parallel sides a, b; height h) | Sum of all sides | A = ½ × (a + b) × h |