Mensuration: Notes और MCQ Practice

आसान भाषा में:

Mensuration वो भाग है जहाँ हम shapes की area, perimeter, surface area, volume निकालते हैं. याद रखो कि units हमेशा square या cubic में हों.

Important Formulas

• Square: Area = side × side, Perimeter = 4 × side
• Rectangle: Area = length × breadth, Perimeter = 2 × (length + breadth)
• Triangle: Area = ½ × base × height
• Circle: Area = π × r², Circumference = 2πr
• Cube: Surface Area = 6 × side², Volume = side³
• Cylinder: Curved Surface Area = 2πr × h, Total Surface Area = 2πr (r + h), Volume = πr² × h
• Cone: Curved Surface Area = πr × l, Total Surface Area = πr (r + l), Volume = ⅓ × πr² × h
• Hemisphere: Surface Area = 3πr², Volume = ⅔ × πr³

Quick Tricks

• अगर radius या side को square करना हो तो (a+b)² = a² + 2ab + b² shortcut use करो.
• Large numbers में π को 22/7 या 3.14 ले लो, और fraction को cross‑cancel करके जल्दी compute करो.

Common Mistakes

• Perimeter को area की तरह square units में लिखना.
• Cylinder/ cone में height और radius को swap कर देना, जिससे volume गलत निकलता है.

Concept

Mensuration is the branch of mathematics that deals with the measurement of geometric figures and their parameters such as length, area, volume and surface area. In SSC CGL, questions are usually based on simple shapes – squares, rectangles, triangles, circles, cubes, cylinders, cones and spheres. The key is to identify the shape, write the appropriate formula and substitute the given values.

Important formulas / rules / facts

• Square: Perimeter = 4 × side, Area = side²
• Rectangle: Perimeter = 2 × (length + breadth), Area = length × breadth
• Triangle: Perimeter = sum of three sides, Area = ½ × base × height
• Right‑angled triangle: Pythagoras theorem – a² + b² = c² (c = hypotenuse)
• Circle: Circumference = 2 × π × r, Area = π × r²
• Cylinder: Curved surface area = 2 × π × r × h, Total surface area = 2 × π × r (r + h), Volume = π × r² × h
• Cone: Curved surface area = π × r × l (l = slant height), Total surface area = π × r (r + l), Volume = ⅓ × π × r² × h
• Sphere: Surface area = 4 × π × r², Volume = ⅘ × π × r³ (or 4/3 × π × r³)
• Conversion: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm, 1 litre = 1000 cm³
• π value: Use 22/7 or 3.14 as per the question.

Solved Examples

Example 1

A rectangular garden measures 30 m by 20 m. A path of uniform width is to be built inside the garden along all four sides, reducing the usable area to 400 m². Find the width of the path.

Solution:

1. Let the width be x metres.
2. Usable length = 30 − 2x, usable breadth = 20 − 2x.
3. Usable area = (30 − 2x) × (20 − 2x) = 400.
4. Expand: 600 − 60x − 40x + 4x² = 400 → 4x² − 100x + 200 = 0.
5. Divide by 4: x² − 25x + 50 = 0.
6. Solve quadratic: x = [25 ± √(25² − 4×50)] / 2 = [25 ± √(625 − 200)] / 2 = [25 ± √425] / 2.
7. √425 ≈ 20.6 → x = (25 − 20.6)/2 ≈ 2.2 m (positive smaller root).

Answer: Width ≈ 2.2 m.

Example 2

The total surface area of a closed cylindrical can is 1500 cm² and its height is equal to its diameter. Find the radius of the can.

Solution:

1. For a closed cylinder, Total surface area = 2 × π × r (r + h).
2. Given h = 2r (diameter = 2r).
3. Substitute: 1500 = 2 × π × r (r + 2r) = 2 × π × r × 3r = 6πr².
4. r² = 1500 ÷ (6π) = 1500 ÷ (6 × 22/7) ≈ 1500 ÷ 18.857 ≈ 79.6.
5. r ≈ √79.6 ≈ 8.93 cm.

Answer: Radius ≈ 8.9 cm.

Example 3

A solid sphere of radius 7 cm is melted to form a solid cube. Find the side of the cube.

Solution:

1. Volume of sphere = 4/3 × π × r³ = 4/3 × 22/7 × 7³ = 4/3 × 22/7 × 343.
2. Simplify: 343 ÷ 7 = 49 → Volume = 4/3 × 22 × 49 = (4 × 22 × 49) / 3 = 4312 / 3 ≈ 1437.33 cm³.
3. Let side of cube = a. Volume of cube = a³ = 1437.33.
4. a = ∛1437.33 ≈ 11.2 cm.

Answer: Side ≈ 11.2 cm.

Example 4

A ladder 12 m long leans against a wall making an angle of 60° with the ground. Find the height reached on the wall.

Solution:

Height = ladder × sin θ = 12 × sin60° = 12 × √3/2 = 6√3 ≈ 10.39 m.

Answer: Height ≈ 10.4 m.

Shortcuts and Tricks

• Quick π: For most SSC problems, use π = 22/7. It gives exact fractions and avoids decimal rounding.
• Area of triangle using sides: When only three sides are given, use Heron’s formula – s = (a+b+c)/2, Area = √[s(s‑a)(s‑b)(s‑c)].
• Common ratio: For a cylinder where height = diameter, total surface area simplifies to 6πr² – remember this pattern.
• Volume‑area conversion: If a solid is melted to form another shape, equate volumes directly; no need to compute intermediate mass.
• Square‑root shortcuts: For numbers close to a perfect square, use (√N ≈ √A + (N‑A)/(2√A)). Helpful for quick cube‑root approximations.

Common Mistakes to Avoid

1. Mixing radius and diameter – always check whether the formula needs r or 2r.
2. For closed cylinders, forgetting the top and bottom circles – total surface area includes both.
3. Using π = 3.14 when the answer expects a fraction – stick to 22/7 unless the question explicitly says otherwise.
4. Incorrect unit conversion – convert all dimensions to the same unit before applying formulas.
5. Skipping the slant height in cones – remember l = √(r² + h²) before using surface‑area formula.