⭐ Sabse Important Points · एक नज़र में
- Triangle Area: Area = ½ × base × height; also Heron's formula √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2.
- Cube & Cuboid: Cube volume = a³, TSA = 6a², diagonal = a√3; Cuboid volume = l×b×h, TSA = 2(lb+bh+hl), diagonal = √(l²+b²+h²).
- Circle: Area = πr², Circumference = 2πr; use π ≈ 22/7 for quick calculation unless specified otherwise.
- Cylinder: Volume = πr²h, CSA = 2πrh, TSA = 2πr(r+h).
- Rectangle & Square: Rectangle area = l×b, perimeter = 2(l+b); Square area = a², perimeter = 4a, diagonal = a√2.
- Cone: Volume = ⅓πr²h, CSA = πrl, TSA = πr(l+r), where slant height l = √(r²+h²).
Introduction · परिचय
Mensuration deals with the measurement of geometric figures—lengths, areas, volumes, and surface areas. For SSC CHSL, focus on standard 2D shapes (triangle, rectangle, square, circle, parallelogram, trapezium, rhombus) and 3D solids (cube, cuboid, cylinder, cone, sphere, hemisphere). Questions test direct formula application, unit conversions, and occasionally combinations of shapes.
2D Shapes: Area and Perimeter
Area measures the region enclosed by a 2D shape; perimeter is the total boundary length. For composite figures, split them into standard shapes, compute each part, and add or subtract as needed. Always keep units consistent (convert cm to m before calculating if required).
📐 Formula / Rule Box · सूत्र याद रखो
Triangle (Heron's)
Area = √[s(s−a)(s−b)(s−c)], s = (a+b+c)/2
Circle
Area = πr²; Circumference = 2πr
Rhombus
Area = ½ × d₁ × d₂
Trapezium
Area = ½ × (a+b) × h
| Shape | Area Formula | Perimeter Formula |
|---|---|---|
| Rectangle | l × b | 2(l + b) |
| Square | a² | 4a |
| Circle | πr² | 2πr |
| Parallelogram | b × h | 2(a + b) |
⭐ Key Points · याद रखने वाली बातें
- Triangle Area: Area = ½ × base × height; also Heron's formula √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2.
- Circle: Area = πr², Circumference = 2πr; use π ≈ 22/7 for quick calculation unless specified otherwise.
- Rectangle & Square: Rectangle area = l×b, perimeter = 2(l+b); Square area = a², perimeter = 4a, diagonal = a√2.
- Parallelogram & Rhombus: Parallelogram area = base × height; Rhombus area = ½ × d₁ × d₂ (diagonals), perimeter = 4 × side.
- Trapezium: Area = ½ × (sum of parallel sides) × height.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Find the area of a triangle with sides 13 cm, 14 cm, and 15 cm.
✅ Solution: 84 cm²
💡 Explanation: s = (13+14+15)/2 = 21. Area = √[21×8×7×6] = √7056 = 84 cm².
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: The perimeter of a square is 48 cm. Find its diagonal.
✅ Solution: 12√2 cm
💡 Explanation: Side a = 48/4 = 12 cm. Diagonal = a√2 = 12√2 cm.
3D Solids: Volume and Surface Area
Volume is the space occupied (cubic units); surface area is the total outer area (square units). For solids, distinguish between total surface area (TSA) and curved/lateral surface area (CSA). For combined solids (e.g., cylinder with hemispherical ends), add volumes and surface areas carefully, excluding hidden faces.
📐 Formula / Rule Box · सूत्र याद रखो
Cylinder
Volume = πr²h; CSA = 2πrh; TSA = 2πr(r+h)
Cone
Volume = ⅓πr²h; CSA = πrl; TSA = πr(l+r); l = √(r²+h²)
Sphere
Volume = (4/3)πr³; Surface Area = 4πr²
Hemisphere
Volume = (2/3)πr³; CSA = 2πr²; TSA = 3πr²
| Solid | Volume | Curved Surface Area | Total Surface Area |
|---|---|---|---|
| Cylinder | πr²h | 2πrh | 2πr(r+h) |
| Cone | ⅓πr²h | πrl | πr(l+r) |
| Sphere | (4/3)πr³ | 4πr² | 4πr² |
| Hemisphere | (2/3)πr³ | 2πr² | 3πr² |
⭐ Key Points · याद रखने वाली बातें
- Cube & Cuboid: Cube volume = a³, TSA = 6a², diagonal = a√3; Cuboid volume = l×b×h, TSA = 2(lb+bh+hl), diagonal = √(l²+b²+h²).
- Cylinder: Volume = πr²h, CSA = 2πrh, TSA = 2πr(r+h).
- Cone: Volume = ⅓πr²h, CSA = πrl, TSA = πr(l+r), where slant height l = √(r²+h²).
- Sphere & Hemisphere: Sphere volume = (4/3)πr³, surface area = 4πr²; Hemisphere volume = (2/3)πr³, CSA = 2πr², TSA = 3πr².
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: A cylinder has radius 7 cm and height 10 cm. Find its volume and total surface area. (Use π = 22/7)
✅ Solution: Volume = 1540 cm³; TSA = 748 cm²
💡 Explanation: Volume = πr²h = (22/7)×49×10 = 1540 cm³. TSA = 2πr(r+h) = 2×(22/7)×7×(7+10) = 44×17 = 748 cm².
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: Find the volume of a cone with radius 3 cm and height 4 cm.
✅ Solution: 12π cm³ (≈ 37.7 cm³)
💡 Explanation: Volume = ⅓πr²h = ⅓×π×9×4 = 12π cm³.
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1The radius of a circle is doubled. By what factor does its area increase?
👀 Answer dekho
✅ Answer: 4 times
💡 Explanation: Area = πr². If radius becomes 2r, new area = π(2r)² = 4πr², which is 4 times the original area.
Q2A cuboid has dimensions 5 cm × 4 cm × 3 cm. Find its total surface area.
👀 Answer dekho
✅ Answer: 94 cm²
💡 Explanation: TSA = 2(lb + bh + hl) = 2(5×4 + 4×3 + 3×5) = 2(20 + 12 + 15) = 2×47 = 94 cm².
Q3The curved surface area of a cylinder is 440 cm² and its height is 10 cm. Find its radius. (Use π = 22/7)
👀 Answer dekho
✅ Answer: 7 cm
💡 Explanation: CSA = 2πrh = 440 ⇒ 2×(22/7)×r×10 = 440 ⇒ (440/7)r = 440 ⇒ r = 7 cm.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Confusing TSA and CSA: For a cylinder, TSA includes the two circular bases (2πr²) while CSA does not; read the question to know which is required.
- Forgetting to convert units: If dimensions are given in different units (e.g., cm and m), convert all to the same unit before applying formulas.
- Using diameter instead of radius: Many questions give diameter; always halve it before substituting into formulas.
- In Heron's formula, miscomputing semi-perimeter: s = (a+b+c)/2, not a+b+c.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- Circle: Area = πr², Circumference = 2πr; use π ≈ 22/7 for quick calculations.
- Cylinder: Volume = πr²h, CSA = 2πrh, TSA = 2πr(r+h).
- Cone: Volume = ⅓πr²h, CSA = πrl, TSA = πr(l+r), l = √(r²+h²).
- Sphere: Volume = (4/3)πr³, Surface Area = 4πr²; Hemisphere TSA = 3πr².
- Cube: Volume = a³, TSA = 6a², diagonal = a√3; Cuboid diagonal = √(l²+b²+h²).