Area Volume
Key Concepts
| # | Concept | Explanation |
|---|---|---|
| 1 | Circle | Area = πr², Circumference = 2πr; take π = 22/7 unless 3.14 is an option |
| 2 | Rectangle | Area = l × b, Perimeter = 2(l + b); diagonal = √(l² + b²) |
| 3 | Triangle | Area = ½ × base × height; Heron: √[s(s–a)(s–b)(s–c)], s = (a+b+c)/2 |
| 4 | Cuboid | Volume = l × b × h; Surface area = 2(lb + bh + lh); longest rod = √(l²+b²+h²) |
| 5 | Cube | Volume = a³, Surface area = 6a², space diagonal = a√3 |
| 6 | Cylinder | Volume = πr²h, Curved surface = 2πrh, Total surface = 2πr(r+h) |
| 7 | Sphere | Volume = 4/3 πr³, Surface = 4πr²; diameter = 2r |
| 8 | Cone | Volume = 1/3 πr²h, Slant height l = √(r²+h²), Curved surface = πrl |
15 Practice MCQs
1. A circular park has circumference 176 m. What is its area?
A) 2464 m² B) 1232 m² C) 616 m² D) 352 m² **Answer: A** 2πr = 176 ⇒ r = 28 m. Area = 22/7 × 28² = 2464 m². **Shortcut:** C → r = 176 × 7/44 = 28; A = 22/7 × 28². **Tag:** Circle2. The length of a rectangle is twice its breadth. If the perimeter is 126 cm, the area is?
A) 882 cm² B) 972 cm² C) 792 cm² D) 756 cm² **Answer: A** 2(2x + x) = 126 ⇒ x = 21; l = 42, b = 21; Area = 882 cm². **Shortcut:** 6x = 126 ⇒ x = 21; area = 2x² = 2 × 441 = 882. **Tag:** Rectangle3. Area of a right triangle is 600 cm² and one leg 30 cm. Find the hypotenuse.
A) 50 cm B) 60 cm C) 40 cm D) 70 cm **Answer: A** ½ × 30 × h = 600 ⇒ h = 40 cm; hypotenuse = √(30²+40²) = 50 cm. **Shortcut:** 3-4-5 triplet scaled by 10. **Tag:** Triangle4. What is the volume of a cube whose diagonal is 17.32 cm (≈10√3)?
A) 1000 cm³ B) 729 cm³ C) 800 cm³ D) 1331 cm³ **Answer: A** a√3 = 10√3 ⇒ a = 10; Volume = 10³ = 1000 cm³. **Shortcut:** Diagonal ÷ √3 = side. **Tag:** Cube5. A cubical tank of side 2 m is half-filled with water. How many litres of water are there?
A) 4000 B) 2000 C) 8000 D) 16000 **Answer: A** Volume = 2³ = 8 m³; half = 4 m³ = 4000 L. **Shortcut:** 1 m³ = 1000 L. **Tag:** Cube6. The curved surface area of a cylinder is 2200 cm² and height 35 cm. Find radius.
A) 10 cm B) 14 cm C) 7 cm D) 21 cm **Answer: A** 2πrh = 2200 ⇒ 2 × 22/7 × r × 35 = 2200 ⇒ r = 10 cm. **Shortcut:** r = CSA / (2πh) = 2200 / 220 = 10. **Tag:** Cylinder7. A sphere of radius 21 cm is melted and recast into a cylinder of radius 7 cm. Find the height of the cylinder.
A) 252 cm B) 126 cm C) 168 cm D) 84 cm **Answer: A** Volumes equal: 4/3 π(21)³ = π(7)²h ⇒ h = 252 cm. **Shortcut:** h = 4/3 × (21³)/(7²) = 4/3 × 9261 / 49 = 252. **Tag:** Sphere & Cylinder8. The slant height of a cone is 26 cm and radius 10 cm. Find its curved surface area.
A) 816.4 cm² B) 820 cm² C) 800 cm² D) 836 cm² **Answer: A** CSA = πrl = 22/7 × 10 × 26 = 816.4 cm². **Tag:** Cone9. A right cylindrical vessel 28 cm in diameter is partly filled with water. A sphere of radius 7 cm is dropped in. By how much does the water level rise?
A) 3.5 cm B) 4 cm C) 7 cm D) 14 cm **Answer: B** Volume of sphere = 4/3 π(7)³; rise h: π(14)²h = 4/3 π(7)³ ⇒ h = 4 cm. **Shortcut:** h = 4r / 3 = 4 × 7 / 3 ≈ 9.33 (here 4 cm). **Tag:** Sphere in Cylinder10. The perimeter of a square field is 176 m. What is the cost of levelling it at ₹15 per m²?
A) ₹29 040 B) ₹36 960 C) ₹18 480 D) ₹23 760 **Answer: A** Side = 176 / 4 = 44 m; area = 44² = 1936 m²; cost = 1936 × 15 = ₹29 040. **Tag:** Square11. A triangle has sides 13 cm, 14 cm, 15 cm. Its area is?
A) 84 cm² B) 91 cm² C) 72 cm² D) 105 cm² **Answer: A** s = 21; area = √[21×8×7×6] = √7056 = 84 cm². **Shortcut:** 13-14-15 is a well-known 84 cm² triangle. **Tag:** Heron12. The total surface area of a hemispherical bowl of radius 14 cm is (π = 22/7)?
A) 1848 cm² B) 1232 cm² C) 2464 cm² D) 2156 cm² **Answer: A** 3πr² = 3 × 22/7 × 14² = 1848 cm². **Tag:** Hemisphere13. If the diagonal of a rectangle is 26 cm and breadth 10 cm, its area is?
A) 240 cm² B) 120 cm² C) 260 cm² D) 180 cm² **Answer: A** l = √(26²–10²) = 24 cm; area = 24 × 10 = 240 cm². **Shortcut:** 5-12-13 triplet × 2. **Tag:** Rectangle14. A cone and a cylinder have the same radius 6 cm and height 7 cm. The ratio of their volumes is?
A) 1 : 3 B) 3 : 1 C) 1 : 1 D) 2 : 3 **Answer: A** Vol cone : Vol cylinder = 1/3 πr²h : πr²h = 1 : 3. **Tag:** Cone vs Cylinder15. A rectangular park 60 m × 40 m has a 5 m wide path inside along the border. Find the area of the path.
A) 900 m² B) 950 m² C) 850 m² D) 800 m² **Answer: A** Inner rectangle 50 × 30 = 1500 m²; path = 2400 – 1500 = 900 m². **Shortcut:** 2×5×(60+40–2×5) = 10×90 = 900. **Tag:** Path AreaSpeed Tricks
| Situation | Shortcut | Example |
|---|---|---|
| Circle from circumference | r = C / 6.28 (approx) | C = 88 m ⇒ r ≈ 14 m |
| Cube diagonal to side | a = diagonal / 1.732 | diagonal 17.32 ⇒ a = 10 |
| Cylinder volume from water rise | 1 L raises 1000 cm³; height = 1000 / base area | 1 L in 20 cm Ø vessel raises ≈ 3.18 cm |
| Sphere→cylinder height | h = 4r_sphere / 3 | 21 cm sphere ⇒ h = 28 cm when r_cyl = 21 cm |
| Path inside rectangle | Area_path = 2w(l + b – 2w) | 5 m path inside 60×40 park = 2×5×(60+40–10) = 900 m² |
Quick Revision
| Point | Detail |
|---|---|
| 1 | Always write units (m, cm, m², cm³) in answers—RRB often traps with unit-less choices. |
| 2 | Take π = 22/7 when radius is multiple of 7; else 3.14 if given in options. |
| 3 | Volume of any prism = base area × height; pyramid/cone = ⅓ × base area × height. |
| 4 | Longest rod in cuboid = √(l²+b²+h²); in cube = a√3. |
| 5 | For same base & height, Vol_cylinder : Vol_cone = 3 : 1. |
| 6 | Hemisphere CSA = 2πr²; TSA = 3πr² (includes base). |
| 7 | Water-rise problems: Volume of dropped object = πr²h_rise. |
| 8 | 1 m = 100 cm, 1 m² = 10 000 cm², 1 m³ = 10⁶ cm³ = 1000 L. |
| 9 | Right triangle triplets: (3,4,5), (5,12,13), (7,24,25), (8,15,17). |
| 10 | If options differ > 10 %, estimate π as 3 and eliminate nearest. |
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