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
-
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: Circle -
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: Rectangle -
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: Triangle -
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: Cube -
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: Cube -
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: Cylinder -
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 & Cylinder -
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: Cone -
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 Cylinder -
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: Square -
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: Heron -
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: Hemisphere -
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: Rectangle -
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 Cylinder -
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 Area
Speed 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. |
BETA
