Prism, Pyramid, Combined and Cut Solids
Prism, Pyramid, Combined and Cut Solids
Prisms and pyramids generalise the cuboid and the cone: a prism is base area × height, a pyramid one-third of that. Combined solids (a cylinder topped by a hemisphere, a capsule) and solids with a piece cut away are added or subtracted part by part. NTPC 2025-26 papers asked a cylinder with a cone and a hemisphere attached and a cylinder with a sector-shaped piece cut out.
1. Formula Box
| Solid | Volume | Surface area |
|---|---|---|
| Right prism (base area B, height h, base perimeter p) | B × h | Lateral = p × h; total = p × h + 2B |
| Pyramid (base area B, height h, slant height s for a regular base) | (1/3) B × h | Lateral = (1/2) × base perimeter × s |
| Combined solid | Sum of the parts | Only the outer surfaces (hidden joins are left out) |
| Cylinder with a θ° sector cut out | (1 − θ/360) × πr²h |
Square pyramid with base side a and height h: slant height s = √(h² + (a/2)²)
Regular hexagon of side a: area = 6 × (√3/4)a²
✗ Surface area of a cylinder topped by a hemisphere = TSA of cylinder + TSA of hemisphere | ✓ The joined circle is inside: use CSA of both plus the bottom base only
✗ Pyramid volume = base area × height | ✓ It is one-third of that, like a cone
हिंदी नोट: प्रिज्म का आयतन = आधार का क्षेत्रफल × ऊँचाई और पिरामिड का उसका एक-तिहाई होता है। संयुक्त ठोस में आयतन जोड़िए, पर पृष्ठीय क्षेत्रफल में अंदर छिपी सतहें मत जोड़िए।
Exam Pointer: Verified NTPC patterns: a cylinder with a cone at one end and a hemisphere at the other (June 2025 Graduate CBT-1) and a solid cylinder with a sector-shaped piece removed (March 2026 Graduate CBT-1). Prism and pyramid questions had no verified NTPC shift and are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Prisms
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A prism has a right-triangular base with legs 6 cm and 8 cm and height 10 cm. Find its volume and lateral surface area.
Base area = 24 cm², volume = 240 cm³. Base perimeter = 6 + 8 + 10 = 24 cm, lateral area = 24 × 10 = 240 cm².
Answer: 240 cm³; 240 cm²
EXAMATLAS LEVEL
Q. A prism has a square base of side 5 cm and height 12 cm. Find its volume and total surface area.
Volume = 25 × 12 = 300 cm³. TSA = 2 × 25 + 4 × (5 × 12) = 50 + 240 = 290 cm².
Answer: 300 cm³; 290 cm²
Pattern 2: Pyramids
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A pyramid has a square base of side 10 cm and height 12 cm. Find its volume and lateral surface area.
Volume = (1/3) × 100 × 12 = 400 cm³. Slant height = √(144 + 25) = 13 cm, lateral area = 4 × (1/2) × 10 × 13 = 260 cm².
Answer: 400 cm³; 260 cm²
EXAMATLAS LEVEL
Q. A pyramid has a regular hexagonal base of side 4 cm and height 6 cm. Find its volume.
Base area = 6 × (√3/4) × 16 = 24√3 cm². Volume = (1/3) × 24√3 × 6 = 48√3 cm³ (about 83.1 cm³).
Answer: 48√3 cm³
Pattern 3: Combined solids
[PYQ: NTPC Graduate CBT-1 24-Jun-2025 Shift-2]
EXAM LEVEL
Q. A solid is a cylinder of radius 7 cm and height 10 cm with a hemisphere of the same radius on top. Find its volume.
Cylinder = 1,540 cm³; hemisphere = (2/3) × 22/7 × 343 = 718 2/3 cm³. Total = 2,258 2/3 cm³.
Answer: 2,258 2/3 cm³
EXAMATLAS LEVEL
Q. A capsule is made of a cylindrical part 14 cm long and 3.5 cm in radius, with a hemisphere at each end (total length 21 cm). Find its volume and surface area.
The two hemispheres make one sphere. Volume = 22/7 × 12.25 × 14 + (4/3) × 22/7 × 42.875 = 539 + 179 2/3 = 718 2/3 cm³. Surface = 2πrh + 4πr² = 308 + 154 = 462 cm².
Answer: 718 2/3 cm³; 462 cm²
Pattern 4: Solids with a part cut away
[PYQ: NTPC Graduate CBT-1 17-Mar-2026 Shift-3]
EXAM LEVEL
Q. A 90° sector-shaped piece is cut along the full height of a solid cylinder of radius 14 cm and height 10 cm. Find the volume of the remaining solid.
Full volume = 22/7 × 196 × 10 = 6,160 cm³. Remaining = (3/4) × 6,160 = 4,620 cm³.
Answer: 4,620 cm³
EXAMATLAS LEVEL
Q. The largest possible cone is carved out of a cube of edge 14 cm. Find the volume of the cone and the volume of wood wasted.
The cone has radius 7 cm and height 14 cm: (1/3) × 22/7 × 49 × 14 = 718 2/3 cm³. Cube = 2,744 cm³. Wasted = 2,025 1/3 cm³ (about 74% of the cube).
Answer: 718 2/3 cm³; 2,025 1/3 cm³
60-Second Revision
- Prism = base area × height; pyramid = one-third of that.
- Combined solids: add volumes; count only exposed surfaces.
- Sector cut from a cylinder: multiply by (1 − θ/360).
- The largest cone in a cube of edge a has r = a/2 and h = a.
Next Step: Mensuration 3D done. Attempt the mensuration questions in the ExamAtlas RRB NTPC 2026 mock tests; with π = 22/7 and radii in multiples of 7, most answers come out as whole numbers.