Study Material — 2 PYQs (2018–2018) · Concept Notes · Shortcuts
SSC GD Constable Prism & Pyramid is a frequently tested subtopic — 2 previous year questions from 2018–2018 papers are included below with concept notes, key rules and shortcut tricks.
Prism: Imagine a 2D shape (circle, triangle, square) stretched straight up. The top and bottom are identical. All sides are rectangles or parallelograms
✏️Examples
cylinder (circular prism), triangular prism, cube
→Pyramid
Imagine a 2D shape as the base, then draw lines from every edge to a single point above. The sides are all triangles
• Volume = Base Area × Height
• Lateral Surface Area = Perimeter of Base × Height
• Total Surface Area = 2(Base Area) + Lateral Surface Area
• The bases are always parallel and congruent (identical)
KEY RULES FOR PYRAMID:
• Volume = (1/3) × Base Area × Height
• Lateral Surface Area = (1/2) × Perimeter of Base × Slant Height
• Total Surface Area = Base Area + Lateral Surface Area
• All lateral faces are triangles meeting at apex
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Exam Patterns
What examiners ask — read before attempting PYQs
📋SSC CGL typically asks
(1) Volume comparisons between prism and pyramid with same base and height. (2) Surface area of square pyramids or triangular prisms. (3) Height/slant height relationships. (4) Word problems combining area and volume
⚡SHORTCUT
For same base and height: Volume of Pyramid = (1/3) × Volume of Prism. This is THE most tested relationship. If a prism has volume 90 cm³, pyramid with same base and height has volume 30 cm³.
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Worked Example
Solve this step-by-step before moving on
A square prism has base side 4 cm and height 10 cm. Find total surface area.
Base Area = 4 × 4 = 16 cm²
Perimeter of Base = 4 × 4 = 16 cm
Lateral Surface Area = 16 × 10 = 160 cm²
Total Surface Area = 2(16) + 160 = 32 + 160 = 192 cm²
COMMON MISTAKE:
Students confuse slant height with perpendicular height in pyramids. Slant height is the distance along the triangular face from base to apex.
Perpendicular height is straight down from apex. Slant height is always longer. Use slant height ONLY for lateral surface area, not volume.
Key Points to Remember
Prism: Two identical parallel bases + rectangular sides. Volume = Base Area × Height
Pyramid: One base + triangular sides meeting at apex. Volume = (1/3) × Base Area × Height
Same base and height: Pyramid volume is always 1/3 of Prism volume (critical exam trick)
Lateral Surface Area of Prism = Perimeter × Height; for Pyramid = (1/2) × Perimeter × Slant Height
Slant height ≠ perpendicular height in pyramids; slant height is for surface area only
Total Surface Area = Base Area(s) + Lateral Surface Area for both shapes
Exam-Specific Tips
Volume of prism formula: V = Base Area × Height (where height is perpendicular distance between bases)
Volume of pyramid formula: V = (1/3) × Base Area × Height (always one-third of prism with same base and height)
Lateral Surface Area of prism: LSA = Perimeter of Base × Height
Lateral Surface Area of pyramid: LSA = (1/2) × Perimeter of Base × Slant Height
A square pyramid with base side a and slant height l has total surface area = a² + 2al
Relationship: If pyramid and prism share same base and height, then Pyramid Volume = (1/3) × Prism Volume
A right prism has bases perpendicular to its lateral edges; an oblique prism is slanted
A regular pyramid has a regular polygon base with apex directly above the center
Practice MCQs
Prism & Pyramid — Practice Questions
8graded MCQs · easy to hard · full solution & trap analysis
A rectangular prism has dimensions 5 cm × 4 cm × 3 cm. A smaller rectangular prism with dimensions 2 cm × 2 cm × 2 cm is removed from one corner. What is the volume of the remaining solid in cm³?
Practice 2medium
A triangular pyramid (tetrahedron) has a triangular base with area 24 cm² and a perpendicular height of 9 cm. What is its volume in cm³?
Practice 3medium
A hexagonal prism has a regular hexagonal base with side length 4 cm. The height of the prism is 10 cm. What is the lateral surface area of the prism in cm²?
Practice 4hard
A triangular prism has an equilateral triangular base with side 6 cm and height 10 cm. If the lateral surface area of the prism is 180 cm², what is the total surface area of the prism (in cm²)?
Practice 5hard
A rectangular prism has dimensions 12 cm × 8 cm × 6 cm. A pyramid with the same rectangular base (12 cm × 8 cm) and height 6 cm is placed inside it such that their bases coincide. What is the volume of the remaining space (in cm³)?
Practice 6hard
A hexagonal prism has a regular hexagonal base with side 4 cm and prism height 10 cm. What is the lateral surface area of the prism (in cm²)?
Practice 7hard
A pentagonal pyramid has a regular pentagonal base with side 5 cm. The slant height of the pyramid is 8 cm. What is the lateral surface area of the pyramid (in cm²)?
Practice 8hard
A triangular pyramid (tetrahedron) has an equilateral triangular base with side 6 cm and height 4 cm. Another similar pyramid is constructed with base side 12 cm. What is the ratio of the volume of the original pyramid to the volume of the larger pyramid?
60-Second Revision — Prism & Pyramid
Remember: Pyramid volume = (1/3) × Base Area × Height. Prism volume = Base Area × Height. Pyramid is always 1/3 of prism with same dimensions.
Formula: Lateral Surface Area of Prism = Perimeter × Height. For Pyramid = (1/2) × Perimeter × Slant Height.
Trap: Don't use perpendicular height for pyramid surface area—always use slant height for lateral area calculations.
Total Surface Area = All faces added together. For prism: 2 bases + lateral. For pyramid: 1 base + triangular sides.
Quick check: In same-base-height comparison, prism volume ÷ pyramid volume = 3:1 ratio always.
Identify the base shape first (triangle, square, pentagon), then apply correct perimeter and area formulas.
Slant height can be found using Pythagoras: slant height² = (height)² + (distance from center to base edge)²