Core ConceptRead this first — the foundation of the topic
CORE CONCEPT
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
Exam PatternsWhat 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.
Worked ExampleSolve 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
Exam TrapsCommon mistakes students make — avoid these
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
10graded MCQs · easy to hard · full solution & trap analysis
A square prism (cuboid with square base) has base side 6 cm and height 10 cm. Find its lateral surface area.
Practice 2easy
A pyramid with rectangular base has length 8 cm, width 6 cm, and height 12 cm. What is its volume?
Practice 3easy
A rectangular prism has length 12 cm, width 8 cm, and height 5 cm. What is its total surface area?
Practice 4easy
A triangular pyramid has a base area of 24 cm² and height 9 cm. What is its volume?
Practice 5medium
A pyramid with a rectangular base of dimensions 8 m × 6 m has a height of 12 m. What is the volume of the pyramid?
Practice 6medium
A right circular cone has a base radius of 7 cm and a slant height of 25 cm. What is the curved surface area of the cone?
Practice 7medium
A rectangular prism has a length of 12 cm, width of 8 cm, and height of 5 cm. What is the total surface area of the prism?
Practice 8hard
A rectangular prism has dimensions 8 cm × 6 cm × 4 cm. A pyramid with the same rectangular base (8 cm × 6 cm) and height 12 cm is placed on top of the prism. What is the total volume of the combined solid in cm³?
Practice 9hard
A hexagonal prism has a regular hexagonal base with side length 5 cm and prism height 15 cm. What is the lateral surface area of the prism in cm²?
Practice 10hard
A right triangular prism has a triangular base with sides 13 cm, 14 cm, and 15 cm. The height of the prism is 20 cm. What is the total surface area of the prism in cm²?
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