A right triangular prism has a base that is an equilateral triangle with side 6 cm. If the height (length) of the prism is 10 cm, what is the total surface area of the prism?

Step 1: Find the area of the equilateral triangular base. Area = (√3/4) × 6² = (√3/4) × 36 = 9√3 cm². Step 2: Find the perimeter of the triangular base. Perimeter = 3 × 6 = 18 cm. Step 3: Lateral surface area = Perimeter × Height = 18 × 10 = 180 cm². Step 4: Total surface area = Lateral surface area + 2 × Base area = 180 + 2 × 9√3 = 180 + 18√3 cm². Using √3 ≈ 1.732: 18 × 1.732 ≈ 31.18, but since exam expects exact form, Total SA = (180 + 18√3) cm². Therefore, the answer is (180 + 18√3) cm².

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IBPS PO Prism & Pyramid

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Concept Notes

Prism & Pyramid— Rules & Concept

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Core Concept

Read 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

✏️Examples

triangular pyramid (tetrahedron), square pyramid, pentagonal pyramid

💡KEY RULES FOR PRISM

• 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.

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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

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

IBPS PO Prism & Pyramid

Prism & Pyramid— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Prism & Pyramid — Practice Questions

60-Second Revision — Prism & Pyramid