The Largest Solid Inside a Cube Is Fixed by Its Side

Duration: 10 min

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Module outline

  1. About Course: About the Course
  2. Paper 1 | Quantitative Aptitude: Speed Mathematics and Basic Mathematics, Percentage, Ratio and Proportion (Ratios), Powers and Exponents (Surds and Indices), Logarithm, Sequence and Series (Series), Mensuration and Geometry, Data Interpretation, Average, Arithmetic
  3. Paper 1 | Logical Reasoning: Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Series (Number and Letter Series) (Numerical Relations and Reasoning), Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Analogy, Cubes & Dices, Syllogisms
  4. Paper 1 | English Language: Vocabulary, Noun, Pronoun, Verb, Adjectives, Adverb, Preposition, Conjunction, Interjection, Articles, Tenses, Subject Verb Agreement (Verb Noun Agreement), Modals, Determiner, Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks), Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Sentence Re-arrangements (Para Jumbles) (Narrative Sequencing), Sentence Construction
  5. Paper 1 | Unit Tests:
  6. Paper 1 | Full-Length Mock Tests:
  7. Paper 2 | Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus, Permutation and Combination, Probability, Statistics
  8. Paper 2 | Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation
  9. Paper 2 | Computer Architecture: Basics of COA, Floating Point Rep, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design
  10. Paper 2 | Programming and Data Structures: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing
  11. Paper 2 | Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  12. Paper 2 | Theory Of Computation/Automata Theory: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Complexity Theory
  13. Paper 2 | Compiler Design: Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
  14. Paper 2 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management
  15. Paper 2 | DataBase Management System/DBMS: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control, Database Recovery
  16. Paper 2 | Computer Networks: Introduction to CN, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware Basics
  17. Paper 2 | CS/IT Sectional Mock Tests:
  18. Complete Paper Full Mock Test:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This lecture explains how to determine the largest sphere, cylinder, and cone that can fit inside a cube of side length 'a'. The instructor establishes that the diameter of the largest sphere equals the side of the cube, not its diagonal. For all three solids, the radius is half the side length (r = a/2) and their height equals the side length (h = a). The volume formulas are derived as V_sphere = πa³/6, V_cylinder = πa³/4, and V_cone = πa³/12. A key teaching point is the 2:3:1 ratio of these volumes, which can be found by comparing denominators when numerators are identical. The lecture then applies this concept to a worked example: finding the volume of the largest sphere carved from a wooden cube with side length 21 cm. Using π = 22/7, the radius is calculated as 10.5 cm, and the volume formula V = πa³/6 is applied. The final section transitions to a different problem involving an inverted pyramid submerged in water, where the volume of displaced water equals the pyramid's volume. The instructor calculates the height H ≈ 20.14 cm by equating 27√3H = 300π, demonstrating the application of volume principles to displacement problems.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The lecture introduces the concept of finding the largest sphere, cylinder, and cone inside a cube. The instructor highlights that for all three solids, the radius is half the side length (r = a/2) and height equals the side length (h = a). The volume formulas are presented: V_sphere = πa³/6, V_cylinder = πa³/4, and V_cone = πa³/12. A common trap is emphasized: the diameter of the largest sphere equals the cube's side, not its diagonal. The 2:3:1 volume ratio is introduced by comparing denominators when numerators are identical.

  2. 2:00 – 5:00 02:00-05:00

    The instructor continues explaining the volume calculations for the largest sphere, cylinder, and cone inside a cube. The formulas are reinforced with visual aids showing how each solid fits within the cube's boundaries. The 2:3:1 ratio is demonstrated through simplified fractions, making it clear how to compare volumes efficiently. The teaching emphasizes using the cube's side length to determine inscribed solid dimensions, avoiding the common mistake of confusing side length with diagonal measurements.

  3. 5:00 – 9:54 05:00-09:54

    A worked example is presented: calculating the volume of the largest sphere carved from a wooden cube with side length 21 cm. The instructor underlines key terms and circles the side length, noting that diameter equals side (a = 21), so radius is 10.5 cm. Using π = 22/7, the volume formula V = πa³/6 is applied. The lecture then transitions to a Hindi word problem about an inverted pyramid submerged in water, where displaced water volume equals pyramid volume. Red annotations show calculations: dh = 28 - 25 = 3cm, displaced water volume = π(10)² × 3 = 300π cm³, base area = √3/4 × 18 × 18 = 81√3 cm², and pyramid volume V_py = 1/3 AH = 27√3H. Equating 27√3H = 300π yields H ≈ 20.14 cm, marking option (ग) as correct.

The lecture systematically builds understanding of inscribed solids in cubes, starting with theoretical foundations and progressing to practical applications. The core principle is that the cube's side length determines all dimensions of inscribed solids, with radius always being half the side and height equal to the side. The 2:3:1 volume ratio provides a quick comparison tool for exam settings. The worked example demonstrates real-world application using specific measurements and standard approximations (π = 22/7). The final displacement problem extends volume concepts to fluid mechanics, showing how geometric principles apply across different contexts. Students should focus on recognizing the side-diameter relationship for spheres, avoiding diagonal confusion, and mastering volume ratio comparisons for efficient problem-solving.

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