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. Section A: General Awareness: Polity, Physics, Biology, Chemistry, Static General Knowledge, History, Geography, Indian Economy, GK- Current Affairs, Evolution of Indian Banking (Banking & Financial Awareness), Environment
  3. Section A: General Intelligence & Reasoning ability: Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Data Sufficiency, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Analytical Reasoning (Counting Figures Reasoning), Blood Relations, Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Seating Arrangements, Directions (Direction Test), Puzzle Solving (Puzzles), Calendar, Clock, Cubes & Dices, Selection Decision Table (Decision Making), Logical Sequence Of Words, Crypt Arithmetic, Venn Diagram, Data Arrangement, Inequality, Analogy, Ranking, Order and Sequence, TCS NQT Advanced Reasoning Ability
  4. Section A: Arithmetical & Numerical Ability: Ratio and Proportion (Ratios), Divisibility Rules, Data Interpretation, Mixture and Alligation, Permutation and Combination, Logarithm, Number System, HCF LCM, Sequence and Series (Series), Speed Time and Distance, Percentage, Average, Profit and Loss, Simple Interest and Compound Interest, Time and Work, Pipe and Cistern, Mensuration and Geometry, Powers and Exponents (Surds and Indices), Algebra, Probability, Mental Ability, Age Problems, Partnership, Statistics, Arithmetic, Speed Mathematics and Basic Mathematics, Miscellaneous
  5. Section A: English Language & Comprehension: Vocabulary, Subject Verb Agreement (Verb Noun Agreement), Adjectives, Tenses, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Articles, Sentence Construction, Preposition, Direct and Indirect Speech, Sentence Completion (Fill in the blanks), Sentence Re-arrangements (Para Jumbles) (Narrative Sequencing), Modals, Active Passive Voice (Voice), Sentence Correction (Error Correction), Adverb, Conjunction, Cloze Test (Cloze Passage)
  6. Section A: Hindi Language & Comprehension: भाषा, लिपि, ध्वनि, वर्ण, अक्षर, स्वर, व्यंजन, मात्रा और संयुक्ताक्षर, वर्तनी, वर्ण-विन्यास, अनुस्वार, अनुनासिक, चंद्रबिंदु और ‘र’ के रूप, शब्द, पद और शब्द-भेद, संज्ञा और सर्वनाम, विशेषण और क्रिया, लिंग, वचन, पुरुष और कारक, काल, वाच्य, वाक्य और वाक्य-भेद, संधि, समास, विलोम शब्द, पर्यायवाची शब्द, सामान्य अशुद्धियाँ, अनेक शब्दों के लिए एक शब्द, मुहावरे और लोकोक्तियाँ, अपठित गद्यांश और अपठित पद्यांश, Multi-Topic Content (Cross-Concept Material), वर्णमाला
  7. Section B: Programming, Data Structures, OOP & Java: 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, Java, OOPS with C++, Language Design, Intro to Compilers, Visual Basics, Syntax Analysis: Bottom-Up, Object Oriented Technology
  8. Section B: Computer Networks, TCP/IP & Network Security: Introduction to CN, Data Communication, DLL: Flow Control, DLL: Access Control, DLL: Error Control, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware basics, Network Access Technologies, Network Security, Society, Law & Ethics, Mobile Computing, Net Layer:Routing Protocol, Cyber Safety and Security
  9. Section B: Teaching Methodology: Child Development, Concept of Pedagogy, Theories, Inclusive Education, Experiential Pedagogy, Assessment & Evaluation, Curriculum Framework & Reforms, National Education Policy (NEP) 2020, Psychology, Teaching-Learning Process, Personality Development, Teaching Methods & Strategies, Learning and Motivation, Learners & Learning Process, Factors Affecting Teaching, Methods of Teaching, Nature, Objectives & Characteristics of Teaching, Teaching-Learning Aids & ICT Integration, Classroom Management & Communication, Evaluation, Assessment & Measurement, Curriculum & School Environment, Inclusive Education & Special Needs, Professional Development & Ethics
  10. Section B: Digital Electronics / Boolean Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Analog-Digital Conversion
  11. Section B: Operating Systems & Linux: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, File Management, Disc Scheduling, Unix OS, Linux OS, Distributed Systems, Windows OS
  12. Section B: Data Base Management System: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, Relational Algebra, SQL, File Organization & Indexing, Transaction Management, Concurrency Control, Database Security & Authorization, Database Recovery, Data Warehousing & Mining
  13. Section B: Internet & Web Technologies: Website, Browser & Services, Internet Fundamentals, Web Design & Development, Protocols & Services, Web Publishing & Hosting, Server Side & Security, HTML Basics, CSS, Emerging Technologies, JavaScript-Basics, .Net Programming, Electronic Data Interchange and E-Commerce, XML
  14. Section B: Computer Fundamentals, IT Basics & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Register Transfer and Microoperations, Control Unit Design, Programming the Basic Computer, Terms & Abbreviations, MS Office Applications, Instr Formats & Modes, Floating Point Rep
  15. Section B: Python Language: Introduction to Python, Tokens, Variables, Data Types, I/O, Operators & Expressions, Control Flow & Looping, Collections & Strings, Functions & Modules, File Handling & Data Analysis
  16. Section B: Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management, extras, Fundamentals of Systems and Information Systems
  17. Section B: Design & Analysis of Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  18. Section B: Mathematics: Linear Algebra, Calculus, Trigonometry, Vectors, Set Theory, Functions, Relations, Propositional and Predicate Logic, Research Process Steps, Graph Theory
  19. Section B: Computer Graphics, Multimedia & Mobile Computing: Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, 3-D Object Representation, Geometric Transformations and Viewing, Multimedia
  20. Section B: Emerging Trends: Emerging Tech Trends, Knowledge Representation, Natural Language Processing
  21. DSSSB Previous Year Paper:
  22. Business Studies and Management: Business Communication, Business Organization and Management, Financial Accounting, Business Economics, Management Information System (MIS), Knowledge Management & New Economy
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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