【数学の名問】円周率は3.05より大きい？【東大入試2003年度】

#UniversityofTokyoEntranceExam #Pi #Mathematics #ProofProblem #BeautyofMathematics Prove that pi is greater than 3.05 University of Tokyo 2003 Science Mathematics This problem asks you to prove that pi (π) is greater than 3.05. At first glance, it seems like you need to calculate pi, but in reality, you can estimate it using the perimeter of a regular polygon inscribed in a circle. 📌 Key Points of the Problem • The perimeter of a regular polygon inscribed in a circle is shorter than the circumference of the circle. • By finding the perimeter of the polygon, you can find the lower limit of the circumference. • It can be proven using only the Pythagorean theorem. 📘 Approach Consider a circle with radius 1. In this case, Circumference = 2π Construct a regular octagon inscribed in the circle. The length of one side of the regular octagon is √(2 − √2) by the Pythagorean theorem. This is the result. 📐 Perimeter of a regular octagon 8 × √(2 − √2) Using √2 1.415 2 − √2 0.585 √(2 − √2) √0.585 √0.585 0.765 Therefore, Perimeter of a regular octagon 8 × 0.765 = 6.12 📊 Comparison with the circumference of a circle Circumference = 2π Since the perimeter of an inscribed polygon is shorter than the circumference of a circle 2π 6.12 Dividing both sides by 2 π 3.06 📌 Conclusion π 3.05 🎯 What you can learn from this problem: You can evaluate pi without directly calculating it. The concept of the "squeeze theorem" using inscribed polygons. A proof combining geometry and inequalities. A simple yet beautiful proof problem, typical of the University of Tokyo. 🔗 Related Videos & Recommended Links [Junior High School Entrance Exam: Math Calculation Problem Explanation Series] ▶    • 灘中入試問題 2005年度1日目｜素因数分解で分数の計算問題   [Junior High School Entrance Exam: Practice Problems on the Properties of Integers] ▶    • 【魔方陣の攻略法】真ん中の数の見つけ方！公務員試験＆中学受験の数的処理対策｜算数過去...   [Kaisei Junior High School Entrance Exam Problems & Past Paper Explanation Series] ▶    • 【開成中学の算数】単位分数の和と整数問題を徹底解説！2010年度過去問に挑戦｜中学受験算数   [Nada Junior High School Entrance Exam Problems & Past Paper Explanation Series] ▶    • 灘中入試問題 2005年度1日目｜素因数分解で分数の計算問題   📢 A Request to Viewers If you found this video helpful, please let us know by liking or commenting! We also welcome requests for explanations of other problems! 📌 About This Channel This channel provides methods for steadily improving the skills necessary for entrance exams while learning arithmetic and mathematics in a fun way. We fully support everyone who wants to "train their thinking skills" and "overcome their weaknesses"! 📩 Business Inquiries/Contact For interview and business inquiries, please contact us here: 📧 [email protected] #JuniorHighSchoolEntranceExam #EntranceExamArithmetic #Mathematics