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    *Filter: "Pentagon"      

      

    The Pentagon - Wikipedia
    The Pentagon is the headquarters of the United States Department of Defense, in Arlington County, Virginia, across the Potomac River from Washington, D.C. The building was constructed on an accelerated schedule during World War II.

    Home | U.S. Department of War
    Our mission is to provide the military forces needed to deter war and ensure our nation's security.

    U.S. Department of War > Pentagon Tours | U.S. Department of War
    The Pentagon, one of the world's largest office buildings, is the headquarters of the Department of War. Located in the Washington, D.C., area, the five-sided building has three times the floor space of the Empire State Building.

    Pentagon | History & Features | Britannica
    Pentagon, large five-sided building in Arlington county, Virginia, near Washington, D.C., that serves as the headquarters of the U.S. Department of Defense, including all three military services— Army, Navy, and Air Force.

    PenFed Credit Union | Checking, Savings, Auto, Credit Cards, Mortgage ...
    We became Pentagon Federal Credit Union, a name that reflected our growing community and expanding reach. As membership increased, so did our commitment to serving more people with the same care and integrity. We began serving all Department of Defense employees and their families nationwide.

    Pentagon - Location, Building Timeline, 9/11 | HISTORY
    The Pentagon is the Virginia headquarters of the U.S. Department of Defense, located in a massive five-sided concrete and steel building that’s a potent symbol of America’s military strength.

    Pentagon - Wikipedia
    In geometry, a pentagon (from Greek πέντε (pente) 'five' and γωνία (gonia) 'angle'[1]) is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

    PFPA | Visit the Pentagon | Plan Your Visit
    Plan Your Visit to the Pentagon The Pentagon is headquarters for the Department of War, a military command center, and a secure facility.

    The Pentagon - Simple English Wikipedia, the free encyclopedia
    The Pentagon is the largest office building in the world, [5][6] but any place in the building can be reached within 7 minutes. Construction began on September 11, 1941.

    PFPA Home
    We provide force protection, security, law enforcement and numerous other functions to safeguard people, facilities, and critical infrastructure from the Pentagon itself to Department of Defense sites across Virginia, Maryland, and Pennsylvania.

     

     



           * Geometry101 LATEST NEWS *

                        *Filter: "Pentagon"

     

     
    You definitely cannot kick this $40,000 football, as Louis Vuitton has turned it into an ingenious mechanical clock, packing 882 components, an eight-day movement, and rotating time displays into a stunning sculpture  Luxurylaunches

    How the Pentagon Is Quietly Turning Laser Communications Into the Backbone of Future Space Warfare  The Debrief

    Cairo pentagon tessellated covalent organic frameworks with mcm topology for near-infrared phototherapy  nature.com

    Not a mathematician, but she solved a problem experts couldn't for 60 years  indiatoday.in

    ‘Sensational’ proof topples decades-old geometry problem  Scientific American

    Geometry Barn Raising at ALCS | Math  St. Bonaventure University

    Geometry can shape our world in unexpected but useful ways  Science News Explores

    Attack on the pentagon results in discovery of new mathematical tile  The Guardian

    Explore the Math Behind the Golden Ratio With Steven Strogatz and The New York Times (Published 2025)  The New York Times

    A Brief History of Tricky Mathematical Tiling  quantamagazine.org

    Why So Many Pandemic Predictions Failed  The Atlantic

    New book dissects mathematical puzzles and their history  Purdue University

    Most People Are Bad At Math, But If You Don't Pass This 9th Grade Geometry Quiz, You're REALLY Bad At Math  BuzzFeed

    'Scissors congruence,' an ancient geometric idea that's still fueling cutting-edge mathematical research  Phys.org

    Creative Duo Blends Art and Architecture at Star Tile  Palm Springs Life

      

     

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    Important Geometry Terms
    • Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
    • Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.
    • Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle ($\tan \theta = \frac{\sin \theta}{\cos \theta}$).
    • Cosecant (csc): The reciprocal of the sine function ($\csc \theta = \frac{1}{\sin \theta}$), representing the ratio of the hypotenuse to the opposite side.
    • Secant (sec): The reciprocal of the cosine function ($\sec \theta = \frac{1}{\cos \theta}$), representing the ratio of the hypotenuse to the adjacent side.
    • Cotangent (cot): The reciprocal of the tangent function ($\cot \theta = \frac{1}{\tan \theta}$), representing the ratio of the adjacent side to the opposite side.
    • Hypotenuse: The longest side of a right-angled triangle, located opposite the right angle ($90^\circ$).
    • Radian: A unit of angle measurement based on arc length, where $2\pi \text{ radians} = 360^\circ$ ($1 \text{ radian} \approx 57.3^\circ$).
    • Unit Circle: A circle with a radius of $1$ centered at the origin $(0,0)$ in the Cartesian coordinate plane, used to extend trigonometric functions to any angle.
    • Pythagorean Identity: The fundamental trigonometric identity derived from the Pythagorean theorem, expressed as $\sin^2\theta + \cos^2\theta = 1$.

     

    Industries That Use Geometry
    • Architecture & Civil Engineering: Calculates structural loads, roof pitches, bridge support angles, and land slopes to design safe buildings and infrastructure.
    • Aeronautics & Aerospace: Determines flight paths, wind drift angles, satellite orbits, and altitude trajectories for aircraft and spacecraft navigation.
    • Video Game Development & CGI: Renders 3D graphics, calculates character raycasting, models object collisions, and animates realistic movement using vector geometry.
    • Land Surveying & Cartography: Measures distances, elevation changes, and geographic boundaries across large areas using triangulation techniques.
    • Acoustics & Audio Engineering: Analyzes sound waves, models frequency harmonics, and designs soundproof spaces using sine and cosine wave functions.
    • Maritime & Ocean Navigation: Computes true heading, compass bearing, ocean current offsets, and celestial navigation coordinates for ships.
    • Electrical & Mechanical Engineering: Models alternating current (AC) voltage waveforms, electromagnetic fields, and rotational forces in motors and machinery.
    • Medical Imaging & Radiography: Reconstructs 3D cross-sectional images in CAT scans and MRI machines by calculating wave projection angles.
    • Astronomy & Astrophysics: Measures interplanetary distances, stellar parallax, and celestial movement relative to Earth.
    • Meteorology & Climate Science: Models global weather patterns, ocean wave dynamics, and solar radiation angles to predict climate and weather systems.
                    

     

     

    Important Geometry Historical Events
    • Plimpton 322 Clay Tablet (c. 1800 BC): Records ancient Babylonian mathematical tables containing advanced Pythagorean triples, marking the earliest known use of right-triangle ratios.
    • Hipparchus Compiles First Chord Table (c. 150 BC): Earns the title "Father of Geometry" by calculating circle chord values to solve astronomical triangles and track planetary motion.
    • Ptolemy Writes the Almagest (c. 150 AD): Expands chord tables and spherical geometry principles, creating a comprehensive geometric framework that dominated astronomy for over a millennium.
    • Aryabhata Introduces the Sine Function (c. 499 AD): Indian mathematician defines jya (half-chord), shifting mathematical focus from full circle chords to the modern concept of sine.
    • Al-Battani and Islamic Golden Age Innovations (c. 900 AD): Islamic scholars formalize tangent, cotangent, secant, and cosecant functions, establishing geometry as an independent mathematical discipline.
    • Regiomontanus Publishes De Triangulis Omnimodis (1464): Produces the first European textbook devoted entirely to plane and spherical geometry, standardizing trigonometric methods for navigation.
    • Rheticus Defines Ratios on Right Triangles (1596): Defines trigonometric functions directly as ratios of triangle sides rather than circle chords, creating the modern textbook approach used today.
    • John Napier Invents Logarithms (1614): Revolutionizes trigonometric computation by simplifying complex multi-digit sine and cosine multiplications into straightforward addition and subtraction.
    • Leonhard Euler Unifies Geometry and Calculus (1748): Introduces standard function notations ($\sin, \cos, \tan$) and formulates Euler's formula ($e^{ix} = \cos x + i \sin x$), connecting geometry with complex analysis.

     

     

    Essential Geometry Formulas
    • Pythagorean Identity: $\sin^2\theta + \cos^2\theta = 1$
    • Tangent Identity: $\tan\theta = \frac{\sin\theta}{\cos\theta}$
    • Reciprocal Identities: $\csc\theta = \frac{1}{\sin\theta}$, $\sec\theta = \frac{1}{\cos\theta}$, $\cot\theta = \frac{1}{\tan\theta}$
    • Sine Angle Sum Formula: $\sin(A + B) = \sin A \cos B + \cos A \sin B$
    • Cosine Angle Sum Formula: $\cos(A + B) = \cos A \cos B - \sin A \sin B$
    • Sine Double-Angle Formula: $\sin(2\theta) = 2\sin\theta\cos\theta$
    • Cosine Double-Angle Formula: $\cos(2\theta) = \cos^2\theta - \sin^2\theta$
    • Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
    • Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos C$
    • Tangential Pythagorean Identity: $1 + \tan^2\theta = \sec^2\theta$

     

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