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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.

    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.

    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 Defense (DOD) | USAGov
    The Department of Defense (DOD) provides the military forces needed to deter war, and to protect the security of the United States.

    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.

    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.

    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°.

    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.

    WELCOME TO THE PENTAGON 2025 - whs.mil
    There are 5 parking spaces to accommodate Pentagon Memorial visitors with disabilities, available 24 hours a day, 7 days a week, in Lane 1 of South Parking, close to the Pentagon Memorial.

    Pentagon: Latest News, Top Stories & Analysis - POLITICO
    Australia wanted more US missiles. The Iran war may make them harder to get. The Pentagon wants to build data centers. Congress would like a word.

     

     

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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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