Geometry

• Where Should Two Parallel Lines Cut a Circle into Thirds?
  Two parallel cuts make three equal areas when they lie about 0.26493 radii on opposite sides of the center; derive the result and equation.
• How to Find a Parallelogram’s Area Using a Determinant
  Find a parallelogram’s area from two adjacent side vectors by taking the absolute value of their 2×2 determinant, with a worked example.
• How to Maximize the Volume of a Cone with Fixed Slant Height
  For a right circular cone with fixed slant height, maximize its volume by using the Pythagorean constraint and differentiating with respect to height.
• Why Does a Polygon Have the Same Number of Vertices and Sides?
  A polygon has the same number of vertices and sides because each side joins consecutive vertices in one closed cycle.
• How to Approximate a Hypotenuse Without a Square Root
  Estimate a hypotenuse without a square root using the larger side plus one third of the smaller side, and see the method's error.
• Earth Curvature and Refraction, Explained Simply
  Estimate Earth’s curvature drop with distance, see how atmospheric refraction changes it, and use a simple horizon formula with worked examples.
• What Equation Describes a Rectangle with a Given Perimeter?
  Use P = 2(l + w) for a rectangle's perimeter. With fixed perimeter P, its area is A = l(P/2 − l), and the sides must be positive.
• Can Two Polygons Form a Triangle, Quadrilateral, and Pentagon?
  Yes. A triangle cut near one corner gives two polygon pieces that can be rejoined along different equal-length sides to form three convex shapes.
• How to Calculate the Partial Volume of a Capsule Tank
  For a horizontal capsule tank with radius r and straight length L, calculate liquid volume at depth h using circular-segment and spherical-cap formulas.
• How to Find the Area of an Isosceles Right Triangle from Its Hypotenuse
  For an isosceles right triangle with hypotenuse c, the area is c²/4. See the derivation, a worked example, and common calculation errors.
• An Intuitive Proof of Heron’s Formula
  See why Heron’s formula works by dropping an altitude, relating the two right triangles, and factoring the area expression into semiperimeter terms.
• How to Find the Volume of a Tetrahedron
  Use one-third of a triangular base’s area times its perpendicular height, or a determinant of four vertex coordinates, to find a tetrahedron’s volume.
• Understanding Euler’s Rotation Theorem
  Euler’s rotation theorem says every proper rotation in three dimensions has a fixed axis. Learn why, how to find that axis, and how to calculate the rotation angle.
• What Fraction of a Sphere Can an External Observer See?
  For an opaque sphere of radius R viewed from distance d from its center, the visible surface-area fraction is (1 − R/d)/2; it approaches one-half far away.