Geometry

• 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.