Area

• 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.
• What Is the Area Bounded by $r=|\sin\theta|$?
  For the polar curve r = |sin θ|, integrating over 0 ≤ θ ≤ 2π gives total area π/2; each lobe encloses π/4.
• How to Find the Area of a Triangle from Its Vertex Coordinates
  Use the shoelace formula or a 2 by 2 determinant to find a triangle’s area from its three vertex coordinates, with a worked example.
• 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.