Math & Trig
Area of a triangle, trapezoid, and parallelogram (plus Heron's 3-side formula)
Compute planar land, roofing, and sheet-metal areas from base and perpendicular height—or from three measured triangle sides.
Base-and-height formulas vs Heron's three-side survey formula
Given perpendicular Height (C2), a parallelogram has area BaseA * Height, a triangle has half that area (0.5 * BaseA * Height), and a trapezoid averages its two parallel bases first: 0.5 * (BaseA + TopB) * Height.
On field surveys where three side lengths (a, b, c) are taped directly without a perpendicular altitude, Heron's formula computes semi-perimeter s = (a + b + c) / 2 and Area = SQRT(s * (s - a) * (s - b) * (s - c)).
Require positive dimensions and all three strict triangle inequalities for a nondegenerate triangle. Equality gives a zero-area degenerate case; nearly degenerate triangles can suffer numerical cancellation. The base/height triangle and the three-side triangle shown are different examples, hence their areas differ. Heights must be perpendicular and units consistent.
| Shape | Dimensions | Formula | Area |
|---|---|---|---|
| Trapezoid | Bases 10 & 6, Height 4 | 0.5 * (A2 + B2) * C2 | 32 |
| Triangle (base/h) | Base 10, Height 4 | 0.5 * A2 * C2 | 20 |
| Parallelogram | Base 10, Height 4 | A2 * C2 | 40 |
| Triangle (3 sides) | Sides 10, 6, 8 (s=12) | Heron SQRT(12*2*6*4) | 24 |
Trapezoid, triangle, parallelogram, and Heron formulas
When TopB equals BaseA (10 and 10), the trapezoid formula naturally reduces to Parallelogram area 0.5 * 20 * 4 = 40.
=0.5 * (A2 + B2) * C2Heron's 3-side triangle area with LET
A 6-8-10 right triangle has semi-perimeter s = 12 and exact area SQRT(12 * 2 * 6 * 4) = SQRT(576) = 24.
=LET(s, (A2 + B2 + D2) / 2, ROUND(SQRT(s * (s - A2) * (s - B2) * (s - D2)), 4))Try the example
Use the original values or change them to test your understanding.