Three Ways to Find Triangle Area
A triangle is a three-sided polygon — the simplest possible polygon. Its area can be calculated using different formulas depending on which measurements are known. Our calculator supports all three main methods.
Method 1: Base and Height (Most Common)
This is the simplest method. The height must be perpendicular (at 90°) to the base.
Example: base=10, height=6 → Area = ½ × 10 × 6 = 30
Why ½? A triangle is exactly half of a parallelogram (or rectangle for right triangles) with the same base and height.
Method 2: Heron's Formula (Three Sides Known)
When you know all three side lengths but not the height, use Heron's Formula — named after Hero of Alexandria (c. 60 AD).
Area = √(s × (s−a) × (s−b) × (s−c))
Example: a=5, b=6, c=7 → s=9
Area = √(9×4×3×2) = √216 ≈ 14.70
Method 3: SAS – Two Sides and Included Angle
When you know two sides and the angle between them (the included angle), use the SAS formula.
Example: a=5, b=7, C=60° → Area = ½×5×7×sin(60°) = 17.5×0.866 ≈ 15.16
Triangle Area Reference Table
| Type | Formula | Example |
|---|---|---|
| Right triangle | ½ × leg₁ × leg₂ | ½ × 3 × 4 = 6 |
| Equilateral (side s) | (√3/4) × s² | s=6 → 15.59 |
| Isosceles | Heron's or ½bh | Use our calculator |
| Scalene | Heron's formula | sides 5,6,7 → 14.70 |
Triangle Inequality Theorem
Not all three lengths form a valid triangle. A valid triangle requires that the sum of any two sides must be greater than the third side:
Example: 3, 4, 5 → 3+4=7>5 ✓ Valid triangle
Example: 1, 2, 10 → 1+2=3 < 10 ✗ Not a valid triangle
Real-World Applications
- Architecture: Triangular rooftops, gable ends, dormer windows, trusses
- Engineering: Structural trusses, bridge supports, triangulated frameworks
- Surveying: Triangulation method to measure land areas accurately
- Navigation: Triangular bearings in GPS and traditional navigation
- Art: Triangular compositions in painting and photography