Formula and Method for the Area of a Right Triangle
Area of a Right Triangle:
A = ½ × leg₁ × leg₂
Since the two legs meet at a 90° angle, one leg can always serve as the base and the other as the height.
A right triangle has one 90° angle formed by two perpendicular sides called legs (or catheti). The third side, opposite the right angle, is the hypotenuse — always the longest side. Because the legs are perpendicular, the standard triangle area formula A = ½ × base × height simplifies directly to A = ½ × leg₁ × leg₂, with no separate "height" measurement needed. This calculator also derives the hypotenuse, perimeter, and the altitude to the hypotenuse from the same two legs.
How the calculation works
Enter the lengths of the two legs and choose the unit they're measured in. The calculator multiplies them together and halves the result to get the area (in square units, such as ft² or m²). It finds the hypotenuse using the Pythagorean theorem, c = √(a² + b²), then adds all three sides for the perimeter: P = a + b + c. Finally, it computes the altitude to the hypotenuse, h = (a × b) ÷ c — the perpendicular distance from the right-angle vertex to the hypotenuse. This falls out of expressing the triangle's area two equivalent ways: A = ½ab (using the legs) and A = ½ch (using the hypotenuse as the base), then solving ½ab = ½ch for h.
Common mistakes
- Using the hypotenuse as a leg: only the two sides that form the right angle are "legs." Plugging the hypotenuse into the area formula instead of a leg gives a result larger than the true area.
- Units: area is reported in square units (ft², m²) while the hypotenuse, perimeter, and altitude are in linear units (ft, m) — don't mix them up when reporting results.
- Mixing unit systems: enter both legs in the same unit — convert inches to feet, or centimeters to meters, before calculating.
Real-world applications
- Carpentry and framing use the 3-4-5 rule (a special case of the Pythagorean theorem) to square corners on job sites.
- Roofing and ramp design use the legs (rise and run) to compute slope length (the hypotenuse) for material estimates.
- Surveying and navigation use right-triangle relationships to find distances that can't be measured directly.
- Engineering drawings use the altitude to the hypotenuse when a right triangle needs to be split into two smaller similar right triangles.