Area of a Right Triangle Calculator
Calculate the area of right triangles using legs, hypotenuse, angles, and various formulas
Right Triangle Area Calculator
Right Triangle Diagram
- All values must be positive
- For leg + hypotenuse: leg must be shorter than hypotenuse
- For angles: must be between 0° and 90° (exclusive)
Example Calculation
Building Construction Example
Problem: A right triangular roof section has legs of 8 feet and 6 feet.
Method: Two legs given
We need to find the area for material estimation.
Solution
Given: Leg A = 8 ft, Leg B = 6 ft
Formula: Area = (a × b) / 2
Calculation: Area = (8 × 6) / 2 = 24 square feet
Hypotenuse: c = √(8² + 6²) = √(64 + 36) = √100 = 10 ft
Result: The roof section has an area of 24 square feet
Calculation Methods
Two Legs
Area = (a × b) / 2
Most common and direct method
Leg + Hypotenuse
Area = a × √(c² - a²) / 2
Uses Pythagorean theorem
Angle + Hypotenuse
Area = c² × sin(α) × cos(α) / 2
Uses trigonometric functions
Angle + Leg
Area = a² × tan(β) / 2
When one angle and adjacent leg known
Formula Reference
Basic Area
Area = (base × height) / 2
Two Legs
Area = (a × b) / 2
Leg + Hypotenuse
Area = a × √(c² - a²) / 2
Angle + Hypotenuse
Area = c² × sin(α) × cos(α) / 2
Angle + Leg
Area = a² × tan(β) / 2
Pythagorean Theorem
a² + b² = c²
Understanding Right Triangle Area
What is a Right Triangle?
A right triangle is a triangle with one 90-degree angle. The two sides that form the right angle are called legs, and the side opposite the right angle is called the hypotenuse. The hypotenuse is always the longest side of a right triangle.
Why These Formulas Work
- •Two legs: The basic triangle area formula where legs act as base and height
- •Pythagorean approach: Find missing leg using a² + b² = c², then apply basic formula
- •Trigonometric approach: Use sine and cosine to find legs from angle and hypotenuse
Real-World Applications
- •Construction and architecture (roof sections, triangular supports)
- •Engineering and structural design
- •Land surveying and property measurement
- •Computer graphics and game development
- •Navigation and GPS calculations
Quick Check: To verify if a triangle is a right triangle, check if the Pythagorean theorem holds: a² + b² = c² where c is the longest side.
Special Right Triangles
Isosceles Right Triangle
Also called 45-45-90 triangle
Both legs equal: a = b
Area = a² / 2
Hypotenuse = a√2
30-60-90 Triangle
Special scalene right triangle
Side ratio: 1 : √3 : 2
If short leg = a:
Area = a² × √3 / 4
3-4-5 Triangle
Common Pythagorean triple
Legs: 3 and 4 units
Hypotenuse: 5 units
Area = (3 × 4) / 2 = 6 units²