Perimeter of a Right Triangle Calculator
Calculate right triangle perimeter using catheti, all sides, or angle measurements
Calculate Right Triangle Perimeter
Length of the first leg (perpendicular side)
Length of the second leg (base side)
Formula Used
Hypotenuse: c = √(a² + b²)
Perimeter: P = a + b + c
Complete: P = a + b + √(a² + b²)
Calculation Results
Step-by-step calculation:
Input Validation
Example Calculations
3-4-5 Right Triangle (Catheti Method)
Given: Leg 1 = 3 cm, Leg 2 = 4 cm
Hypotenuse: c = √(3² + 4²) = √25 = 5 cm
Perimeter: P = 3 + 4 + 5 = 12 cm
Area: A = (3 × 4) ÷ 2 = 6 cm²
5-12-13 Right Triangle (All Sides)
Given: Sides = 5, 12, 13 cm
Validation: 13² = 169, 5² + 12² = 25 + 144 = 169 ✓
Perimeter: P = 5 + 12 + 13 = 30 cm
Area: A = (5 × 12) ÷ 2 = 30 cm²
Leg + Hypotenuse Method
Given: Leg = 8 cm, Hypotenuse = 10 cm
Other leg: b = √(10² - 8²) = √36 = 6 cm
Perimeter: P = 8 + 6 + 10 = 24 cm
Area: A = (8 × 6) ÷ 2 = 24 cm²
Angle + Leg Method
Given: Leg = 6 cm, Opposite angle = 30°
Other leg: b = 6 / tan(30°) = 6 / 0.577 ≈ 10.39 cm
Hypotenuse: c = 6 / sin(30°) = 6 / 0.5 = 12 cm
Perimeter: P = 6 + 10.39 + 12 ≈ 28.39 cm
Right Triangle Properties
One angle is exactly 90 degrees
Follows Pythagorean theorem: a² + b² = c²
Hypotenuse is always the longest side
Sum of all angles is 180°
Area = (leg₁ × leg₂) ÷ 2
Essential Formulas
Pythagorean Theorem
a² + b² = c²
Fundamental relationship
Perimeter (Catheti)
P = a + b + √(a² + b²)
When two legs known
Perimeter (All Sides)
P = a + b + c
When all sides known
Area
A = (a × b) ÷ 2
Half base times height
Common Pythagorean Triples
Calculation Tips
Hypotenuse is always the longest side
Verify using Pythagorean theorem
Use consistent units throughout
Check that angles sum to 180°
Understanding Right Triangle Perimeter
What is a Right Triangle?
A right triangle is a triangle with one angle measuring exactly 90 degrees (a right angle). The side opposite to the right angle is called the hypotenuse, and it's always the longest side. The other two sides are called legs or catheti.
Pythagorean Theorem
The fundamental relationship in right triangles is the Pythagorean theorem: a² + b² = c², where a and b are the legs, and c is the hypotenuse. This theorem allows us to find any missing side when two sides are known.
Perimeter Calculation
The perimeter is simply the sum of all three sides. When only the legs are known, we use the Pythagorean theorem to find the hypotenuse first.
Calculation Methods
Catheti Method
When both legs are known
P = a + b + √(a² + b²)
All Sides Method
When all three sides are known
P = a + b + c
Leg + Hypotenuse
When one leg and hypotenuse are known
P = a + √(c² - a²) + c
Trigonometric
Using angle and one side
Uses sin, cos, tan functions
Special Right Triangles
- •45-45-90 triangle: Sides in ratio 1:1:√2
- •30-60-90 triangle: Sides in ratio 1:√3:2
- •Pythagorean triples: Integer solutions like 3-4-5
Real-World Applications
- •Construction: Foundation layouts and roof calculations
- •Navigation: Distance and direction calculations
- •Engineering: Structural design and analysis
- •Surveying: Land measurement and mapping