Right Triangle Side and Angle Calculator
Calculate missing sides and angles of right triangles using Pythagorean theorem and trigonometry
Calculate Right Triangle
Example Calculation
Two Sides Given
Given: Side a = 3, Side b = 4
Find: Hypotenuse c and angles α, β
Solution
c = √(a² + b²) = √(3² + 4²) = √(9 + 16) = √25 = 5
α = arctan(a/b) = arctan(3/4) = 36.87°
β = 90° - α = 90° - 36.87° = 53.13°
Area = (a × b) / 2 = (3 × 4) / 2 = 6 units²
Perimeter = a + b + c = 3 + 4 + 5 = 12 units
Right Triangle Properties
Pythagorean Theorem
a² + b² = c²
Foundation for all calculations
Angle Sum
α + β + 90° = 180°
Two acute angles are complementary
Area Formula
Area = (a × b) / 2
Half the product of the legs
Trigonometric Ratios
sin(α) = opposite/hypotenuse = a/c
cos(α) = adjacent/hypotenuse = b/c
tan(α) = opposite/adjacent = a/b
sin(β) = opposite/hypotenuse = b/c
cos(β) = adjacent/hypotenuse = a/c
tan(β) = opposite/adjacent = b/a
Special Right Triangles
Ratio: 1 : 1 : √2
If legs = 1, hypotenuse = √2 ≈ 1.414
Ratio: 1 : √3 : 2
If short leg = 1, long leg = √3, hypotenuse = 2
Understanding Right Triangle Calculations
Calculation Methods
1. Two Sides Given
When you know two sides, use the Pythagorean theorem to find the third side, then use inverse trigonometric functions to find the angles.
- • If a and b known: c = √(a² + b²)
- • If a and c known: b = √(c² - a²)
- • If b and c known: a = √(c² - b²)
2. Angle and Side Given
Use trigonometric ratios to find the missing sides and the complementary angle.
- • sin(α) = a/c, so a = c × sin(α)
- • cos(α) = b/c, so b = c × cos(α)
- • tan(α) = a/b, so a = b × tan(α)
Key Formulas
Pythagorean Theorem
a² + b² = c²
The fundamental relationship in right triangles
Area Formula
Area = (a × b) / 2
Half the product of the two legs
Perimeter Formula
P = a + b + c
Sum of all three sides
Applications
Engineering & Construction
- • Roof design and structural analysis
- • Bridge construction calculations
- • Foundation layout and surveying
Navigation & Physics
- • Distance and bearing calculations
- • Vector analysis and force resolution
- • Projectile motion problems