Right Triangle Calculator
Calculate sides, angles, area, and perimeter of right triangles using the Pythagorean theorem. Find hypotenuse or missing leg.
Triangle Results
Hypotenuse
5.0000
Side c (longest)
Triangle Input
Triangle Diagram
Complete Results
Sides
Angles
Area
6.0000
Perimeter
12.0000
Inradius
1.0000
Circumradius
2.5000
Pythagorean Theorem Verification
a² + b² = c²
3.00² + 4.00² = 5.00²
9.0000 + 16.0000 = 25.0000
25.0000 ≈ 25.0000 ✓
Special Right Triangles
45-45-90 Triangle
Isoceles right triangle
- Angles: 45°, 45°, 90°
- Side ratio: 1 : 1 : √2
- If leg = a, hypotenuse = a√2
30-60-90 Triangle
Half of equilateral triangle
- Angles: 30°, 60°, 90°
- Side ratio: 1 : √3 : 2
- Short leg : Long leg : Hypotenuse
Right Triangle Formulas
Pythagorean Theorem
c = √(a² + b²)
Area
A = (a × b) / 2
Angle from sides
θ = arctan(opposite/adjacent)
Altitude to hypotenuse
h = (a × b) / c
Inradius
r = (a + b - c) / 2
Circumradius
R = c / 2
Triangle Results
Hypotenuse
5.0000
Side c (longest)
?How Do You Solve a Right Triangle?
A right triangle has one 90-degree angle. Use Pythagorean theorem (a^2 + b^2 = c^2) to find sides, and trigonometry (SOHCAHTOA) for angles. sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Given two values (sides or angles), you can find all others.
What is a Right Triangle?
A right triangle is a triangle containing one 90-degree (right) angle. The side opposite the right angle is called the hypotenuse and is always the longest side. Right triangles are fundamental to trigonometry, and their properties are used extensively in engineering, physics, architecture, and navigation.
Key Facts About Right Triangles
- One angle is always 90 degrees; other two sum to 90
- Pythagorean theorem: a^2 + b^2 = c^2 (c is hypotenuse)
- SOHCAHTOA: Sin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj
- Hypotenuse is the longest side (opposite the right angle)
- Area = (1/2) * base * height = (1/2) * leg1 * leg2
- Special triangles: 30-60-90 (1:sqrt(3):2) and 45-45-90 (1:1:sqrt(2))
- Inverse trig functions find angles: arcsin, arccos, arctan
- Given any two sides, or one side and one angle, solve the whole triangle
Quick Answer
A right triangle has one 90-degree angle. Use Pythagorean theorem (a^2 + b^2 = c^2) to find sides, and trigonometry (SOHCAHTOA) for angles. sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Given two values (sides or angles), you can find all others.
Frequently Asked Questions
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of squares of the other two sides: a² + b² = c². This only applies to right triangles (triangles with one 90° angle).
A right triangle is a triangle with one angle measuring exactly 90 degrees (a right angle). The two sides forming the right angle are called legs, and the side opposite the right angle is called the hypotenuse, which is always the longest side.
To find the hypotenuse when you know both legs: c = √(a² + b²). Square each leg, add them together, then take the square root. For example, if legs are 3 and 4: c = √(9 + 16) = √25 = 5.
To find a missing leg when you know the hypotenuse and one leg: a = √(c² - b²). Square the hypotenuse, subtract the square of the known leg, then take the square root. Example: c=10, b=6, so a = √(100 - 36) = √64 = 8.
Two special right triangles: 45-45-90 triangle (isoceles right triangle) with sides in ratio 1:1:√2, and 30-60-90 triangle with sides in ratio 1:√3:2. These ratios help solve problems quickly without calculation.
Last updated: 2025-01-15
Triangle Results
Hypotenuse
5.0000
Side c (longest)