Right Triangle Calculator
Calculate missing sides, hypotenuse, angles, area, perimeter, and altitudes of any right triangle step-by-step.
Right Triangle Geometry Solver
Select given information and enter known values.
Right Triangle Solution Summary
Quick Summary
Our right triangle calculator hypotenuse legs area perimeter steps tool solves all missing elements of any right triangle given any pair of sides or angles. It outputs hypotenuse $c$, legs $a, b$, acute angles $A, B$, area $\1 / 2ab$, perimeter $a+b+c$, altitude $h_c$, inradius $r$, and circumradius $R$.
How It Works: Pythagorean Theorem & Trigonometric Ratios
A right triangle contains one 90-degree angle ($C = 90^\circ$). Solving it relies on fundamental geometric rules:
1. **Pythagorean Theorem:** $a^2 + b^2 = c^2$, where $c$ is the longest side opposite the right angle (hypotenuse).
2. **SOH CAH TOA Rules:** $\sin A = \a / c$, $\cos A = \b / c$, $\tan A = \a / b$.
3. **Complementary Acute Angles:** $A + B = 90^\circ \implies B = 90^\circ - A$.
4. **Altitude to Hypotenuse:** $h_c = \ab / c$.
5. **Inradius & Circumradius:** Inradius $r = \a+b-c / 2$, Circumradius $R = \c / 2$.
Formula Explanation
Your right triangle calculations use these standard formulas:
Step-by-Step Worked Example
Here is a detailed 5-step breakdown for a right triangle with legs $a = 3$ and $b = 4$:
- Step 1 (Find Hypotenuse c): Apply Pythagorean theorem: $c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = \mathbf{5}$.
- Step 2 (Find Acute Angle A): $A = \arctan\left(\a / b\right) = \arctan\left(\3 / 4\right) \approx \mathbf{36.87^\circ}$.
- Step 3 (Find Acute Angle B): $B = 90^\circ - 36.87^\circ = \mathbf{53.13^\circ}$.
- Step 4 (Calculate Area & Perimeter): $\text{Area} = \1 / 2 \times 3 \times 4 = \mathbf{6}$, $\text{Perimeter} = 3 + 4 + 5 = \mathbf{12}$.
- Step 5 (Calculate Altitude & Inradius): Altitude $h_c = \3 \times 4 / 5 = \mathbf{2.4}$, Inradius $r = \3 + 4 - 5 / 2 = \mathbf{1}$, delivering **c = 5, Area = 6, P = 12**!
Calculation Examples: Real-World Scenario Comparison
Compare classic Pythagorean triple triangles and special angle right triangles:
| Triangle Type / Inputs | Legs (a, b) | Hypotenuse c | Angle A | Area | Perimeter |
|---|---|---|---|---|---|
| 3-4-5 Classic Triple | a=3, b=4 | 5.0000 | 36.87° | 6.0000 | 12.0000 |
| 5-12-13 Triple | a=5, b=12 | 13.0000 | 22.62° | 30.0000 | 30.0000 |
| 45-45-90 Isosceles | a=1, b=1 | 1.4142 | 45.00° | 0.5000 | 3.4142 |
| 30-60-90 Special | a=1, b=1.7321 | 2.0000 | 30.00° | 0.8660 | 4.7321 |
Benefits of Using the Right Triangle Calculator
Utilizing this calculator provides essential construction roof rafter calculations, land surveying, and carpentry framing advantages:
- Construction & Carpentry Framing: Solves exact stair riser/tread stringer lengths and diagonal bracing measurements.
- Roof Pitch & Rafter Lengths: Calculates rafter hypotenuse length from building run and roof rise.
- Land Surveying & Triangulation: Computes inaccessible heights and distances using baseline measurements.
- Complete Geometric Output: Computes inradius, circumradius, and altitude alongside standard side and area values.
Frequently Asked Questions (FAQ)
What is a right triangle?
A right triangle (or right-angled triangle) is a triangle in which one angle measures exactly 90 degrees.
What is the Pythagorean theorem formula?
a^2 + b^2 = c^2, where a and b are the two perpendicular legs, and c is the hypotenuse.
What is the hypotenuse?
The hypotenuse is the longest side of a right triangle, located directly opposite the 90-degree right angle.
How do you find the area of a right triangle?
Area = 1/2 * base * height = 1/2 * a * b.
What are common Pythagorean triples?
Common integer leg/hypotenuse triples include 3-4-5, 5-12-13, 8-15-17, 7-24-25, and 9-40-41.
What is a 45-45-90 right triangle?
A 45-45-90 triangle is an isosceles right triangle with equal legs a = b and hypotenuse c = a * sqrt(2).
What is a 30-60-90 right triangle?
A 30-60-90 triangle has side ratio 1 : sqrt(3) : 2 (short leg : long leg : hypotenuse).
What is the altitude to the hypotenuse (h_c)?
The altitude h_c is the perpendicular height drawn from the right angle to the hypotenuse, calculated as h_c = (a * b) / c.
What is the circumradius of a right triangle?
The circumradius R is half of the hypotenuse length (R = c / 2). The midpoint of the hypotenuse is the center of the circumscribed circle.
How do you find an angle if you only know two sides?
Use inverse trigonometric functions: A = arcsin(a/c), A = arccos(b/c), or A = arctan(a/b).