Hypotenuse (c)
5.0000
Area
6.00
Perim
12.00
The hypotenuse (c) is always the longest side, opposite the 90° angle
Angle C is always 90°. Angles A + B must equal 90°. Fill in any 2 values total (sides or angles) to solve.
3.0000
4.0000
5.0000
36.8699°
53.1301°
90°
6.0000
12.0000
c = √(a² + b²) = √(3² + 4²) = √(9 + 16) = √25 = 5.0000
Angle A = arctan(a/b) = arctan(3/4) = 36.8699°
Angle B = 90° - 36.8699° = 53.1301°
Area = (a × b) / 2 = (3.0000 × 4.0000) / 2 = 6.0000
Perimeter = a + b + c = 3.0000 + 4.0000 + 5.0000 = 12.0000
c = √(a² + b²)Find the hypotenuse from the two legs. Rearrange to find a leg: a = √(c² - b²).
Where:
a= First leg (side opposite angle A)b= Second leg (side opposite angle B)c= Hypotenuse (longest side, opposite the 90° angle)Area = (a × b) / 2The area equals half the product of the two legs. Since the legs are perpendicular, they serve as base and height.
Where:
a= First leg lengthb= Second leg lengthA = arctan(a / b)Find an acute angle from the two legs using the inverse tangent function. The other acute angle is 90° - A.
Where:
A= Acute angle opposite side a (in degrees)a= Side opposite angle Ab= Side adjacent to angle AInputs
Result
The 3-4-5 triangle is the smallest Pythagorean triple. c = √(9 + 16) = √25 = 5. Area = (3 × 4)/2 = 6.
Inputs
Result
b = √(13² - 5²) = √(169 - 25) = √144 = 12. This is the 5-12-13 Pythagorean triple.
Inputs
Result
With a = 10 and A = 30°: c = a/sin(A) = 10/sin(30°) = 10/0.5 = 20. b = a/tan(A) = 10/tan(30°) ≈ 17.32.
Use the Pythagorean theorem (a² + b² = c²) to find the missing side, then use inverse trig functions to find the angles. For example, with sides 3 and 4: c = √(9+16) = 5, and angle A = arctan(3/4) = 36.87°.
| Known | Find c | Find Angle A |
|---|---|---|
| a=3, b=4 | √(9+16) = 5 | arctan(3/4) = 36.87° |
| a=5, c=13 | Given: 13 | arcsin(5/13) = 22.62° |
| a=8, b=15 | √(64+225) = 17 | arctan(8/15) = 28.07° |
Use trigonometric ratios: sin, cos, or tan. If you know side a and angle A, then b = a/tan(A) and c = a/sin(A). The other angle B = 90° - A. Any one side plus one acute angle is enough to solve completely.
| Known | Missing Side | Formula Used |
|---|---|---|
| a=10, A=30° | c = 20 | c = a/sin(A) |
| b=8, A=45° | a = 8 | a = b×tan(A) |
| c=15, A=60° | a = 12.99 | a = c×sin(A) |
The two special right triangles are the 45-45-90 and the 30-60-90 triangles. A 45-45-90 triangle has legs in ratio 1:1:√2. A 30-60-90 triangle has sides in ratio 1:√3:2. These appear frequently in geometry and standardized tests.
| Triangle | Sides Ratio | Example |
|---|---|---|
| 45-45-90 | 1 : 1 : √2 | 5, 5, 7.07 |
| 30-60-90 | 1 : √3 : 2 | 5, 8.66, 10 |
| 3-4-5 | 3 : 4 : 5 | 6, 8, 10 |
Area = (a × b) / 2, where a and b are the two legs (not the hypotenuse). This works because the two legs are already perpendicular, so they serve as the base and height. For a 3-4-5 triangle: area = (3×4)/2 = 6 square units.
| Triangle | Area Formula | Result |
|---|---|---|
| 3-4-5 | (3×4)/2 | 6 |
| 5-12-13 | (5×12)/2 | 30 |
| 8-15-17 | (8×15)/2 | 60 |
| 7-24-25 | (7×24)/2 | 84 |
A right triangle has one 90-degree angle, making it the most studied triangle in mathematics. The side opposite the right angle is the hypotenuse, always the longest side. The other two sides are called legs. Given any two measurements (sides or angles), you can solve for all remaining values.
The Pythagorean theorem (a² + b² = c²) relates the three sides, while trigonometric functions (sin, cos, tan) connect sides to angles. Together, these tools let you solve any right triangle completely from just two known values.
Our right triangle calculator accepts any valid combination of inputs — two sides, one side and one angle, or any other pair — and computes all missing measurements. Every solution includes step-by-step work showing the exact formulas and arithmetic used.
Last Updated: Mar 11, 2026
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