Give this tool any two known values — sides and/or an acute angle — and it solves the rest of a right triangle: the missing sides via the Pythagorean theorem, both acute angles via SOHCAHTOA, plus the area and perimeter. Useful for geometry, trigonometry homework, construction layout, and surveying.
How it works
The right angle sits at vertex C, with angle A opposite leg a, angle B opposite leg b, and the hypotenuse c opposite the right angle. Depending on the two values you supply, the solver applies:
- Two legs: c = √(a² + b²); angle A = arctan(a ÷ b).
- Leg and hypotenuse: the other leg = √(c² − leg²); angles from arcsin/arccos.
- Side and acute angle: the missing sides come from sin, cos, or tan of the angle (SOHCAHTOA).
It always uses that the two acute angles sum to 90°, then computes area = ½ × a × b and perimeter = a + b + c.
Example
Legs a = 3 and b = 4:
- Hypotenuse c = √(3² + 4²) = √25 = 5
- Angle A = arctan(3 ÷ 4) ≈ 36.87°
- Angle B = 90 − 36.87 = 53.13°
- Area = ½ × 3 × 4 = 6
- Perimeter = 3 + 4 + 5 = 12
| Known | Find sides with | Find angles with |
|---|---|---|
| Two legs | Pythagoras | arctan |
| Leg + hypotenuse | Pythagoras | arcsin / arccos |
| Side + angle | sin / cos / tan | 90° − known angle |
Everything is computed in your browser — nothing is uploaded.
Why right triangles come up so often in real life
The right triangle is the geometry workhorse precisely because any straight-line measurement problem involving a known angle can be reduced to one. Practical applications:
- Construction and carpentry: checking that a wall is plumb (vertical), laying out a square foundation (the 3-4-5 rule), calculating the pitch of a roof (rise over run, which is a tangent ratio)
- Navigation and surveying: triangulating distances to inaccessible points, finding the height of a tree or building without climbing it
- Engineering drawing: resolving forces into horizontal and vertical components, calculating cable lengths and deflection angles
- Screen and display work: computing diagonal sizes from width and height (the hypotenuse of a right triangle with the two dimensions as legs — which is how screen sizes are measured)
The full set of formulas this solver uses
Given legs a and b, hypotenuse c, and angles A (opposite a) and B (opposite b), with C = 90°:
Pythagorean theorem: c = sqrt(a² + b²)
a = sqrt(c² - b²)
SOHCAHTOA:
sin(A) = a / c => A = arcsin(a / c)
cos(A) = b / c => A = arccos(b / c)
tan(A) = a / b => A = arctan(a / b)
Complement: B = 90° - A
Area: area = (1/2) * a * b
Perimeter: p = a + b + cThe solver detects which pair of values you provide and selects the shortest path to the full solution — for example, two legs use Pythagoras for the hypotenuse and arctan for angle A; a leg and an angle use the trig ratio directly for the other leg.
Common examples
The 3-4-5 triangle: legs 3 and 4, hypotenuse 5, angles approximately 36.87° and 53.13°. This is the simplest integer right triangle and a staple of construction layout — multiply all three sides by any constant and the right angle is guaranteed.
A 30-60-90 triangle: if one acute angle is 30° and the hypotenuse is known, the short leg is half the hypotenuse and the long leg is the short leg times √3. Enter the hypotenuse and 30° into the solver to verify.
Screen diagonal: a display that is 16 inches wide and 9 inches tall (16:9 ratio) has a diagonal of √(16² + 9²) = √(256 + 81) = √337 ≈ 18.4 inches.