Solve any right triangle instantly. Enter any two values — at least one must be a side — to find every side, angle, area and trig ratio, with steps shown.
A right triangle has one 90° angle (always angle C) and needs just two other values to be fully solved — as long as one of them is a side length. Two sides use the Pythagorean theorem: c = √(a² + b²). A side and an angle use trigonometry: SOH-CAH-TOA. Enter any two values below and the calculator finds the rest, plus the area, perimeter, altitude and all six trig ratios.
Angle C is always 90°. Leave the value you want solved for blank — at least one entry must be a side.
Keyboard: Enter solves, Esc clears.
Common Right Triangles
| Property | Value |
|---|
| Ratio | Angle A | Angle B |
|---|
A right triangle contains exactly one 90° angle. By convention on this page that is angle C, so the other two angles, A and B, always add to 90°. The side opposite the right angle is the hypotenuse (c) and is always the longest side; the other two, a and b, are the legs.
Right triangles are the working core of trigonometry — every sine, cosine and tangent value is originally defined from one. They are also the shape behind the Pythagorean theorem, covered in full on the Pythagorean theorem calculator; this page focuses on solving the whole triangle from any two known values, sides or angles.
| Ratio | Definition | Mnemonic |
|---|---|---|
| sin(θ) | Opposite / Hypotenuse | SOH |
| cos(θ) | Adjacent / Hypotenuse | CAH |
| tan(θ) | Opposite / Adjacent | TOA |
| csc(θ) | Hypotenuse / Opposite | 1 / sin |
| sec(θ) | Hypotenuse / Adjacent | 1 / cos |
| cot(θ) | Adjacent / Opposite | 1 / tan |
"Opposite" and "adjacent" depend on which angle you are looking from. For angle A, side a is opposite and side b is adjacent; for angle B those swap. The hypotenuse never changes — it is always c, regardless of which angle is in question.
| You know | Use | Formula |
|---|---|---|
| Two legs (a, b) | Pythagorean theorem | c = √(a² + b²) |
| A leg and the hypotenuse | Pythagorean theorem | a = √(c² − b²) |
| One side and one angle | Trigonometry (SOH-CAH-TOA) | e.g. a = c × sin A |
| Two angles only | Not solvable for size | Angles fix the shape, not the scale — at least one side is required |
| Type | Angles | Side ratio | Note |
|---|---|---|---|
| 45-45-90 | 45°, 45°, 90° | 1 : 1 : √2 | Isosceles — half a square cut along its diagonal |
| 30-60-90 | 30°, 60°, 90° | 1 : √3 : 2 | Half an equilateral triangle |
Recognising these instantly gives every side without a calculator. A 45-45-90 triangle with legs of 5 has a hypotenuse of 5√2 ≈ 7.071. A 30-60-90 triangle with a short leg of 4 has a long leg of 4√3 ≈ 6.928 and a hypotenuse of 8.
| Field | Use |
|---|---|
| Construction | Rafter and stair lengths from rise and run, squaring corners |
| Surveying | Height and distance from a measured angle (theodolite work) |
| Navigation | Resolving a course into north/east distance components |
| Engineering | Resolving forces and velocities into perpendicular components |
| Astronomy | Parallax and distance calculations using tiny angles |
| Mistake | Fix |
|---|---|
| Confusing opposite and adjacent | They depend on which angle you're using — relabel per angle |
| Using degrees where the calculator expects radians (or vice versa) | Check your calculator's angle mode |
| Assuming angles alone can size a triangle | At least one side length is always required |
| Mixing units between sides | Convert everything to one unit first |
| Treating a leg as the hypotenuse | c is always the longest side, opposite the 90° angle |
1) √(36+64) = 10 2) √(169−25) = 12, A = arcsin(5/13) ≈ 22.62°, B ≈ 67.38° 3) c = 7/sin40° ≈ 10.89, b = 7/tan40° ≈ 8.34
What is a right triangle?
A right triangle has exactly one 90° angle. The side opposite that angle is the hypotenuse — always the longest side — and the other two sides are the legs. The two non-right angles always add to 90°.
How do I solve a right triangle with two values?
If both values are sides, use the Pythagorean theorem. If one is a side and one is an angle, use SOH-CAH-TOA to find the rest. Two angles alone are not enough, since they fix the shape but not the size — at least one side is required.
What is SOH-CAH-TOA?
A memory aid for the three basic trig ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Which side counts as "opposite" or "adjacent" depends on which angle you're measuring from.
Which side is the hypotenuse?
The side opposite the right angle, and always the longest of the three. On this calculator it is labelled c, opposite angle C, which is fixed at 90°.
Can I solve a right triangle with only two angles?
No. Two angles determine the triangle's shape but not its size — infinitely many similar triangles share the same angles. At least one side length must be given.
What are the special right triangles?
The 45-45-90 (isosceles, sides in ratio 1:1:√2) and the 30-60-90 (sides in ratio 1:√3:2). Recognising them lets you find every side without a calculator.
How is this different from the Pythagorean theorem calculator?
The Pythagorean theorem calculator focuses purely on a² + b² = c² from two sides. This calculator solves the whole triangle — sides, all three angles, area, altitude and every trig ratio — from any two known values, including a side and an angle.
What is the altitude to the hypotenuse?
The perpendicular distance from the right angle to the hypotenuse, equal to ab/c. It also splits the hypotenuse into two segments, a²/c and b²/c, which together equal c.
What units does the calculator use?
Whatever you enter — the maths is unit-independent as long as every side uses the same one. A unit label is available purely so results display clearly.
Does the calculator show working?
Yes — every solve shows the formula used, the substituted values, and a final check that a² + b² = c².