📐 Right Triangle Trigonometry & Pythagorean Calculator
A right triangle trig calculator solving from any two values: sides, angles, area, altitude, inradius and circumradius, with the working shown.
What Right Triangle Trigonometry & Pythagorean Calculator Does
A right triangle has three unknowns once the 90° angle is accounted for, and any two of them fix the rest. That is the whole subject. Give it two legs and Pythagoras returns the hypotenuse; give it a leg and the hypotenuse and the same equation runs backwards; give it a side and an acute angle and the trigonometric ratios finish the job.
Most calculators for this term promise exactly that, and Google now prints the Pythagorean theorem above the results anyway. What almost none of them compute is what comes next in a geometry course — the altitude drawn to the hypotenuse, the radius of the circle that fits inside the triangle, and the radius of the circle that passes through all three corners.
Those are not decorative. The altitude to the hypotenuse splits the triangle into two smaller triangles both similar to the original, which is the basis of the geometric mean relations. The circumradius is exactly half the hypotenuse — always, for every right triangle — because the hypotenuse is a diameter of the circumscribed circle. That is Thales’ theorem, and it is one of the oldest results in geometry.
This calculator solves from any of five input pairs and reports all of it, including whether what you entered is one of the classic triangles: 45-45-90, 30-60-90, or a Pythagorean triple with three whole-number sides.
How to Use Right Triangle Trigonometry & Pythagorean Calculator
- Pick which two values you know — two legs, a leg and the hypotenuse, or a side and an acute angle
- Enter those two numbers; the other inputs disappear so you cannot over-specify the triangle
- Read the complete solution: both legs, hypotenuse, both acute angles, area and perimeter
- Check the derived geometry — altitude to the hypotenuse, inradius, circumradius and any special-triangle match
Formula Used by Right Triangle Trigonometry & Pythagorean Calculator
Pythagorean theorem — both directions
c = √(a² + b²) and a = √(c² − b²)
- a, b
- The two legs — the sides forming the right angle
- c
- The hypotenuse, opposite the right angle and always the longest side
Worked example
Legs of 3 and 4, then a hypotenuse of 13 with one leg of 5
- c = √(9 + 16) = √25 = 5
- a = √(169 − 25) = √144 = 12
Result: The 3-4-5 and 5-12-13 triangles — the two smallest Pythagorean triples
Solving from a side and an angle
sin A = a/c · cos A = b/c · tan A = a/b
- A
- The acute angle opposite leg a
- B
- The other acute angle, always 90° − A
Worked example
Leg a = 5 opposite an angle of 30°
- c = a ÷ sin 30° = 5 ÷ 0.5
- b = a ÷ tan 30° = 5 ÷ 0.57735
Result: c = 10 and b = 8.660254 — the 30-60-90 ratio 1 : √3 : 2, scaled by 5
The values the other calculators skip
h = ab/c · r = (a + b − c)/2 · R = c/2
- h
- Altitude from the right angle to the hypotenuse
- r
- Inradius — the largest circle that fits inside
- R
- Circumradius — half the hypotenuse, by Thales’ theorem
Worked example
The 3-4-5 triangle
- h = (3 × 4) ÷ 5 = 2.4
- r = (3 + 4 − 5) ÷ 2 = 1
- R = 5 ÷ 2 = 2.5
Result: A unit inradius, which is part of why the 3-4-5 is so convenient in practice
Five ways in — any two values solve it
A right triangle is fixed by any two independent values. Which two you have determines the route, not the answer.
| If you know | The missing side comes from | The angles come from |
|---|---|---|
| Both legs a and b | c = √(a² + b²) | A = arctan(a/b) |
| Leg a and hypotenuse c | b = √(c² − a²) | A = arcsin(a/c) |
| Leg b and hypotenuse c | a = √(c² − b²) | A = arccos(b/c) |
| Leg a and angle A | c = a ÷ sin A, b = a ÷ tan A | B = 90° − A |
| Hypotenuse c and angle A | a = c × sin A, b = c × cos A | B = 90° − A |
The six smallest Pythagorean triples
All three sides whole numbers. Note that the inradius comes out an integer in every one of them — a property of these triples, not a coincidence.
| a | b | c | Angle A | Area | Altitude to c | Inradius |
|---|---|---|---|---|---|---|
| 3 | 4 | 5 | 36.8699° | 6 | 2.400000 | 1 |
| 5 | 12 | 13 | 22.6199° | 30 | 4.615385 | 2 |
| 8 | 15 | 17 | 28.0725° | 60 | 7.058824 | 3 |
| 7 | 24 | 25 | 16.2602° | 84 | 6.720000 | 3 |
| 20 | 21 | 29 | 43.6028° | 210 | 14.482759 | 6 |
| 9 | 40 | 41 | 12.6804° | 180 | 8.780488 | 4 |
The two special triangles
These appear constantly because their values are exact surds rather than decimals. The calculator flags them when your input matches.
| Triangle | Side ratio | Exact values | Where it comes from |
|---|---|---|---|
| 45-45-90 | 1 : 1 : √2 | sin 45° = cos 45° = √2/2 ≈ 0.7071 | A square cut along its diagonal |
| 30-60-90 | 1 : √3 : 2 | sin 30° = 0.5, sin 60° = √3/2 ≈ 0.8660 | An equilateral triangle cut down its height |
How to Read Your Result
The altitude creates two similar triangles
Drop a perpendicular from the right angle to the hypotenuse and you get two smaller right triangles, each similar to the original and to each other. That similarity produces the geometric mean relations: the altitude squared equals the product of the two hypotenuse segments it creates. In the 3-4-5 triangle the altitude is 2.4 and it splits the hypotenuse into 1.8 and 3.2 — and 2.4² = 5.76 = 1.8 × 3.2 exactly. This is the most useful fact about right triangles that the general calculators leave out.
Why the circumradius is always half the hypotenuse
Draw a circle with the hypotenuse as its diameter and the right-angle vertex lands exactly on that circle, every time. The converse holds too: any point on a circle sees the diameter at exactly 90°. This is Thales’ theorem, and it means the circumradius needs no real calculation — it is c ÷ 2 and nothing else. It also explains why the midpoint of the hypotenuse is equidistant from all three vertices.
The hypotenuse must be longest, and the check is not optional
The largest angle in any triangle faces the longest side, and in a right triangle the largest angle is the 90° one. So the hypotenuse always exceeds either leg. Enter a hypotenuse shorter than the leg you paired it with and there is no triangle to solve — c² − a² goes negative and its square root is imaginary. This calculator refuses the input rather than returning NaN, which is a more useful answer than a blank field.
The two acute angles are locked together
They sum to exactly 90°, because the three angles of a triangle total 180° and one is already spoken for. Knowing one acute angle is therefore the same as knowing both, which is why a side plus any angle is enough to solve the whole triangle. It is also where the co-function identities get their name: sin A = cos B whenever A and B are complementary, so sin 30° and cos 60° are both 0.5.
Why 3-4-5 shows up on building sites
Measure 3 units along one line, 4 along another, and adjust until the diagonal reads exactly 5 — the corner is then square to within your measuring accuracy, with no instrument beyond a tape. It is the smallest set of whole numbers satisfying a² + b² = c², which is why it became a standard field method for squaring foundations and remains in use. Multiples behave identically: 6-8-10 and 12-16-20 are the same triangle scaled up.
Limitations & Accuracy Notes
- Results are unitless. Enter meters and the sides come back in meters with the area in square meters; the arithmetic makes no assumption about scale, but do not mix units within one problem.
- Angles are entered and reported in degrees. Radian input is not accepted here — multiply by 180/π to convert first.
- The acute angle must be strictly between 0° and 90°. At either endpoint the triangle degenerates into a straight line and has no interior.
- Only right triangles are handled. A general triangle with no 90° angle needs the law of sines or the law of cosines, which is a different calculation.
- Special-triangle detection uses a small tolerance on the angle, so a triangle within about a twentieth of a degree of 45-45-90 is flagged as one. Check the exact ratio if it matters.
Frequently Asked Questions
Can I solve a right triangle if I only know one side and one angle?
What is the Pythagorean theorem?
Why must the hypotenuse be the longest side?
How do I find the altitude to the hypotenuse?
What are the special right triangles?
Do the two acute angles always add to 90 degrees?
References & Further Reading
- NIST Digital Library of Mathematical Functions — Trigonometric functions — US National Institute of Standards and Technology reference for the sine, cosine and tangent definitions used throughout
- Calculator.net — Right Triangle Calculator — The most complete competing treatment, and the source of the any-two-values approach adopted here