Right Triangle Calculator
Enter any 2 of 7 values — legs, hypotenuse, either acute angle, altitude, area or perimeter — and get every remaining measurement, plus the inradius, circumradius and a to-scale diagram.
What is a right triangle?
A right triangle is any triangle that contains one 90° angle. The two sides that meet at the right angle are the legs (usually written a and b); the side opposite the right angle — always the longest — is the hypotenuse, c. The two non-right angles are acute and, because the interior angles of a triangle sum to 180°, they always add up to 90°: α + β = 90°.
That single 90° angle is what makes right triangles special. The Pythagorean theorem a² + b² = c² holds only here, and the sine, cosine and tangent ratios are defined on the sides of a right triangle. In practice this means any two independent measurements are enough to lock the whole triangle down — the calculator above uses that fact to solve for every remaining side, angle, area, perimeter, altitude, inradius and circumradius.
Right triangles, case by case
Find a missing side with the Pythagorean theorem
If you know both legs, the hypotenuse is the square root of the sum of their squares. If you know the hypotenuse and one leg, subtract that leg-square from the hypotenuse-square before taking the root.
Find a side from one side + an acute angle (trigonometry)
When you only know one side plus one acute angle, use sine, cosine or tangent. sin α = opposite/hypotenuse, cos α = adjacent/hypotenuse, tan α = opposite/adjacent. Pick the ratio that pairs your known side with the one you need.
Find a missing angle with inverse trigonometry
Given any two sides, form the appropriate ratio and take its inverse trig function. The other acute angle is always 90° − α.
Altitude from the right angle to the hypotenuse
The altitude h drops from the 90° corner perpendicular to the hypotenuse. It splits the triangle into two smaller triangles that are similar to the whole. Because the area can be written as ½·a·b OR ½·c·h, the two must match — giving h = a·b / c.
Circumradius = half the hypotenuse (Thales' theorem)
The circle passing through all three vertices — the circumcircle — always has the hypotenuse as its diameter for a right triangle. This is Thales' theorem: an angle inscribed in a semicircle is a right angle. So the circumradius is exactly half the hypotenuse, no matter the shape of the right triangle.
Trigonometric ratios — the cheat sheet
For a right triangle with acute angle α opposite leg a, the three primary ratios and their inverses cover every side-and-angle problem.
| Ratio | Definition | Solves for a side | Solves for α |
|---|---|---|---|
| sin α | opposite / hypotenuse = a / c | a = c · sin α | α = sin⁻¹(a / c) |
| cos α | adjacent / hypotenuse = b / c | b = c · cos α | α = cos⁻¹(b / c) |
| tan α | opposite / adjacent = a / b | a = b · tan α | α = tan⁻¹(a / b) |
Special right triangles
Two right triangles come up so often in trigonometry that their side ratios are worth memorising. Given a single side, you can write the other two straight from the ratio — no calculator required.
30°-60°-90°
Side ratio 1 : √3 : 2. The short leg faces the 30° angle; the long leg faces the 60° angle; the hypotenuse is twice the short leg.
45°-45°-90°
Side ratio 1 : 1 : √2. Because two angles are equal, the two legs are equal — it's an isosceles right triangle, and the hypotenuse is √2 times a leg.
Where h = a·b / c actually comes from
The altitude formula falls straight out of computing the same area two different ways. Using the two legs as base and height gives A = ½·a·b. Using the hypotenuse as the base with h as its perpendicular height gives A = ½·c·h. Setting the two equal:
The same picture also produces the geometric-mean relationship h² = p·q, where p and q are the two segments the altitude cuts the hypotenuse into. That's another consequence of the three similar triangles the altitude creates.
Features of this calculator
- Accepts any 2 of: leg a, leg b, hypotenuse c, angle α, angle β, altitude h, area A, perimeter P.
- Degrees or radians toggle for every angle input and output.
- Live to-scale SVG showing labelled sides, angles, the right-angle marker and the altitude h.
- One-click presets for the 3-4-5, 30-60-90 and 45-45-90 triangles plus a ladder-against-wall example.
- Step-by-step derivation using your actual numbers — every substitution shown.
- Reports the inradius r = (a + b − c) / 2 and circumradius R = c / 2 (half the hypotenuse, always).
Frequently asked questions
+What is a right triangle?
A right triangle is a triangle with exactly one 90° angle. The side opposite the right angle is the hypotenuse (always the longest side), and the two sides that form the right angle are called the legs.
+How do I find a missing side of a right triangle?
If you know both legs or the hypotenuse and one leg, use the Pythagorean theorem: a² + b² = c². If you know one side and one acute angle, use the trigonometric ratios: sin α = opposite/hypotenuse, cos α = adjacent/hypotenuse, tan α = opposite/adjacent.
+How do I find a missing angle?
Take any two known sides and apply an inverse trig function. For example, α = tan⁻¹(a/b) if you know both legs, α = sin⁻¹(a/c) if you know a leg and the hypotenuse. Because the angles in any triangle add to 180° and one is 90°, the other acute angle is simply β = 90° − α.
+What is the altitude of a right triangle?
The altitude h is the perpendicular distance from the right-angle vertex to the hypotenuse. It splits the triangle into two smaller triangles similar to the original. The compact formula is h = a·b / c — the product of the legs divided by the hypotenuse.
+Why is the circumradius always half the hypotenuse?
By Thales' theorem, any triangle inscribed in a circle with one side as a diameter is a right triangle, and the right angle sits on the circle. Turned around: for a right triangle, the hypotenuse IS a diameter of the circumscribed circle. So R = c / 2 exactly.
+What are the 30-60-90 and 45-45-90 special triangles?
They are the two most useful right triangles in trigonometry. A 30-60-90 triangle has sides in the fixed ratio 1 : √3 : 2 — if the short leg is 1, the long leg is √3 and the hypotenuse is 2. A 45-45-90 triangle (an isosceles right triangle) has sides 1 : 1 : √2. Knowing just one side lets you write down the other two without a calculator.
+How is this different from your Triangle and Pythagorean calculators?
The Triangle Calculator solves any triangle (not necessarily right-angled) from SSS, SAS, ASA, AAS or SSA. The Pythagorean Theorem Calculator only handles the single equation a² + b² = c². This Right Triangle Calculator is specialised: it accepts any two of seven measurements — legs, hypotenuse, either acute angle, altitude, area or perimeter — and returns every one of them along with the inradius and circumradius.