Right Triangle Calculator
A right triangle needs only two facts. Enter any two of leg a, leg b, hypotenuse c, either acute angle, or the area — and get every remaining side, angle, the altitude to the hypotenuse, perimeter, inradius, and circumradius, with special-triangle patterns flagged automatically.
Steps & special-triangle notes
Two facts fix everything
Because one angle is locked at 90°, a right triangle has only two degrees of freedom. Two sides resolve through the Pythagorean theorem; a side plus an angle resolves through sine, cosine, or tangent; a side plus the area works because area = ab/2 ties the legs together. The two acute angles always sum to 90°, the hypotenuse is always the longest side, and a few famous shapes recur constantly: the 45-45-90 (sides 1 : 1 : √2), the 30-60-90 (1 : √3 : 2), and the 3-4-5 family of whole-number triples — the steps panel calls them out when your triangle matches.
How it’s calculated
Pythagorean theorem: a² + b² = c². Trig relations: a = c·sin α, b = c·cos α, tan α = a/b, β = 90° − α. Area = ab/2; altitude to the hypotenuse h = ab/c; inradius = (a + b − c)/2; circumradius = c/2. Given c and area: a + b = √(c² + 4·Area) and a − b = √(c² − 4·Area).
Results update as you type and are for education, not professional advice — double-check any number that matters.
Worked example
Legs a = 3 and b = 4: c = √(9 + 16) = 5, angles α = 36.87° and β = 53.13°, area = 6, perimeter = 12. The altitude to the hypotenuse is 3 × 4 ÷ 5 = 2.4, the inradius is (3 + 4 − 5)/2 = 1, and the circumradius is 5/2 = 2.5. The steps panel flags it as a 3-4-5 Pythagorean triple.
Common mistakes
- Entering both acute angles — they fix the shape but not the size; pair an angle with a side or the area instead.
- Putting a leg in the hypotenuse box — c must be the longest side, opposite the right angle.
- Assuming the altitude h equals a leg — it is the perpendicular dropped onto the hypotenuse, ab÷c.
Where it is used
- Carpentry and framing — rafters, stair stringers, squaring corners.
- Ladder-against-wall and ramp problems.
- Vector and screen-geometry work — splitting a magnitude into components.
Frequently asked questions
Which two values can I enter?
Any two that include at least one length: two sides, a side and an acute angle, or a side (or hypotenuse) and the area. Two angles alone cannot set the size, and the calculator will say so.
What makes 3-4-5 special?
It is the smallest whole-number Pythagorean triple (9 + 16 = 25), and any multiple — 6-8-10, 9-12-15 — is also right-angled. Builders exploit it to square corners with just a tape measure.
What are the 45-45-90 and 30-60-90 shortcuts?
In a 45-45-90 the legs are equal and the hypotenuse is leg × √2. In a 30-60-90 the sides run 1 : √3 : 2 — the short leg is half the hypotenuse. Recognizing them avoids trig entirely.
How is the altitude to the hypotenuse found?
h = ab ÷ c, because the area can be written both as ab/2 and as ch/2. For 3-4-5, h = 12 ÷ 5 = 2.4, and it splits the hypotenuse into segments of 1.8 and 3.2.
Given the hypotenuse and area, how are the legs recovered?
From ab = 2·Area and a² + b² = c²: (a+b)² = c² + 4·Area and (a−b)² = c² − 4·Area. If c² < 4·Area no right triangle exists — the maximum area for a given c is c²/4.