Right Triangle FAQ

A right triangle is any triangle with one interior angle equal to exactly 90°. The side opposite the right angle is called the hypotenuse and is always the longest side; the other two sides are called legs. The two remaining angles are acute and always add up to 90°.

Enter any two known values from the eight a right triangle can have — the two legs, the hypotenuse, the two acute angles, the altitude to the hypotenuse, the area, or the perimeter. The solver picks the matching closed-form formula (or, for the two combinations with no clean algebraic formula, a numerical bisection search) to recover the full triangle, then shows every step of the derivation alongside a scaled diagram.

Any pair works except two angles alone, since two angles fix a triangle’s shape but not its size — you also need one length, the altitude, the area, or the perimeter to pin down scale. Every other pairing among the two legs, the hypotenuse, one angle, the altitude to the hypotenuse, the area, and the perimeter is supported.

Exactly two independent values, at least one of which carries scale (a length, an area, or a perimeter). Because the right angle is already fixed at 90° and the two acute angles must sum to 90°, a right triangle has only two remaining independent degrees of freedom — shape and size — so two well-chosen values are always enough.

The Pythagorean theorem states a² + b² = c², where a and b are the legs and c is the hypotenuse. It works because the areas of the squares built on the two legs sum exactly to the area of the square built on the hypotenuse — a relationship that can be proven with several classic dissection arguments dating back over two thousand years.

If you know both legs, take the square root of the sum of their squares: c = √(a² + b²). If you instead know one leg and an acute angle, divide the leg by the sine or cosine of that angle depending on which leg you have. This calculator handles both cases (and several others) automatically.

With two sides known, use the Pythagorean theorem: c = √(a² + b²) for the hypotenuse, or a = √(c² − b²) for a leg. With one side and one acute angle, use trigonometry: for example, a = c·sin α and b = c·cos α. The right triangle calculator picks the correct formula for whichever two values you enter.

Check the sides with the converse of the Pythagorean theorem: if the squares of the two shorter sides add up to the square of the longest side, it is a right triangle. For example, 8, 15, 17 works because 64 + 225 = 289 = 17². You can also check whether any angle measures exactly 90°.

No. It is not a triangle at all. Any two sides of a triangle must add up to more than the third, but 1 + 2 = 3, so the three segments lie flat along a straight line. It also fails the Pythagorean test, since 1² + 2² = 5, not 3² = 9.

Use Area = ½·c²·sin α·cos α. For a hypotenuse of 10 and an angle of 30°, that is ½ × 100 × 0.5 × 0.866 ≈ 21.65. The legs are c·sin α = 5 and c·cos α ≈ 8.66, and half their product gives the same area.

It is the perpendicular segment drawn from the right-angle vertex to the hypotenuse. It splits the original triangle into two smaller right triangles, both similar to the original and to each other, and equals (a·b)/c — the product of the legs divided by the hypotenuse.

The inradius r is the radius of the circle inscribed inside the triangle, tangent to all three sides, and equals (a + b − c) / 2. The circumradius R is the radius of the circle passing through all three vertices; for a right triangle it always equals half the hypotenuse, c / 2, because the hypotenuse is a diameter of that circle.

A Pythagorean triple is a set of three positive integers (a, b, c) satisfying a² + b² = c², such as (3, 4, 5) or (5, 12, 13). Any right triangle whose sides are proportional to one of these triples — including non-integer scaled versions like (6, 8, 10) — is flagged by this calculator’s classification badge.

The 45-45-90 triangle (isosceles right triangle, legs equal, hypotenuse = leg×√2) and the 30-60-90 triangle (sides in ratio 1 : √3 : 2) are called "special" because their side ratios are exact and memorable, letting you solve them from a single known side without a calculator.

No. A right triangle only needs one 90° angle, and the other two can be any pair of acute angles that add up to 90°, such as 30° and 60° or 36.87° and 53.13°. The 45-45-90 triangle is the special case where both legs are equal, and its sides follow the ratio 1 : 1 : √2.

No. The interior angles of any triangle always sum to 180°. Two 90° angles would already total 180° by themselves, leaving nothing for the third angle — which is impossible for a valid triangle.

Millimeters, centimeters, meters, inches, and feet. Switch units at any time with the unit selector; previously entered values and results are converted automatically without needing to re-solve from scratch.

Yes. Every solve updates the page URL with your inputs encoded as query parameters. Copy that URL and share it, or bookmark it — opening it again restores the exact same known values, unit, and angle mode.

Either — use the angle-mode toggle to switch. Internally the calculator always works in exact radians and converts for display, so switching modes mid-solve never changes the underlying triangle, only how the angle is displayed and entered.