The Right Triangle Calculator solves a right-angled triangle from any two measurements that fix the shape: two sides, or one side and one acute angle. It returns the remaining sides and angles, the area, the perimeter, the altitude to the hypotenuse, and a labelled diagram with the right angle marked by a square.
A right triangle has one 90-degree angle, so two further measurements determine everything else. Enter what is known and the calculator applies the Pythagorean theorem or the trigonometric ratios, showing each substituted step.
Solve a right triangle from two measurements
A right triangle is fixed once two independent measurements are known, because the right angle already supplies the third. Valid pairs include two legs, a leg and the hypotenuse, or any side paired with one of the acute angles. Combinations that leave the triangle free to scale are rejected with a clear message rather than a guessed number.
Enter the known values in the fields that match them. Lengths accept the same unit set used across the geometry suite. Angles accept degrees or radians. Live recalculation updates the diagram and the result panel as soon as a valid pair is present.
The case card names whether the solution used the Pythagorean theorem, a sine or cosine ratio, or both in sequence, so the method is never hidden.
Find the hypotenuse and legs
The two sides that meet at the right angle are the legs. The side opposite the right angle is the hypotenuse, and it is always the longest side. The Pythagorean theorem relates all three:
a² + b² = c²Solving for the hypotenuse when both legs are known:
c = √(a² + b²)Solving for a missing leg when the hypotenuse and the other leg are known:
b = √(c² - a²)The calculator applies the matching form automatically. If the entered hypotenuse is shorter than a leg, the configuration is impossible and the tool says so instead of inventing a negative length under a square root.
Use the trigonometric ratios
When one acute angle and one side are known, the three primary ratios finish the triangle. With the right angle fixed, each acute angle has a clear opposite side, adjacent side, and hypotenuse.
sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacentSOH CAH TOA is the usual memory aid: sine takes opposite over hypotenuse, cosine takes adjacent over hypotenuse, and tangent takes opposite over adjacent. Rearranging any ratio finds the unknown length once the angle and one side are set.
The second acute angle is then 90 degrees minus the first, because the two acute angles in a right triangle are complementary. The calculator reports both angles with the sides so the complementary check is visible.
Calculate the area of a right triangle
Area for a right triangle is half the product of the two legs, because each leg is a base and the other is the matching perpendicular height. That shortcut is the reason right-triangle area rarely needs a separate height measurement when both legs are already known from the solver.
A = (1/2) × a × bIf the known pair is a leg and the hypotenuse, the missing leg is found first with the Pythagorean theorem, then the area formula runs on the two legs. The result uses square units matching the side units. A labelled altitude from the right-angle vertex to the hypotenuse is also available once all three sides are known.
Solve a right triangle with legs 3 and 4
Take the classic case with legs a = 3 and b = 4. This is the smallest integer Pythagorean triple, and every step is a clean whole number until the acute angles, which are reported in degrees to two decimal places.
1. Find the hypotenuse. `` c = √(3² + 4²) = √(9 + 16) = √25 = 5 ``
2. Find the area. `` A = (1/2) × 3 × 4 = 6 ``
3. Find the acute angles. `` tan(A) = 3/4 = 0.75 → A ≈ 36.87° B = 90 - 36.87 = 53.13° ``
The diagram marks the right angle with a square and labels each side and angle with its solved value.
Find the heights and other measures
Once all three sides are known, several secondary measures follow without new inputs. The altitude to the hypotenuse is h = ab / c. For legs 3 and 4 with hypotenuse 5, that altitude is 12 / 5 = 2.4. The perimeter is the sum of the three sides: 3 + 4 + 5 = 12.
The inradius of a right triangle is (a + b - c) / 2, which for 3-4-5 equals 1. The circumradius is c / 2 = 2.5, because the hypotenuse is the diameter of the circumcircle. These extras sit in the result panel beside the primary sides and angles so a full solution is available from the same two inputs.
Frequently asked questions
How do I solve a right triangle?
Enter any two measurements that fix the triangle: two sides, or one side and one acute angle. The calculator applies the Pythagorean theorem or the trigonometric ratios, returns the remaining sides and angles, and shows the substituted working beside a labelled diagram.
What is the hypotenuse?
The hypotenuse is the side opposite the right angle and the longest side of the triangle. In the relation a² + b² = c², the hypotenuse is c. For legs 3 and 4 it equals 5.
When do I use sine, cosine or tangent?
Use sine when the known side and the unknown side are opposite the angle and the hypotenuse. Use cosine when they are adjacent and the hypotenuse. Use tangent when both sides in the ratio are legs. The calculator selects the matching ratio from the known pair.
How is the area of a right triangle calculated?
Area equals half the product of the two legs: A = (1/2) × a × b. For legs 3 and 4 the area is 6. If a leg is missing, find it first, then apply the same formula.
Can I enter an angle instead of a second side?
Yes. One acute angle plus any side determines the similar shape and the scale. The complementary angle is 90 degrees minus the entered angle, and the remaining sides follow from sine, cosine or tangent.
What if my sides do not form a right triangle?
If three sides are entered and a² + b² does not equal c² for the longest side, the triangle is not right-angled. This page solves the right-angled case. For a general triangle use the Triangle Calculator, which applies the law of cosines and law of sines.
How do the acute angles relate?
The two acute angles always add to 90 degrees. Once one is known, the other is its complement. Checking that sum is a fast way to verify an answer by hand.
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers a, b and c that satisfy a² + b² = c². The smallest is 3-4-5. Scaled copies such as 6-8-10 are also right triangles. The Pythagorean Theorem Calculator lists common triples in more depth.
Does the calculator show the working?
Yes. Every result includes the substituted arithmetic for the theorem or ratio used, rounded only for display. The same path the engine uses appears in the breakdown so a hand check is possible.
Summary
The Right Triangle Calculator solves any right-angled triangle from two measurements, returning the missing sides and angles, the area, and secondary measures such as the altitude to the hypotenuse. Legs and hypotenuse connect through a² + b² = c². Acute angles connect through sine, cosine and tangent. The worked case with legs 3 and 4 yields hypotenuse 5, area 6, and angles about 36.87° and 53.13°.