Introduction
A right triangle is a triangle that has one angle equal to 90 degrees. The longest side, called the hypotenuse, sits across from the right angle. The other two sides are called legs. If you know just two measurements of a right triangle, you can figure out everything else: all three sides, all three angles, the area, the perimeter, and more.
This Right Triangle Calculator makes that process fast and simple. Choose the values you already know (like two sides, a side and an angle, or even the area and perimeter), and the calculator finds the rest. It uses the Pythagorean theorem, trigonometry, and basic geometry formulas to find every missing measurement. You'll also get a labeled diagram of your triangle, a step-by-step solution that shows the math behind each answer, and a bar chart comparing the side lengths. This tool gives you accurate results for homework, for checking your work, or for solving a real-world problem.
How to use our Right Triangle Calculator
Enter any two known measurements of a right triangle, and this calculator will find all missing sides, angles, area, perimeter, altitude, and more. It also shows a diagram, a step-by-step solution, and a bar chart comparing the side lengths.
Select Known Values: Pick which pair of measurements you already know. You can choose from nine options: Two Sides, Side + Angle, Side + Area, Side + Perimeter, Side + Altitude h, Angle + Area, Angle + Perimeter, Angle + Altitude h, or Area + Perimeter.
Side a (opposite α): Enter the length of leg a, which is the side across from angle alpha. Use the dropdown next to it to pick your unit, such as mm, cm, m, km, in, ft, yd, or mi.
Side b (opposite β): Enter the length of leg b, which is the side across from angle beta. Choose the matching unit from the dropdown.
Side c (hypotenuse): Enter the length of the hypotenuse, which is the longest side of the right triangle. Select your preferred unit from the dropdown.
Angle α (alpha): Enter one of the two acute angles in degrees or radians. This field also accepts expressions like pi/3, pi/4, or 2pi/5 for exact radian values. The angle must be between 0° and 90°.
Angle β (beta): Enter the other acute angle in degrees or radians. It works the same way as the alpha field and also supports pi expressions.
Altitude h (to hypotenuse): Enter the length of the line drawn from the right-angle corner straight down to the hypotenuse. Pick your unit from the dropdown.
Area (A): Enter the area of the triangle. Choose a square unit from the dropdown, such as mm², cm², m², in², ft², or yd².
Perimeter (P): Enter the total distance around the triangle. Select the matching length unit from the dropdown.
Which side / angle is known: When a mode asks for one side or one angle, use the selector dropdown to tell the calculator whether you are entering side a, side b, side c, angle α, or angle β.
Calculate and Reset: Press the Calculate button to see your results, or press Reset to clear all fields and start over with the default 3-4-5 right triangle.
What Is a Right Triangle?
A right triangle is a triangle that has one angle that measures exactly 90 degrees. The side across from the right angle is always the longest side and is called the hypotenuse. The other two sides are called legs. Right triangles show up everywhere in real life: in buildings, ramps, screen sizes, and even in navigation.
Key Formulas for Right Triangles
The most important formula for any right triangle is the Pythagorean theorem: a² + b² = c², where a and b are the two legs and c is the hypotenuse. This means if you know any two sides, you can always find the third. For example, a triangle with legs of 3 and 4 has a hypotenuse of 5, because 3² + 4² = 9 + 16 = 25, and √25 = 5.
The two acute angles in a right triangle always add up to 90 degrees. If one acute angle is 30°, the other must be 60°. You can find unknown angles using trigonometric functions: sine, cosine, and tangent. These ratios relate the angles to the side lengths:
- sin(α) = opposite / hypotenuse = a / c
- cos(α) = adjacent / hypotenuse = b / c
- tan(α) = opposite / adjacent = a / b
If you need to find the slope of a line or ramp that forms a right triangle, the tangent ratio is essentially the same calculation: rise over run. The distance between two points on a coordinate plane is found the same way, using the Pythagorean theorem to get the straight-line length.
Area, Perimeter, and Altitude
The area of a right triangle is simple to calculate: Area = ½ × a × b. Because the two legs meet at a right angle, they act as the base and height. The perimeter is just the sum of all three sides: P = a + b + c.
The altitude to the hypotenuse (often labeled h) is the perpendicular line drawn from the right-angle vertex to the hypotenuse. It can be found using the formula h = (a × b) / c. This altitude creates two smaller triangles inside the original, and both of those smaller triangles are also right triangles that are similar to the original.
Other Useful Properties
A right triangle also has a few special properties worth knowing:
- The circumradius (radius of the circle that passes through all three vertices) is always exactly half the hypotenuse: R = c / 2.
- The inradius (radius of the circle that fits perfectly inside the triangle) is calculated as r = (a + b − c) / 2.
- The median to the hypotenuse is also equal to half the hypotenuse, which means it equals the circumradius.
The center of the hypotenuse is exactly where the circumscribed circle is centered.
How to Use This Calculator
This right triangle calculator lets you find every measurement of a right triangle from just two known values. You can enter two sides, a side and an angle, a side and the area, or several other combinations. The calculator then uses the Pythagorean theorem and trigonometry to solve for all missing sides, angles, area, perimeter, altitude, inradius, and circumradius. It also shows a step-by-step solution so you can follow along with the math and a diagram of the triangle drawn to scale.