Every formula for solving a right triangle — sides, angles, area
Reviewed by [email protected], Geometry Calculator Developer & Online Math Educator Last updated May 21, 2026
A right triangle is a triangle with one 90° angle. That single constraint enables far more formulas than any other triangle type — the Pythagorean theorem, the trigonometric ratios (sine, cosine, tangent), and the special triangle ratios (30-60-90 and 45-45-90) all apply. This page lists every formula you need to fully solve a right triangle from any 2 known pieces of information.
| Name | Formula | Notes |
|---|---|---|
| Pythagorean Theorem | a² + b² = c² |
a, b = legs (sides adjacent to the right angle); c = hypotenuse (opposite the right angle). Deep dive on the Pythagorean theorem page. |
| Find Hypotenuse | c = √(a² + b²) |
When both legs are known. |
| Find a Leg | a = √(c² − b²) |
When hypotenuse and one leg are known. |
| Area | A = ½ × a × b |
The two legs ARE the base and height (perpendicular to each other) — no extra calculation needed. |
| Area from Hypotenuse + Leg | A = ½ × b × √(c² − b²) |
Substitute the missing leg via Pythagoras. |
| Perimeter | P = a + b + c = a + b + √(a² + b²) |
Standard triangle perimeter; substitute c via Pythagoras. |
| Sine of an Angle | sin(θ) = opposite / hypotenuse |
For angle θ (not the right angle): opposite leg ÷ hypotenuse. Always between 0 and 1. |
| Cosine of an Angle | cos(θ) = adjacent / hypotenuse |
Adjacent leg ÷ hypotenuse. Always between 0 and 1. |
| Tangent of an Angle | tan(θ) = opposite / adjacent |
Opposite leg ÷ adjacent leg. SOH-CAH-TOA mnemonic: Sin-Opp-Hyp, Cos-Adj-Hyp, Tan-Opp-Adj. |
| Inverse Trig (find angle) | θ = arcsin(o/h), arccos(a/h), arctan(o/a) |
When you know two sides and need to find the angle, use the inverse functions on a calculator (sin⁻¹, cos⁻¹, tan⁻¹). |
| Sum of Acute Angles | A + B = 90° (since A + B + 90° = 180°) |
The two non-right angles are complementary. Knowing one gives the other immediately. |
| 30-60-90 Triangle Sides | 1 : √3 : 2 |
Opposite 30°, 60°, 90° respectively. Full derivation + examples: special right triangles. |
| 45-45-90 Triangle Sides | 1 : 1 : √2 |
Right isosceles triangle. Legs equal; hypotenuse = leg × √2. See special right triangles for the square-derivation. |
| Altitude from Right Angle | h = (a × b) / c |
The altitude drawn from the right-angle vertex perpendicular to the hypotenuse. |
| Inradius | r = (a + b − c) / 2 |
Radius of the inscribed circle. Simpler than the general triangle formula. |
| Circumradius | R = c / 2 |
In a right triangle the hypotenuse is the diameter of the circumscribed circle — so R = c/2. |
Using the right triangle calculator requires understanding which inputs determine the output. The tool is designed to solve for missing variables based on the geometric properties of a triangle containing a right angle. To begin, identify the two known values from your problem statement. These can be combinations of side lengths or angle measurements. For instance, you might know the lengths of both legs, or perhaps one leg and the hypotenuse. Alternatively, you may have one side length and an acute angle.
Once you have selected your input fields, choose the specific calculation mode required by your task. The available modes typically include calculating the area, determining the perimeter, or solving for unknown side lengths. Selecting the correct mode ensures the algorithm applies the appropriate mathematical relationships, such as the Pythagorean theorem for sides or basic trigonometry for angles. After entering your data, initiate the computation.
The calculator will then display the results along with a detailed, step-by-step solution. This feature is particularly useful for educational purposes, as it shows the formula derivation used to reach the final answer. By reviewing these steps, users can verify their manual calculations and understand how the sine, cosine, and tangent ratios relate to the sides of the triangle. This transparency helps reinforce the connection between abstract formulas and practical application, ensuring accuracy in homework, engineering projects, or architectural planning.
Understanding the fundamental formulas is essential for solving problems involving right triangles. The most well-known relationship is the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, expressed as a² + b² = c². Here, c represents the hypotenuse, the longest side opposite the right angle, while a and b are the legs adjacent to the right angle.
The area of a right triangle is calculated using the formula (1/2) * base * height. In this context, the two legs serve as the base and height because they are perpendicular to each other. This simplifies the calculation compared to general triangles where the altitude might need to be derived separately. The perimeter is straightforward, defined as the sum of all three sides: a + b + c.
Trigonometric ratios link the angles to the side lengths. The sine of an acute angle is the ratio of the opposite side to the hypotenuse (sin(θ) = opposite/hypotenuse). The cosine is the ratio of the adjacent side to the hypotenuse (cos(θ) = adjacent/hypotenuse), and the tangent is the ratio of the opposite side to the adjacent side (tan(θ) = opposite/adjacent). Mastery of these formulas allows for quick determination of any missing component within the triangle structure.
Consider a scenario where the two legs of a right triangle are given as 3 units and 4 units. To find the hypotenuse, apply the Pythagorean theorem: 3² + 4² = c², which simplifies to 9 + 16 = 25. Taking the square root yields a hypotenuse of 5 units. The area is calculated as (1/2) * 3 * 4 = 6 square units. This classic 3-4-5 triangle is a common example in geometry.
In another case, suppose the hypotenuse is 10 and one leg is 6. To find the other leg, rearrange the Pythagorean theorem: b² = c² - a². Substituting the values gives b² = 10² - 6², or 100 - 36 = 64. The square root of 64 is 8, so the missing leg is 8 units long. The perimeter would then be 6 + 8 + 10 = 24 units.
Finally, if one side and an acute angle are known, trigonometric ratios are necessary. For example, if the adjacent side is 5 and the angle is 30 degrees, the opposite side can be found using the tangent function: opposite = 5 * tan(30°). This approach demonstrates how different formulas apply depending on the available data, allowing for comprehensive problem-solving across various geometric configurations.
The hypotenuse is calculated using the Pythagorean theorem. If a and b are the lengths of the legs, the hypotenuse c is found by taking the square root of the sum of the squares of the legs: c = √(a² + b²). This formula only applies to triangles containing a right angle.
The area is determined by multiplying the lengths of the two legs and dividing by two. Since the legs form the right angle, one acts as the base and the other as the height. The formula is Area = (leg₁ × leg₂) / 2. This method is specific to right triangles; for other triangles, the general altitude must be identified.
The primary trigonometric functions are sine, cosine, and tangent. Sine is the ratio of the opposite side to the hypotenuse. Cosine is the ratio of the adjacent side to the hypotenuse. Tangent is the ratio of the opposite side to the adjacent side. These functions allow you to calculate missing sides or angles when one side and one acute angle are known.
No, this calculator is specifically designed for right triangles. It relies on the Pythagorean theorem and basic trigonometric ratios that require a 90-degree angle. For non-right triangles, you would need tools that utilize the Law of Sines or the Law of Cosines, which handle arbitrary triangular shapes without a right angle constraint.
Plug in your numbers and get instant step-by-step results.