Please provide 2 values below to calculate the other values of a right triangle. If radians are selected as the angle unit, it can take values such as pi/3, pi/4, etc.
A right triangle has one angle equal to 90° (γ = 90° or π/2 rad). To solve all properties, enter any 2 independent values (such as two sides, one side and one angle, or a combination of side/angle with area, perimeter, or altitude). The calculator will automatically identify the geometric theorem and calculate the remaining parameters.
A right triangle (also termed a right-angled triangle or orthogonal triangle) is a polygon with three sides and three interior angles where exactly one angle measures 90 degrees (a right angle, equal to π/2 radians).
In Euclidean plane geometry, the side situated directly opposite the right angle is called the hypotenuse (conventionally denoted as c), which is always the longest side of the triangle. The remaining two perpendicular sides that form the 90° angle are called the legs or catheti (denoted as a and b). The two non-right angles (α and β) are acute angles and are strictly complementary (α + β = 90°).
The fundamental trigonometric functions are defined as the ratios of the lengths of the sides of a right triangle relative to an acute angle α:
| Property | Primary Formula | Description & Notes |
|---|---|---|
| Pythagorean Theorem | a² + b² = c² | Relates the squares of the two legs to the hypotenuse square. |
| Area (A) | A = ½ · a · b = ½ · c · h | Equal to half the product of the legs, or half the hypotenuse times altitude. |
| Perimeter (P) | P = a + b + c | Sum of all three boundary lengths. |
| Altitude to Hypotenuse (h) | h = (a · b) / c | Perpendicular distance from the 90° vertex C to hypotenuse c. |
| Inradius (r) | r = (a + b - c) / 2 | Radius of the inscribed circle tangent to all three sides. |
| Circumradius (R) | R = c / 2 | Radius of circumscribed circle passing through vertices A, B, and C (Thales theorem). |
| Median to Hypotenuse (m_c) | m_c = c / 2 = R | Length from right-angle vertex C to midpoint of hypotenuse. |
An isosceles right triangle where both legs are equal in length (a = b). The hypotenuse is exactly c = a√2. The acute angles are both 45° (π/4 rad).
Formed by bisecting an equilateral triangle. The shorter leg opposite 30° is x, the longer leg opposite 60° is x√3, and the hypotenuse is 2x.
A Pythagorean triple consists of three positive integers (a, b, c) satisfying a² + b² = c². The most frequent primitive triples include:
Builders use the 3-4-5 rule to verify square 90-degree corners when laying building foundations, decks, walls, and framing.
Surveyors measure distances to inaccessible objects and determine elevation changes across terrain using theodolite angles and right-triangle trigonometry.
Pilots and navigators calculate ground speed, crosswind drift angles, bearing intercepts, and true heading using vector resolution into orthogonal components.
3D game engines, ray tracing shaders, and collision detection algorithms rely on right-triangle dot products, trigonometric projections, and Euclidean distances.
Because one angle is already fixed at 90 degrees (γ = 90°), you only need 2 additional independent values (with at least one linear measurement such as a side, perimeter, altitude, or area) to compute all remaining sides, acute angles, and geometric properties.
No. Entering only angles determines the shape of the triangle (AAA similarity) but not its physical scale or size. At least one linear side, perimeter, altitude, or area is required to fix its exact dimensions.
Select "radian" from the unit dropdown next to angle α or β, and type mathematical expressions directly into the field, such as pi/3, pi/4, 2*pi/5, or decimal values like 0.5236.
By equating two expressions for the area of the triangle (Area = ½ · a · b = ½ · c · h), the altitude h is given by h = (a · b) / c.