Right Triangles Calculator -

Solve for sides, angles, area, perimeter, and more.

Calculate using:

Leg a

Leg b

Units (optional)

Triangle Properties

Side a (Leg):

Side b (Leg):

Side c (Hypotenuse):

Angle A (α):

Angle B (β):

Angle C (γ):90°

Area (A):

Perimeter (P):

Inradius (r):

Circumradius (R):

Altitude to Hypotenuse (h_c):

Altitude to Leg a (h_a):

Altitude to Leg b (h_b):

Visual Representation

How to Use the Right Triangle Calculator

This calculator helps determine all unknown sides, angles, area, perimeter, and other properties of a right-angled triangle based on any two known values (excluding Angle C, which is always 90°).

Select Calculation Method:
- Use the “Calculate using:” dropdown menu to choose which two properties of the right triangle you already know. “Leg a” and “Leg b” are the sides forming the right angle, and “Hypotenuse c” is the side opposite the right angle. “Angle A” is opposite Leg a, and “Angle B” is opposite Leg b.
Enter Known Values:
- Based on your selection, two input fields will appear, labeled appropriately (e.g., “Leg a”, “Angle A (°)”).
- Enter the numerical values for these known properties. Lengths must be positive. Angles must be greater than 0 and less than 90 degrees.
Enter Units (Optional):
- Specify the units of your length measurements (e.g., cm, m, inches) in the “Units” field. This is for display purposes; results will be shown with appropriate units (e.g., area in units², angles in degrees).
Calculate: Click the “Calculate” button.
Review the Results: The calculator will display:
- Sides (a, b, c): Lengths of both legs and the hypotenuse.
- Angles (A, B, C): Measures of all three interior angles (Angle C is always 90°).
- Area (A) and Perimeter (P) of the triangle.
- Inradius (r): Radius of the circle inscribed within the triangle.
- Circumradius (R): Radius of the circle that circumscribes the triangle (passes through all vertices). For a right triangle, this is half the hypotenuse.
- Altitudes (h_a, h_b, h_c): The perpendicular distance from each vertex to its opposite side. For a right triangle, h_a = b, h_b = a, and h_c is the altitude to the hypotenuse.
Visual Representation: A diagram of the right triangle will be shown, scaled according to the calculated dimensions, illustrating its sides, angles, and the right-angle symbol.
Clear: Click the “Clear” button to reset all input fields and results.
Error Messages: If your input is invalid or leads to a geometrically impossible triangle (e.g., a leg longer than the hypotenuse, sum of acute angles not 90°), an error message will appear.

Right Triangles Calculator: Mastering the 90-Degree Angle

What is a Right Triangle? The Cornerstone of Geometry

The right triangle, also known as a right-angled triangle, is one of the most fundamental and widely applied shapes in all of geometry. Its defining characteristic is simple yet profound: one of its three interior angles is exactly 90 degrees – a perfect “right” angle. This single property unlocks a cascade of unique relationships between its sides and other angles, making it a cornerstone for fields ranging from architecture and engineering to navigation and art.

The side opposite the right angle is always the longest side and is called the hypotenuse. The other two sides, which form the right angle, are called the legs (often denoted as ‘a’ and ‘b’). The two angles other than the right angle are always acute (less than 90 degrees) and their sum is always 90 degrees, a direct consequence of the fact that the sum of all angles in any triangle is 180 degrees.

Key Properties and Formulas of a Right Triangle

Understanding a right triangle involves several key properties and the formulas that connect them:

Sides:
- Legs (a, b): The two shorter sides that form the 90° angle.
- Hypotenuse (c): The longest side, opposite the right angle.
Pythagorean Theorem: Perhaps the most famous theorem in mathematics, it states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the two legs (a and b): a² + b² = c². This allows us to find any side if the other two are known.
Angles (A, B, C):
- Angle C is always 90°.
- Angles A and B are acute, and A + B = 90°. Angle A is typically opposite leg a, and Angle B opposite leg b.
Trigonometric Ratios (SOHCAHTOA): These relate the angles to the ratios of the sides:
- sin(A) = Opposite/Hypotenuse = a/c; sin(B) = b/c
- cos(A) = Adjacent/Hypotenuse = b/c; cos(B) = a/c
- tan(A) = Opposite/Adjacent = a/b; tan(B) = b/a
Area (A): Since the legs are perpendicular, one can be considered the base and the other the height. So, Area = (1/2) × base × height = (1/2) × a × b.
Perimeter (P): The sum of the lengths of all three sides: P = a + b + c.
Inradius (r): The radius of the largest circle that can be inscribed within the triangle (the incircle). For a right triangle, r = (a + b - c) / 2.
Circumradius (R): The radius of the circle that passes through all three vertices of the triangle (the circumcircle). For a right triangle, the circumcenter is the midpoint of the hypotenuse, so R = c / 2.
Altitudes:
- Altitude to leg a (h_a) is simply leg b.
- Altitude to leg b (h_b) is simply leg a.
- Altitude to the hypotenuse (h_c) can be found by h_c = (a × b) / c or by using area: Area = (1/2) × c × h_c.

This calculator uses these formulas to solve for all unknown properties based on your inputs.

Real-World Applications: Why Right Triangles Matter

The right triangle isn’t just a textbook concept; its principles are applied constantly in the real world:

Construction & Carpentry: Ensuring walls are square, cutting roof rafters, building stairs – all rely on right triangle geometry and the Pythagorean theorem.
Navigation & Surveying: Used to determine distances and heights using angles of elevation or depression and known distances (trigonometry).
Engineering: Analyzing forces, designing structures, and in various mechanical applications.
Art & Design: Creating perspective, understanding proportions, and in graphic design layouts.
Physics: Resolving vectors into components (e.g., force, velocity).
Computer Graphics: Triangles, especially right triangles, are fundamental units for rendering 3D objects and scenes.

How This Calculator Solves for Unknowns

This Right Triangle Calculator is designed to be versatile. You only need to know two pieces of information (besides the 90° angle) to find all other properties. Here’s a glimpse of the logic:

If you provide two sides:
- Legs a and b: Hypotenuse c = √(a² + b²). Angles A and B from atan(a/b) and atan(b/a).
- Leg a and Hypotenuse c: Leg b = √(c² - a²). Angles from asin(a/c) and acos(a/c).
If you provide one side and one acute angle:
- Leg a and Angle A: c = a / sin(A), b = a / tan(A). Angle B = 90° – A.
- Hypotenuse c and Angle A: a = c × sin(A), b = c × cos(A). Angle B = 90° – A.
If you provide Area and a leg:
- Area and Leg a: b = (2 × Area) / a. Then use a and b to find c and angles.
If you provide Perimeter and a side:
- Perimeter and Leg a: P = a + b + c. We also know c = √(a² + b²). Substituting c: P = a + b + √(a² + b²). Rearranging: (P - a - b)² = a² + b². This becomes a quadratic equation in ‘b’ if ‘P’ and ‘a’ are known. Solving for ‘b’ gives the other leg. P - a - b = √(a² + b²) (P-a)² - 2b(P-a) + b² = a² + b² (P-a)² - a² = 2b(P-a) b = ((P-a)² - a²) / (2(P-a)). We must ensure P-a > 0 and the resulting ‘b’ is positive and that a+b and c .
- Perimeter and Hypotenuse c: P = a + b + c, so a + b = P - c. Also a² + b² = c². Squaring a+b: (a+b)² = (P-c)² => a² + b² + 2ab = (P-c)². Substitute a²+b²=c²: c² + 2ab = (P-c)². So, 2ab = (P-c)² - c². This gives us ab. We have a+b and ab, allowing us to form a quadratic equation x² - (a+b)x + ab = 0 whose roots are a and b.

Once the three sides (a, b, c) and two acute angles (A, B) are determined, all other properties like Area, Perimeter, Inradius, Circumradius, and Altitudes are calculated using their respective direct formulas.

"There is geometry in the humming of the strings, there is music in the spacing of the spheres." - Pythagoras. The right triangle is a testament to this inherent harmony in mathematics.

Visualizing Your Right Triangle

Understanding geometry is often enhanced by visualization. This calculator provides a dynamic SVG (Scalable Vector Graphics) drawing of the right triangle based on the calculated dimensions. The sides will be scaled proportionally, and the right angle will be clearly marked. This visual aid helps in relating the numerical results to an actual shape, making the concepts more intuitive.

Conclusion: Your Essential Tool for Right Triangle Problems

The right triangle is a simple shape with profound implications and uses. This calculator aims to be your comprehensive companion for exploring all aspects of right triangle geometry. Whether you're a student tackling trigonometry, a builder ensuring square corners, an engineer designing components, or simply someone curious about the properties of this fundamental shape, this tool provides accurate calculations and clear visualizations to aid your understanding and application of right triangle principles.