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Right Triangle Calculator

Solve all dimensions of a right triangle. Enter any two parameters (such as leg lengths, hypotenuse, or acute angles) to calculate sides, angles, area, perimeter, and altitude.

Triangle Parameters (Enter Any 2)

Leg a (Opposite to Angle α)

Leg b (Opposite to Angle β)

Hypotenuse c (Longest Side)

Angle α (Degrees)

Angle β (Degrees)

* Note: Entering only angles (α and β) is mathematically insufficient to scale the triangle. At least one side length must be provided.

Pre-loaded Geometric Presets

Trigonometric Slider Explorer

Drag the slider to adjust angle α and hypotenuse to observe how sides scale dynamically.

Angle α: 30°Angle β: 60°

Hypotenuse c: 10

Solving Case Category

Solved via Leg a & Leg b

Valid Right Triangle

Leg a

Leg b

Hypotenuse c

Angle α

36.869898° (0.6435 rad)

Angle β

53.130102° (0.9273 rad)

Right Angle γ

90.00° (1.5708 rad)

Proportionally Scaled Diagram

Geometric & Secondary Metrics

Perimeter (P)

12 units

Formula: a + b + c

Area (A)

6 sq units

Formula: 0.5 * a * b

Altitude to Hypotenuse (h_c)

2.4 units

Formula: (a * b) / c

Inradius (r_in)

1 units

Formula: (a + b - c) / 2

Circumradius (R_circum)

2.5 units

Formula: c / 2 (Center of circumcircle is always hypotenuse midpoint)

Step-by-Step Mathematical Solver

1. Solve Hypotenuse (c) using Pythagorean Theorem:

c = √(a² + b²) = √(3² + 4²)

c = √(9.00 + 16.00) = √(25.00) = 5

2. Solve Angle α using Tangent (tan) ratio:

tan(α) = a / b = 3 / 4 = 0.7500

α = arctan(0.7500) ≈ 36.869898°

3. Solve Angle β (complementary):

β = 90° - α = 90° - 36.869898° = 53.130102°

3. Secondary Attributes:

Perimeter = a + b + c = 3 + 4 + 5 = 12

Area = 0.5 * a * b = 0.5 * 3 * 4 = 6

Altitude to Hypotenuse = (a * b) / c = (3 * 4) / 5 = 2.4

Related Trigonometry & Math Tools

Pythagorean TheoremFocus strictly on hypotenuse and leg lengths a² + b² = c².Slope CalculatorFind coordinates slope, inclination angles, and intercepts.Distance CalculatorSolve 2D & 3D coordinate lengths based on triangulation.

Solving Right-Angled Triangles

A right-angled triangle (or right triangle) is a triangle in which one angle is exactly 90 degrees (a right angle). The side opposite to the right angle is called the hypotenuse (c), and it is always the longest side. The other two sides are the legs (a and b).

Trigonometric Ratios (SOH CAH TOA)

For any acute angle alpha in the triangle, we define the core trigonometric functions:

Sine (sin): Ratio of the opposite leg to the hypotenuse (sin alpha = a/c).
Cosine (cos): Ratio of the adjacent leg to the hypotenuse (cos alpha = b/c).
Tangent (tan): Ratio of the opposite leg to the adjacent leg (tan alpha = a/b).

Pythagorean Theorem

The fundamental relationship between the sides of a right triangle is defined by: a^2 + b^2 = c^2. This allows us to find any third side if two side lengths are already known.

Secondary Characteristics

Aside from sides and angles, several geometric parameters are associated with right triangles:

Area (A): Calculated as A = 0.5 * a * b.
Perimeter (P): Sum of all sides P = a + b + c.
Altitude to Hypotenuse (hc): The shortest distance from the right angle to the hypotenuse, solved by hc = (a * b) / c.
Inradius: The radius of the incircle, given by r_in = (a + b - c) / 2.
Circumradius: The radius of the circumcircle, which is always half the hypotenuse (R_circum = c / 2).

Frequently Asked Questions About Right Triangles

What parameters are needed to solve a right triangle?

To solve a right triangle, you need to provide at least two parameters, and at least one of them must be a side length (a, b, or c). Providing only angles is insufficient because it defines similarity but not the absolute size.

How is the altitude to the hypotenuse calculated?

The altitude (hc) of a right triangle is the perpendicular line segment drawn from the right angle to the hypotenuse. It is calculated with the formula hc = (a * b) / c.

How do I calculate the sides of a right triangle if I know one side and an angle?

You can use trigonometric functions: sine (sin), cosine (cos), or tangent (tan). For example, if the side opposite to the angle is known, then the hypotenuse can be calculated as hypotenuse = opposite / sin(angle).

Can this solve any right triangle property?

Yes. Calculate sides, angles, area, perimeter, altitude, inradius, and circumradius instantly.

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