Right Triangle Calculator | Solve Sides, Angles & Area
Solve all sides, acute angles, area, perimeter, altitude, inradius, and circumradius of any right triangle. Enter any two independent parameters to trigger instant Euclidean geometry and trigonometric derivations.
Drag the slider to adjust angle α and hypotenuse to observe how sides scale dynamically.
Solved via Leg a & Leg b
3
4
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36.869898° (0.6435 rad)
53.130102° (0.9273 rad)
90.00° (1.5708 rad)
Geometric & Secondary Metrics
12 units
Formula: a + b + c
6 sq units
Formula: 0.5 * a * b
2.4 units
Formula: (a * b) / c
1 units
Formula: (a + b - c) / 2
2.5 units
Formula: c / 2 (Center of circumcircle is always hypotenuse midpoint)
Step-by-Step Mathematical Solver
The Mathematical Foundations of Right Triangle Geometry
Pythagorean theorem, trigonometric ratios, altitude derivations, and circle osculations
A right-angled triangle (or right triangle) is an elementary polygon in two-dimensional Euclidean geometry characterized by one interior angle measuring exactly 90 degrees (a right angle, denoted as γ = 90° or π/2 radians). The side opposite this right angle is the hypotenuse (traditionally designated c), which is strictly the longest side of the polygon. The two remaining intersecting sides that form the right angle are the legs or catheti (designated a and b).
Derivation of Derived Geometric Metrics
Beyond the three sides and angles, right triangles feature elegant analytical solutions for area, perimeter, and osculating circles:
- Area (A):Because the two legs serve as mutually perpendicular base and height vectors, area simplifies directly to A = (a × b) / 2.
- Perimeter (P): The total boundary length is the scalar sum P = a + b + c.
- Altitude to Hypotenuse (h_c):Expressing area using the hypotenuse as base (A = ½ × c × h_c) and equating it to ½ × a × b gives h_c = (a × b) / c.
- Inradius (r): The radius of the incircle tangent to all three sides is given by r = (a + b - c) / 2 = Area / s, where s is the semi-perimeter (a + b + c)/2.
- Circumradius (R):According to Thales's Theorem, any triangle inscribed in a semicircle with the hypotenuse as diameter has a right angle opposite that diameter. Thus, the circumradius is always R = c / 2.
Right Triangle Solving Matrix Across Input Cases
Systematic classification of known parameters and corresponding analytic solution formulas
| Given Parameters | Case Type | Primary Side Formula | Angle Derivations | Area & Altitude |
|---|---|---|---|---|
| Two Legs (a, b) | SAS (Right Angle) | c = √(a^2 + b^2) | α = arctan(a/b), β = 90° - α | A = ½ ab, h_c = ab/c |
| Leg & Hypotenuse (a, c) | Hypotenuse-Leg (HL) | b = √(c^2 - a^2) | α = arcsin(a/c), β = 90° - α | A = ½ ab, h_c = ab/c |
| Leg & Opposite Angle (a, α) | AAS Case | c = a / sin(α), b = a / tan(α) | β = 90° - α | A = ½ a^2 / tan(α) |
| Leg & Adjacent Angle (a, β) | ASA Case | c = a / cos(β), b = a × tan(β) | α = 90° - β | A = ½ a^2 × tan(β) |
| Hypotenuse & Angle (c, α) | Hypotenuse-Angle (HA) | a = c × sin(α), b = c × cos(α) | β = 90° - α | A = ¼ c^2 × sin(2α) |
Frequently Asked Questions
Authoritative technical answers to right triangle geometry, trigonometry, and dimension solvability
What minimum information is required to fully solve a right triangle?
Because one angle is fixed at 90 degrees (γ = 90°), you only need two additional independent values, provided at least one value is a side length. Valid input pairs include two legs (a, b), one leg and hypotenuse (a, c), or one side and one acute angle (a, α or c, β). Providing only two angles defines triangle shape (similarity) but cannot determine absolute scale or perimeter.
How does the Pythagorean theorem derive the missing sides of a right triangle?
The theorem establishes that the square of the hypotenuse equals the sum of squares of both legs: a^2 + b^2 = c^2. If both legs are known, c = √(a^2 + b^2). If hypotenuse c and leg a are known, the unknown leg is calculated as b = √(c^2 - a^2). The hypotenuse must always be strictly greater than either leg.
How are the acute angles α and β calculated from side lengths?
Using basic trigonometric inverse ratios: α = arcsin(a / c) = arccos(b / c) = arctan(a / b) in degrees. Because the sum of internal angles in Euclidean space is always 180°, the second acute angle is complementary: β = 90° - α.
How are altitude, inradius, and circumradius computed for a right triangle?
The altitude to the hypotenuse is derived by equating area formulations: h_c = (a × b) / c. The inradius (radius of the inscribed circle tangent to all three sides) equals r = (a + b - c) / 2. By Thales's theorem, the hypotenuse serves as the diameter of the circumscribed circle, meaning the circumradius is exactly half the hypotenuse: R = c / 2.
What constitutes an invalid or geometrically impossible input combination?
Inputs are rejected if: any length is less than or equal to zero; an acute angle is less than or equal to 0° or greater than or equal to 90°; the specified leg a or b is greater than or equal to hypotenuse c; or two conflicting acute angles are supplied that do not sum to 90° (α + β ≠ 90°).