Right Triangle Calculator

What the Right Triangle Calculator Solves

A Right Triangle Calculator is designed for the most common triangle-solving case in geometry: a triangle with one 90° angle. Right triangles appear everywhere — construction layouts, ramps, roofs, screen measurements, navigation, engineering drawings, and classroom geometry problems — because the math is clean and predictable.

When you have a right triangle, you can often solve the entire shape from just two pieces of information, as long as at least one of them is a side. This calculator supports multiple input modes (legs, hypotenuse, and an acute angle) and returns a full solution: all three sides, both acute angles, plus measurements that are frequently needed in real work: area, perimeter, altitude to the hypotenuse, inradius, and circumradius.

Right Triangle Names: Legs, Hypotenuse, and Acute Angles

Every right triangle has:

• Two legs (here named a and b): they meet at the 90° angle.
• The hypotenuse (c): the side opposite the 90° angle, always the longest side.
• Two acute angles (here θ and φ): they are both less than 90° and together add up to 90°.

This tool uses a consistent angle definition to avoid confusion: θ is the acute angle adjacent to leg a and opposite leg b. That means: sinθ = b/c, cosθ = a/c, tanθ = b/a. The other acute angle is φ = 90° − θ (or φ = π/2 − θ if using radians).

The Pythagorean Theorem: The Core Right Triangle Shortcut

The defining relationship in a right triangle is the Pythagorean theorem:

If you know both legs, you can compute the hypotenuse as c = √(a² + b²). If you know the hypotenuse and one leg, you can compute the other leg: missing = √(c² − known²). This is why right triangles are so “solvable” — you don’t need advanced triangle rules to get started.

A practical tip: the hypotenuse must be longer than either leg. If you enter a leg value larger than the hypotenuse in a “leg + hypotenuse” mode, the calculation would require √(negative), which is impossible for real triangle side lengths. The calculator flags these cases.

Trigonometry: Solving a Right Triangle with One Side and One Angle

When you know one acute angle and one side, trigonometric ratios let you solve the remaining sides immediately. For the angle θ (adjacent to a, opposite b), the key identities are:

From these:

• If you know c and θ, then a = c·cosθ and b = c·sinθ.
• If you know a and θ, then c = a / cosθ and b = a·tanθ.
• If you know b and θ, then c = b / sinθ and a = b / tanθ.

The calculator includes a degrees/radians toggle because angle input errors are one of the most common causes of “impossible” results. A 30° angle is not the same thing as 30 radians — 30 radians is far larger than a full circle. If a trig-based calculation looks wildly off, confirm the angle unit first.

Area, Perimeter, and the Altitude to the Hypotenuse

Once the triangle’s sides are known, several practical measurements become easy:

• Area: A = (a·b)/2 because the legs form the base and height at a right angle.
• Perimeter: P = a + b + c, useful for fencing, trim length, and boundary totals.
• Altitude to the hypotenuse: h = (a·b)/c, the perpendicular drop from the right angle to side c.

The altitude-to-hypotenuse value is especially useful in geometry proofs and in some engineering contexts where you care about the “height” relative to the longest span. It also gives you a quick consistency check: if your triangle is extremely “flat” (one acute angle very small), the altitude becomes small too.

Inradius and Circumradius: Two Useful Circles

Right triangles have two notable circles:

• Circumcircle (passes through all three vertices). For a right triangle, the circumradius is always: R = c/2. This is a special and elegant property of right triangles.
• Incircle (tangent to all three sides). For a right triangle, the inradius is: r = (a + b − c)/2.

These values show up in geometry problems, design checks, and any work where you need a circle that “fits” the triangle (incircle) or a circle that “contains” its corners (circumcircle). They also help verify that your triangle is physically plausible because r must be positive (which fails if c is too large for the chosen legs).

Special Right Triangles: 3–4–5, 5–12–13, and Angle Patterns

Some right triangles are famous because their side lengths are clean integers. These are called Pythagorean triples. The calculator’s “Load 5-12-13” example is one of the classic triples:

• 3–4–5
• 5–12–13
• 6–8–10 (a scaled 3–4–5)

In practical work, these are used for layout and squareness checks because they avoid decimals. In construction, for example, a 3–4–5 triangle can verify a 90° corner: measure 3 units on one side, 4 units on the other, and the diagonal should be exactly 5 units if the corner is square.

Two angle-based patterns also come up repeatedly:

• 45°–45°–90°: legs equal, and c = a√2.
• 30°–60°–90°: the side opposite 30° is half the hypotenuse.

If you recognize one of these, you can often estimate results even before calculating — a great way to sanity-check answers.

How to Avoid Common Right Triangle Mistakes

Mixing units across sides

Side lengths must be in one unit system. If one measurement is in meters and another is in centimeters, convert before calculating. Otherwise the Pythagorean theorem and area/perimeter computations won’t represent real dimensions.

Confusing opposite vs adjacent

“Opposite” and “adjacent” depend on which acute angle you’re using. This calculator fixes the definition by tying θ to leg a (adjacent) and leg b (opposite). If your mental picture uses the other angle, simply remember that φ = 90° − θ and the roles swap.

Angle unit errors (degrees vs radians)

A quick rule: if you’re typing an angle like 25, 35, or 60, that’s almost always degrees. Radians are typically small decimals for acute angles (for example, 0.5236 rad ≈ 30°). Use the toggle to match your input.

Entering an impossible combination

For “leg + hypotenuse,” the hypotenuse must be larger than the leg. For “angle modes,” θ must be acute (strictly between 0 and 90°). The calculator validates these constraints and warns you when something doesn’t describe a real right triangle.

When to Use Each Calculator Mode

Different problems provide different information. Here’s a quick way to choose:

• Given legs (a, b): Best when you’ve measured two perpendicular sides, like width and height.
• Given leg + hypotenuse: Great for diagonal checks, ladders, and distance-to-corner problems.
• Given hypotenuse + angle: Useful when you know a slope angle and a diagonal length (e.g., a ramp).
• Given adjacent leg + angle: Common when you know the “run” and the incline angle.
• Given opposite leg + angle: Common when you know the “rise” and the incline angle.