Right Triangle Calculator

Solve a right triangle from any two known values. Find all sides, angles, and trigonometric ratios.

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Enter any 2 known values (right angle C = 90° is fixed):

Side a (opposite angle A)

Side b (opposite angle B)

Hypotenuse c

Angle A (degrees)

Area

Enter 2 known values to solve

Side a

Side b

Hypotenuse c

Angle A

Angle B

Perimeter

Ratio	Value	Definition

Solve a complete right triangle — all sides, both acute angles, area, perimeter, and every trig ratio — from any two known values.

How It Works

How Right Triangle Calculator Works

A right triangle has six measurements: two legs (a and b), the hypotenuse (c), and two acute angles (A and B, since the right angle is fixed at 90°). Because these values are all related through the Pythagorean theorem and basic trigonometry, knowing just two of them (in addition to the right angle) is enough to solve for everything else.

Depending on which two values you enter, the calculator picks the appropriate relationship — for two sides it applies the Pythagorean theorem directly; for a side and an angle it uses sine, cosine, or tangent to solve for the rest. Once side a, side b, and the hypotenuse are known, area (½ab) and perimeter (a + b + c) follow immediately.

The results table also lists all six trigonometric ratios for angle A — sine, cosine, tangent, and their reciprocals cosecant, secant, and cotangent — along with the triangle's inradius and circumradius, giving a full picture of the triangle's geometry in one place.

Worked Example

See It In Action

Given leg a = 5 and angle A = 30°: side b = a ÷ tan(30°) = 5 ÷ 0.5774 ≈ 8.66, and the hypotenuse c = a ÷ sin(30°) = 5 ÷ 0.5 = 10. Angle B = 90° − 30° = 60°. The area is ½ × 5 × 8.66 ≈ 21.65 square units, and the perimeter is 5 + 8.66 + 10 ≈ 23.66 units.

Real-World Use Cases

Who Uses Right Triangle Calculator and Why

Solving a full right triangle from just one side and one angle, common in surveying or navigation problems.
Getting every trig ratio (sine, cosine, tangent, and their reciprocals) for a given angle without looking up a reference table.
Computing a right triangle's inradius and circumradius for a geometry proof or design problem.
Solving for every missing side and angle when only two of the triangle's six measurements are known.

Common Mistakes

Mistakes to Avoid

Entering two angles without any side value — two acute angles alone (plus the fixed 90°) only fix the triangle's shape, not its actual size, so the calculator can't solve it.
Mixing up which acute angle (A or B) corresponds to which leg when interpreting the trig ratio results.
Assuming this calculator's inradius formula, (a + b − c)/2, applies to a non-right triangle — it's specific to right triangles.

Pro Tips

Tips for Best Results

Remember the right angle is always fixed at 90°, so any two other independent values (two sides, or one side plus one angle) are enough to solve everything else.
Use the reciprocal trig ratios (cosecant, secant, cotangent) directly from the results table instead of manually inverting sine, cosine, and tangent.

Troubleshooting

Fixing Common Problems

The calculator won't produce a full solution. — A right triangle needs exactly two known values besides the fixed 90° angle — check that you've entered either two sides, or one side plus one acute angle, not just a single value or two angles alone.

Glossary

Terms Explained

Inradius: The radius of the largest circle that fits entirely inside the triangle, calculated as (a + b − c)/2 for a right triangle.

Circumradius: The radius of the circle passing through all three of a triangle's vertices — for a right triangle, always exactly half the hypotenuse.

FAQ

Frequently Asked Questions

How many values do I need to enter to solve the triangle?

Exactly two known values (besides the fixed 90° angle) are enough — any combination of two sides, or one side and one acute angle. The calculator automatically detects which pair you entered and solves for the rest.

What is the difference between this and the Pythagorean theorem calculator?

The Pythagorean theorem calculator only solves for a side length using two other sides. This calculator solves the full triangle — every side, angle, trig ratio, area, and perimeter — even when you only know an angle and one side.

What do the reciprocal ratios (cosecant, secant, cotangent) mean?

They are simply the reciprocals of sine, cosine, and tangent — cosecant = 1/sine, secant = 1/cosine, cotangent = 1/tangent. They show up often in calculus and physics problems as alternative ways to express the same triangle relationships.

What are the inradius and circumradius shown in the results?

The inradius is the radius of the largest circle that fits inside the triangle, calculated as (a + b − c)/2 for a right triangle. The circumradius is the radius of the circle passing through all three vertices, which for a right triangle is always exactly half the hypotenuse.