Right Triangle Calculator

Solve a right triangle from any two known values: two sides, or one side and one acute angle. The calculator returns all three sides, both acute angles, the area, perimeter, the altitude to the hypotenuse, and the inradius and circumradius, with the steps shown. Example: legs a = 3 and b = 4 give a hypotenuse c = 5, angles A = 36.87° and B = 53.13°, an area of 6 and a perimeter of 12.

How to use the Right Triangle Calculator

1. Fill in exactly two of the five boxes: sides a, b, c or angles A, B. Leave the others empty.
2. Side a is opposite angle A, side b is opposite angle B, and side c is the hypotenuse opposite the 90° angle.
3. Press Calculate to get every side, both acute angles, the area, perimeter, altitude, inradius and circumradius.
4. Open Show steps to see which formula was used.

Formula and method

Pythagorean theorem: a² + b² = c²

The trigonometric ratios give the rest: sin A = a / c, cos A = b / c, tan A = a / b. The acute angles add up to 90°.

Area = ½ab. Perimeter = a + b + c. Altitude to the hypotenuse = ab / c. Inradius = (a + b − c) / 2. Circumradius = c / 2.

Example: a = 3, b = 4 gives c = √(9 + 16) = 5, A = arctan(3/4) = 36.87°, area = 6, altitude to the hypotenuse = 12 / 5 = 2.4.

Which two values can I enter?

Pythagorean triples

A triple is a set of whole numbers that satisfies a² + b² = c². Multiples such as 6-8-10 and 9-12-15 also work. The calculator flags when your triangle is one.

Special right triangles

The 45°-45°-90° triangle has sides in the ratio 1 : 1 : √2. The 30°-60°-90° triangle has sides 1 : √3 : 2. Knowing these shortcuts helps with tests where calculators are not allowed. The calculator labels both types when it finds them.

Worked example: a ladder

A 10 ft ladder leans against a wall at a 70° angle with the ground. The hypotenuse is c = 10 and the angle is 70°, so the height on the wall is a = 10 sin 70° = 9.40 ft and the distance from the wall is b = 10 cos 70° = 3.42 ft. Enter c = 10 and A = 70 to check: a is the side opposite the ground angle.

Altitude, inradius and circumradius

The altitude to the hypotenuse splits the triangle into two smaller right triangles similar to the original, and equals ab / c. The circumcircle of a right triangle has the hypotenuse as its diameter, so its radius is c / 2. The incircle touches all three sides and has radius (a + b − c) / 2.

Common mistakes

Check that the hypotenuse is the longest side; it can never be shorter than a leg. Make sure the angle is in degrees, not radians. Remember that side a is opposite angle A, so mixing up the labels swaps sine and cosine.

Right triangles in real life

Surveyors, carpenters and navigators use right triangles to find heights and distances: roof pitch, ramp length, the diagonal of a screen, and the height of a tree from its shadow.

Accuracy and limits

Results are shown to four decimals (angles to three) using double-precision arithmetic. Very thin triangles with a tiny angle may show small rounding differences.

Frequently asked questions

How do I find the hypotenuse?

Square both legs, add them and take the square root: c = √(a² + b²).

How do I find a missing leg?

Subtract the square of the known leg from the square of the hypotenuse and take the square root.

How do I find the angles of a right triangle?

Use arctan(a / b) for angle A, then B = 90° − A.

Can I enter two angles?

No. Two angles give the shape but not the size. Add one side length.

What is a 3-4-5 triangle?

The smallest Pythagorean triple: sides 3, 4 and 5 satisfy 3² + 4² = 5².

Sources

• Right triangle (Wikipedia)

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