Triangle Calculator

The Triangle Calculator solves missing sides and angles when enough independent measurements are known. It also finds perimeter and area while checking basic validity, including angle sum and the triangle inequality.

The ambiguous SSA case

Right triangles use the Pythagorean theorem and trigonometric ratios. General triangles use the law of sines or law of cosines. Heron’s formula finds area from three sides using semiperimeter s=(a+b+c)/2.

Triangle relationships include angle sum A+B+C=180°, Pythagorean a²+b²=c² for right triangles and Heron area √(s(s−a)(s−b)(s−c)).

Enter the values in their mathematical roles

Match each side with its opposite angle and enter the known values. Choose the case—SSS, SAS, ASA, AAS, or right triangle—when available. Keep angle mode in degrees unless the problem explicitly uses radians.

The governing relationship

Triangle relationships include angle sum A+B+C=180°, Pythagorean a²+b²=c² for right triangles and Heron area √(s(s−a)(s−b)(s−c)).

A compact example

A right triangle with legs 3 and 4 has hypotenuse 5.

Valid inputs and boundary cases

Side lengths must be positive, the interior angles sum to 180°, and any two sides must total more than the third. The SSA case can produce zero, one, or two valid triangles, so both solutions may need review.

Read the answer in context

Side lengths must satisfy the triangle inequality, and ambiguous input may allow more than one triangle.

An independent check

Add the angles, test the triangle inequality, and substitute the results into the law of sines or cosines. The longest side should face the largest angle, and a computed area must be positive.

Errors that change the answer

• Pairing a side with the wrong opposite angle
• Assuming SSA has only one solution
• Using radians for degree inputs
• Rounding angles before finding later sides
• Accepting sides that violate the triangle inequality

Continue with a related calculator

Continue with the Volume Calculator or Probability Calculator. For everyday arithmetic and order-of-operations checks, use the Basic Calculator.

What this numerical result cannot decide

A diagram is not necessarily drawn to scale. Surveying, structural layout, and navigation also require measurement uncertainty, coordinate conventions, and professional methods beyond solving an ideal plane triangle.

A second way to understand the method

When two sides and a nonincluded angle are known, the sine relationship may allow an acute angle θ or its supplement 180°−θ. Both must be tested against the remaining angle sum and side data. This is why an SSA input can return two valid triangles.

Select the law that matches the known data

SSS and SAS cases naturally use the law of cosines first. ASA and AAS cases use the angle sum and then the law of sines. A right triangle may be solved more directly with sine, cosine, tangent, and the Pythagorean theorem. Matching the method to the known data reduces unnecessary rounding.

Area can be found in several ways

Use bh/2 when a perpendicular height is known, Heron’s formula for three sides, or ab sin(C)/2 for two sides and their included angle. These formulas should agree for the same valid triangle. Comparing two methods is a strong check when all dimensions are available.

Right-triangle shortcuts

For legs a and b and hypotenuse c, a²+b²=c². The acute angles can be checked with tan(A)=a/b when the side labels match the chosen angle. Special 45-45-90 and 30-60-90 triangles also have fixed side ratios that provide quick exact checks.

Round only after the triangle is solved

Angles and side lengths feed into later steps. Rounding an early sine, cosine, or intermediate side can shift the final area or remaining angle enough to violate the 180° sum. Retain guard digits, then round all reported values consistently at the end.

Triangle Calculator FAQ

How many measurements are needed?

Usually three independent values, including at least one side.

Can three angles determine the side lengths?

No. AAA fixes shape but not scale.

What is the triangle inequality?

Each side must be shorter than the sum of the other two.

For additional worked mathematics, see the free OpenStax Prealgebra 2e reference.