Triangle Calculator

Triangle calculator at a glance#

A triangle calculator finds the missing parts of a triangle (side lengths, angles, area and perimeter) from a few known measurements. The two formulas it uses most are the area formula, area = 1/2 x base x height, and the Pythagorean theorem, a² + b² = c², which finds a missing side in a right triangle.

Area of a triangle#

Multiply the base by the height, then halve the result: area = 1/2 x base x height. For a triangle with a base of 6 and a height of 4, the area is 1/2 x 6 x 4 = 12 square units.

Find the hypotenuse of a right triangle#

For a right triangle, the Pythagorean theorem relates the two legs (a and b) to the hypotenuse (c): a² + b² = c². With legs of 3 and 4, c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5.

When you know all three side lengths instead, Heron’s formula gives the area without needing the height: with s = (a + b + c) / 2, area = √(s(s − a)(s − b)(s − c)).

Enter any three known measurements in the calculator above for the exact sides, angles, area and perimeter. Results are rounded, and at least one input must be a side length so the triangle has a fixed size.

Types of triangles#

A triangle has three sides, three angles, and angles that always add up to 180 degrees. The main types are equilateral (all sides and angles equal), isosceles (two equal sides and two equal base angles), scalene (all different), and right (one 90-degree angle).

How the calculator solves a triangle#

Enter any three known measurements and at least one must be a side length, so the triangle has a fixed size. From there the tool finds the remaining sides, angles, area, and perimeter. Pick degrees or radians for angle inputs, since mixing the two gives wrong results.

Formulas it uses#

For a missing side when you know two sides and the angle between them, it uses the law of cosines: c² = a² + b² − 2ab·cos(C). For a side or angle when you know a side and its opposite angle, it uses the law of sines: a / sin(A) = b / sin(B) = c / sin(C). When all three sides are known, Heron’s formula gives the area: with s = (a + b + c) / 2, area = √(s(s − a)(s − b)(s − c)).

Worked example: third side from two sides and the included angle#

Given sides of 4 cm and 5 cm with a 60-degree angle between them, the law of cosines gives c² = 4² + 5² − 2 × 4 × 5 × cos(60°). Since cos(60°) = 0.5, that is 16 + 25 − 20 = 21, so c = √21 = about 4.58 cm.

Common input errors#

Every triangle must satisfy the triangle inequality: the two shorter sides added together must be longer than the longest side. If the calculator rejects your input, the side lengths cannot form a triangle, or an angle and side pair leave no valid shape. Check that all angles you entered are positive and add to less than 180 degrees.

FAQ#

What can a triangle calculator solve?#

It finds missing side lengths, angles, area, and perimeter from three known values, as long as one is a side. It works for equilateral, isosceles, scalene, and right triangles.

How do I find the area of a triangle?#

If you know the base and height, area is half the base times the height. If you know all three sides, use Heron’s formula. The calculator picks the right method from your inputs.

How does it handle right triangles?#

For a right triangle with two known legs, it uses the Pythagorean theorem, c = √(a² + b²), to find the hypotenuse. With one side and one acute angle, it uses sine, cosine, or tangent.

Can it find the angles?#

Yes. With three sides it uses the law of cosines to find each angle, and the three angles always sum to 180 degrees, so it derives the third from the other two.

What is an exterior angle?#

The exterior angle at a vertex is 180 degrees minus the interior angle there. It also equals the sum of the two non-adjacent interior angles.

Can it solve oblique triangles?#

Yes. Oblique triangles have no right angle, so the calculator uses the law of sines and the law of cosines instead of the Pythagorean theorem.