Triangle Calculator

Triangle Calculator

Given the lengths of all three sides, every other property of a triangle is fixed — there is exactly one triangle with those sides, up to reflection and rotation. This calculator uses that fact to derive area, all three interior angles, the perimeter, and the classification as right, acute or obtuse. Area comes from Heron's formula, which needs only the side lengths and no height, and the angles come from the law of cosines. It also checks the triangle inequality first: any side longer than the sum of the other two describes a shape that cannot close, and the calculator will tell you so rather than returning a meaningless number. That check is the most common reason an input is rejected, and it usually means a typo rather than an unusual triangle.

How this is calculated

Heron's formula gives the area from the sides alone. Let s be the semi-perimeter, (a + b + c) ÷ 2. Then area = √(s(s−a)(s−b)(s−c)).

The law of cosines gives each angle: cos(A) = (b² + c² − a²) ÷ (2bc), and similarly for the others. The three angles must sum to 180 degrees, which is a useful check on any hand calculation.

Worked example on sides 3, 4 and 5: s is 6, so the area is √(6 × 3 × 2 × 1) = √36 = 6. The angle opposite the side of length 5 has cosine (9 + 16 − 25) ÷ 24 = 0, so it is exactly 90 degrees — this is the familiar right triangle.

Why some side lengths cannot form a triangle

The triangle inequality states that the sum of any two sides must exceed the third. Sides of 1, 2 and 10 fail it: the two short sides laid end to end reach only 3, and cannot bridge the gap to close against a side of 10.

The boundary case is degenerate. Sides of 3, 4 and 7 satisfy the inequality only as an equality, producing a flattened triangle with zero area and one angle of exactly 180 degrees. Heron's formula returns zero, which is arithmetically correct and geometrically the signal that you have a straight line rather than a triangle.

Right, acute and obtuse

Compare the square of the longest side against the sum of the squares of the other two. If they are equal the triangle is right — that is the Pythagorean theorem. If the longest side squared is smaller, all angles are under 90 degrees and the triangle is acute. If larger, one angle exceeds 90 degrees and it is obtuse.

A triangle can have at most one right or obtuse angle, since the three must total 180 degrees. This is why the classification depends entirely on the longest side: only the angle opposite it can possibly be 90 degrees or more.

How to use the triangle calculator

Enter the three side lengths. Any consistent unit. The results are in those units, with area in units squared.
Check the classification. Right, acute or obtuse follows from comparing the longest side squared against the sum of the other two squared.
Verify the angles sum to 180. They always will here, but it is the standard check when working a triangle by hand.

Frequently asked questions

Why is my triangle "impossible"?

The triangle inequality: each side must be shorter than the other two combined. Sides 1, 2 and 5 cannot close into a triangle.

How do I find the area of a triangle without the height?

Heron's formula. Take the semi-perimeter s = (a + b + c) ÷ 2, then area = √(s(s−a)(s−b)(s−c)). It needs only the three side lengths, which is what makes it useful when you cannot measure a perpendicular height.

How do I know if a triangle is right-angled?

Square the longest side and compare it against the sum of the squares of the other two. Equal means right-angled. Smaller means acute, larger means obtuse. For 3, 4, 5: 25 = 9 + 16, so it is right-angled.

Can three sides make more than one triangle?

No. Three side lengths determine a triangle uniquely, up to rotation and reflection — this is the side-side-side congruence rule. It is why every other property can be derived from the sides alone.