Triangle

Solving a Triangle From Any Three Known Values

A triangle is fully determined by any 3 of its 6 defining values — its 3 side lengths and 3 interior angles — as long as at least one of them is a side. Enter whichever 3 you already know (in whatever unit is convenient for the sides — the calculator converts internally) and it solves for the rest: every side, every angle, the perimeter, the area, and two classifications — by side lengths (equilateral, isosceles, or scalene) and by angles (acute, right, or obtuse). Use the “Solving For” dropdown to choose which single value is shown as the headline answer; every value is still calculated and listed under “The Numbers” regardless of which one is headlined.

These are the classic geometry cases: three sides (SSS), two sides and the angle between them (SAS), two angles and any one side (ASA/AAS), or two sides and an angle that isn’t between them (SSA). That last case is genuinely ambiguous in general — the same two side lengths and an angle outside them can sometimes be completed into two different valid triangles — so this calculator flags it and asks for the included angle or the third side instead, rather than silently guessing.

Not every three side lengths can actually form a triangle, either. The “triangle inequality” requires each side to be shorter than the sum of the other two — three drinking straws of length 1, 2, and 10 simply can’t be arranged to meet at three corners, no matter how you angle them. This calculator checks that first and explains clearly if the lengths you entered don’t work.

The Formula

Area comes from Heron’s formula, using the semi-perimeter $\vD{s}$ (half the perimeter):

$$\text{Area} = \sqrt{\vD{s}(\vD{s} - \vA{a})(\vD{s} - \vB{b})(\vD{s} - \vC{c})}$$

Each angle comes from the law of cosines, solved for the angle opposite a given side — for example, the angle opposite side a:

$$\text{Angle A} = \cos^{-1}\left(\frac{\vB{b}^2 + \vC{c}^2 - \vA{a}^2}{2\vB{b}\vC{c}}\right)$$

The same formula, with the sides rotated, gives angles B and C. All three angles always add up to exactly 180°.

Worked Example

A triangle with sides 7, 8, and 9:

1. Semi-perimeter: $(7 + 8 + 9) \div 2 = 12$.
2. Area: $\sqrt{12 \times 5 \times 4 \times 3} = \sqrt{720} \approx 26.83$.
3. Angle opposite the 7-side: $\cos^{-1}\left(\frac{8^2 + 9^2 - 7^2}{2 \times 8 \times 9}\right) \approx 48.19°$.
4. Angle opposite the 8-side: $\approx 58.41°$.
5. Angle opposite the 9-side: $\approx 73.40°$ — the largest angle, but still under 90°.

All three sides are different lengths and every angle is under 90°, so this triangle is scalene and acute.

Key Factors to Consider

• The two triangle classification systems (by sides and by angles) are completely independent of each other. A triangle is always exactly one of equilateral/isosceles/scalene AND exactly one of acute/right/obtuse — these two labels describe genuinely different properties, so a triangle correctly gets both labels at once (e.g. “isosceles and obtuse” is a perfectly valid combination).
• The SSA (side-side-angle) case is a well-known genuine ambiguity in triangle geometry, not a quirk of this calculator. Given two sides and a non-included angle, it’s mathematically possible for zero, one, or two distinct triangles to satisfy those exact measurements — this is a real feature of triangle geometry (sometimes called the “ambiguous case” in trigonometry courses), which is why this calculator asks for a different combination of inputs instead of silently guessing.
• The triangle inequality is a fast way to check whether three given lengths can form a real triangle before doing any further math. Each side must be strictly shorter than the sum of the other two — if this check fails, no valid triangle exists with those three lengths, regardless of how they’re arranged.
• The three interior angles of ANY triangle always sum to exactly 180 degrees, a fixed geometric fact used as an internal consistency check. Once two angles are known, the third is automatically determined by subtracting from 180° — this fixed-sum property is exactly why knowing just two angles (plus any one side) is enough information to fully solve a triangle.

Common Mistakes

• Entering three angles with no side length at all. Three angles alone only fix a triangle’s shape, not its size — infinitely many triangles share the same three angles but scale up or down — so at least one side length is always required to get an actual area and set of side lengths, not just a shape.
• Mixing up which angle is “included” in the SAS case. The angle between the two given sides is the one that matters for the law of cosines — an angle from elsewhere in the triangle produces the genuinely ambiguous SSA case instead, which this calculator flags rather than guessing at.
• Assuming any three positive numbers can form a triangle. A set of lengths that fails the triangle inequality (one side as long as or longer than the other two combined) doesn’t describe a real triangle at all, no matter how the numbers are arranged.

Useful to Know

• Working with a triangle’s area or side lengths as part of a bigger shape? Area Calculator covers area for a range of common 2D shapes beyond just triangles.
• Need the distance or slope between two coordinate points instead of three known triangle values? Distance, Midpoint & Slope Calculator handles that directly.