📐 Triangle Calculator

The Triangle Calculator solves any triangle from three known measurements, in the four configurations geometry actually uses: SSS, SAS, ASA, and SSA. Enter what you know, and it returns all three sides, all three angles in degrees, the area, and the perimeter, with a scaled diagram showing the resulting shape. It is for checking geometry homework, resolving field measurements in surveying and engineering, and working out triangular cuts in carpentry.

How to Use

1. Pick the mode that matches your known values — SSS, SAS, ASA, or SSA. The four tabs relabel the inputs accordingly.
2. Enter three values. Sides are unitless positive numbers; angles are in degrees between 0 and 180.
3. Read the full solution: every side, every angle, area, and perimeter, computed instantly.
4. Check the diagram. The sketch is scaled to the true side proportions, which is a quick way to catch a mistyped number.
5. Watch the SSA case. For two sides and a non-included angle, the tool reports whether there are zero, one, or two valid solutions.

The Four Input Modes

Each mode corresponds to a standard congruence criterion, and the reason there are exactly four is that they are the combinations that determine a triangle uniquely.

• SSS — three sides. The most constrained case. Once all three sides are fixed, the triangle is rigid: there is exactly one shape. The tool solves for the angles using the law of cosines.
• SAS — two sides and the included angle. The angle between the two known sides fixes the third side by the law of cosines, after which the remaining angles follow.
• ASA — two angles and the included side. Because the angles of a triangle sum to 180°, knowing two determines the third. The known side is then used with the law of sines to find the other two.
• SSA — two sides and a non-included angle. The one configuration that can be ambiguous, described below.

The Ambiguous SSA Case

SSA is the case worth understanding, because it is the only one where a set of measurements can describe more than one triangle. Picture a known side anchored at one vertex with a known angle at the other end; the second known side can swing out and meet the base in two positions — one producing an acute angle, one an obtuse angle. Both triangles satisfy the given data.

There are three possible outcomes. The swing may miss entirely (no triangle), touch at exactly one point (one, right-angled solution), or cross twice (two solutions). When you have measured a physical object, one of the two will be geometrically impossible in context — a side that would have to pass through the object, or an angle that contradicts a second observation. Use the diagram to see which shape is plausible, and note both in your work if the problem calls for it. Silently returning only the acute solution is how calculators produce “wrong” answers for this configuration.

Worked Examples

Example 1 — an SSS check. Sides of 3, 4, and 5. The tool confirms a right triangle by returning angles of about 36.87°, 53.13°, and 90°, an area of 6, and a perimeter of 12. Students can use this to verify that their own working matches, and the diagram makes the right angle visible.

Example 2 — a surveying measurement. Two sides of a plot meet at an angle of 62°, with lengths of 40 m and 55 m. In SAS mode, the tool returns the third side (about 51.1 m) along with the remaining angles, which is exactly what is needed to compute the area of the parcel and mark the boundary.

Example 3 — an SSA ambiguity in carpentry. A brace is cut with one side of 300 mm, another of 200 mm, and a 30° angle opposite the shorter side. SSA reports two possible shapes — a shallow triangle and a steeper one. The drawing shows which matches the intended joint, so the cut is made to the right geometry rather than the first solution printed.

The Formulas Behind the Answers

The calculator applies the standard identities, in the order each mode requires:

• Law of cosines — for SSS and SAS: c² = a² + b² − 2ab·cos(C). This finds the missing side from two sides and the included angle, or an angle from all three sides.
• Law of sines — for ASA and SSA: a / sin(A) = b / sin(B) = c / sin(C). With two angles known, the third follows from the 180° sum, and the ratio yields the remaining sides.
• Area from three sides — Heron’s formula: Area = √(s(s−a)(s−b)(s−c)), where s is half the perimeter.
• Area from two sides and an included angle — Area = ½ · a · b · sin(C).

Both area methods return the same value for the same triangle; the tool picks whichever the input mode makes available.

Common Mistakes

• Choosing the wrong mode. SSA and SAS differ by whether the angle sits between the known sides. Picking the wrong tab gives a solution to a different problem that still looks plausible.
• Entering an angle in radians. All inputs and outputs are in degrees. An angle of 1 radian is about 57.3°, so a value of 1 entered as degrees creates a very different triangle.
• Ignoring the triangle inequality. Sides of 1, 2, and 10 cannot close; the tool reports it, but knowing the rule lets you spot a bad measurement before calculating.
• Forgetting the second SSA solution. If your context knows the triangle is obtuse, the acute result alone is not the answer.
• Rounding intermediate values by hand. Work from the calculator’s full-precision output rather than re-entering rounded numbers, which compounds error across several steps.

Features

• Four input modes: SSS, SAS, ASA, and SSA with relabelled inputs
• Explicit ambiguous-case handling for SSA — zero, one, or two solutions listed
• Complete solution: all sides, all angles in degrees, area, and perimeter
• Area via Heron’s formula or ½ab·sin(C), matching the known values
• Scaled SVG diagram drawn from the computed side proportions
• Clear validation for impossible inputs, and fully client-side processing

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