Triangle Calculator

Triangle Calculator for SSS, SAS, ASA, AAS, SSA and Right Triangles

A complete triangle solver needs more than one formula. Three sides call for the Law of Cosines; two sides with their included angle also start with the Law of Cosines; two angles and one side are naturally solved with the angle-sum rule and Law of Sines; and SSA is special because the same inputs can sometimes describe two different triangles. SonoCalculator separates those cases instead of pretending every set of three numbers behaves the same way.

Triangle Notation: Which Side Is Opposite Which Angle?

The calculator follows standard notation for triangle ABC: side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. OpenStax uses the same opposite-side convention when introducing triangle geometry and trigonometric laws. Keeping the pairing correct is essential because the Law of Sines matches each side only with its opposite angle.

All angle inputs in this calculator are degrees. Side units are optional: choose a real unit such as cm, m, in or ft, or stay unitless for textbook exercises. Area automatically uses the squared form of the selected side unit.

How to Use the Triangle Calculator

1. Choose the known-information case. Pick SSS, SAS, ASA/AAS, SSA or Right triangle.
2. Enter only the fields shown for that case. Labels tell you which angle is opposite or included between which sides.
3. Use positive side lengths and angles between 0° and 180°.
4. Press Solve Triangle. The solver validates whether a Euclidean triangle exists.
5. For SSA, check for a second solution. If the ambiguous case produces two valid triangles, both are displayed.

SSS: Solve a Triangle from Three Sides

With all three sides known, the triangle is determined provided the triangle inequality holds. The Law of Cosines finds the angles:

Computing two angles with inverse cosine and taking the third from the 180° angle sum reduces small floating-point inconsistencies. The calculator also uses Heron’s formula for area once the sides are known.

SAS: Two Sides and the Included Angle

For sides b and c with included angle A, first solve the opposite side a:

Once the third side is known, the triangle becomes an SSS problem. This avoids the ambiguity associated with SSA because the included angle fixes how the two known sides open away from one another.

SAS also gives a direct area formula, K = ½bc sin A. The calculator solves all three sides and then checks area from the solved geometry.

ASA and AAS: Two Angles and One Side

In Euclidean geometry, the three interior angles of a triangle sum to 180°. Therefore two known angles immediately determine the third:

With side a opposite angle A, the Law of Sines gives the remaining sides:

ASA and AAS are grouped in one mode because the solving mathematics is the same once two angles and one opposite side-angle pair are known.

SSA and the Ambiguous Case

SSA means two sides and an angle that is not included between them. This is the one common triangle case where valid inputs do not always produce exactly one triangle. OpenStax explicitly describes SSA as the ambiguous case: depending on the geometry, there may be zero, one or two solutions.

Given side a opposite known angle A and another side b, the Law of Sines gives:

If that ratio is greater than 1, no triangle exists. If it is valid, inverse sine gives a principal angle B₁. But sine has the same positive value at B₁ and 180° − B₁, so a second candidate B₂ may also exist. Each candidate is valid only if A + B < 180°.

Many simple triangle calculators return only the principal arcsine result and silently lose the second triangle. This calculator checks both branches and displays the second valid solution explicitly.

Right Triangle Mode

A right triangle has one 90° angle. In this calculator, C = 90° and side c is the hypotenuse. If both legs are known:

If the hypotenuse c and one leg a are known:

The acute angles can then be found from inverse trigonometric ratios or from the Law of Cosines. The hypotenuse must be longer than either leg; otherwise the entered measurements cannot form a non-degenerate right triangle.

Triangle Area: Heron’s Formula and Trigonometric Area

Once all three sides are known, Heron’s formula is universal for ordinary Euclidean triangles:

When two sides and their included angle are known, an equivalent trigonometric form is:

For a right triangle with legs a and b, this reduces to K = ab/2.

Altitude, Inradius and Circumradius

After area K is known, the altitude to side c is:

The inradius—the radius of the circle tangent to all three sides—is:

The circumradius—the radius of the circle passing through all three vertices—is:

These are useful derived measures, but they do not replace the primary solved sides and angles. For a nearly degenerate triangle, area approaches zero and the circumradius can become very large.

How the Calculator Detects Impossible Triangles

Three positive side lengths must satisfy the triangle inequality: the sum of any two sides must be greater than the third. For sorted sides x ≤ y ≤ z, it is enough to test x + y > z. Equality would make a flat, zero-area degenerate triangle rather than an ordinary triangle.

Angle inputs must also leave room for a positive third angle. For ASA/AAS, A + B must be less than 180°. For SAS, the included angle must be strictly between 0° and 180°. For SSA, every arcsine candidate is tested against the same angle-sum condition.

Worked Triangle Examples

SSA example with two valid triangles

Let a = 10, b = 12 and A = 40°. Then sin B = 12 sin40° / 10 ≈ 0.7713. One solution uses B ≈ 50.48°. The supplementary possibility B ≈ 129.52° also leaves a positive third angle because 40° + 129.52° is still below 180°. Therefore two distinct triangles satisfy the same SSA data.

SSA example with no triangle

If a = 5, b = 10 and A = 40°, then b sin A / a ≈ 1.286, which is greater than 1. No real angle B has a sine greater than 1, so no Euclidean triangle matches those measurements.

Common Triangle Solver Mistakes

• Pairing a side with the wrong opposite angle. In the Law of Sines, a ↔ A, b ↔ B and c ↔ C.
• Using the Law of Sines first for SSS. Without a known side-angle pair, start with the Law of Cosines.
• Using SAS formulas on an SSA problem. “Included angle” means the angle physically between the two known sides.
• Ignoring the SSA second solution. arcsin returns a principal angle, but a supplementary angle may also be valid.
• Accepting sides that violate the triangle inequality. Three positive numbers do not automatically make a triangle.
• Letting two known angles sum to 180° or more. The third angle must remain positive.
• Mixing degrees and radians. This calculator accepts degree inputs and converts internally for trigonometric functions.
• Rounding an intermediate angle too early. Small angle errors can noticeably change the final side lengths.
• Confusing altitude with a side. An altitude is perpendicular to its base and may lie outside an obtuse triangle.
• Assuming AAA determines size. Three angles determine shape but not scale; infinitely many similar triangles share the same three angles.

Triangle Calculator FAQs