Triangle Calculator

Q: Can you solve a triangle from three angles alone?

No. Three angles fix the triangle's shape but not its size, since infinitely many triangles share the same angles at different scales. You need at least one side length to determine the actual dimensions.

Q: What is the ambiguous case (SSA)?

It occurs when you know two sides and an angle that is not between them. Depending on the values there may be no triangle, exactly one, or two valid triangles. The calculator detects this, shows one solution, and lets you switch to the alternate when a second exists.

Q: When do you use the law of sines versus the law of cosines?

Use the law of cosines when you have three sides or two sides and the included angle. Use the law of sines when you already have a complete angle and opposite-side pair, as with two angles and a side.

Q: How do you find an angle from three sides?

Rearrange the law of cosines: cos A = (b squared + c squared minus a squared) divided by (2bc), then take the inverse cosine. Repeat for a second angle and subtract both from 180 degrees for the third.

Enter any three measurements — including at least one side — to get every length, angle and property, drawn to scale. You can also click a value on the drawing to change it.

Known values

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Each side is named after the corner it faces — side a is opposite vertex A, b opposite B, c opposite C. The drawing labels each one.

Sides

Side a

Side b

Side c

Angles (degrees)

Angle A

Angle B

Angle C

Two triangles fit these values.

Units Decimals

Click an amber value to edit it · not to printed scale

Solution

How to solve a triangle

A triangle is fully determined by any three of its six measurements, as long as at least one is a side — three angles alone fix the shape but not the size. From there the calculator applies two standard rules.

Law of cosines · c² = a² + b² − 2ab·cos C

Law of sines · a / sin A = b / sin B = c / sin C

The ambiguous case (SSA)

If you give two sides and an angle that isn't between them, there can be zero, one or two triangles. The calculator detects this, shows the first, and lets you switch to the alternate when a second exists.

What it calculates

All three sides and angles, plus area (Heron's formula), perimeter, the three heights and medians, the inscribed-circle radius and the circumscribed-circle radius — with the triangle classified by its sides and its angles.

Solving a triangle, step by step

A triangle has six measurements — three sides and three angles. "Solving" it means finding all six from the few you already know. You need exactly three, and at least one of them has to be a side: three angles on their own set the triangle's shape but not its size. Which three you have decides which rule comes first.

The four cases

SSS — three sides. Every angle is found with the law of cosines, rearranged to solve for the angle.
SAS — two sides and the angle between them. The law of cosines gives the third side, then the law of sines (or cosines) gives the remaining angles.
ASA or AAS — two angles and a side. The third angle comes from the 180° sum, then the law of sines gives the other two sides.
SSA — two sides and an angle not between them. This is the ambiguous case: depending on the numbers there can be zero, one, or two valid triangles.

Worked example: two sides and the included angle (SAS)

Suppose you know two sides and the angle between them: b = 8, c = 5, and the included angle A = 60°. Here is how the calculator solves it.

Step 1 — third side, by the law of cosines

The side opposite the known angle is a:

a² = b² + c² − 2bc·cos A = 8² + 5² − 2·8·5·cos 60° = 89 − 40 = 49 → a = 7

Step 2 — a second angle, by the law of cosines

Rearranged to solve for angle B:

cos B = (a² + c² − b²) / (2ac) = (49 + 25 − 64) / 70 = 0.1429 → B = 81.79°

Step 3 — the last angle, by the 180° sum

C = 180° − A − B = 180° − 60° − 81.79° = 38.21°

Step 4 — area, from two sides and their included angle

Area = ½·b·c·sin A = ½·8·5·sin 60° = 17.32

Result

Sides 7, 8 and 5; angles 60°, 81.79° and 38.21°; area 17.32; perimeter 20. Enter those three values above and you'll see the same triangle drawn to scale.

Formulas this calculator uses

Law of cosines · c² = a² + b² − 2ab·cos C

Law of sines · a / sin A = b / sin B = c / sin C

Area (Heron's) · √[s(s−a)(s−b)(s−c)], s = ½(a+b+c)

Angle sum · A + B + C = 180°

Beyond the sides, angles, area and perimeter, the tool also reports the three heights, the three medians, the inradius (inscribed circle) and the circumradius (circumscribed circle), and classifies the triangle by both its sides and its angles.

Frequently asked questions

How do you solve a triangle?

Start with any three measurements that include at least one side. If you have all three sides, use the law of cosines to find the angles. If you have two sides and the angle between them, the law of cosines gives the third side. If you have two angles and a side, subtract from 180° for the third angle and use the law of sines for the sides.

Can you solve a triangle from three angles alone?

No. Three angles (the AAA case) fix the triangle's shape but not its size — infinitely many triangles share the same angles at different scales. You need at least one side length to pin down the actual dimensions.

What is the ambiguous case (SSA)?

It happens when you know two sides and an angle that is not between them. Depending on the values there may be no triangle, exactly one, or two different valid triangles. This calculator detects the situation, shows one solution, and lets you switch to the alternate when a second one exists.

How do you find the area without knowing the height?

Two ways. If you know all three sides, use Heron's formula. If you know two sides and the angle between them, the area is one half of their product times the sine of that angle (½·a·b·sin C). Both avoid needing a measured height.

When do you use the law of sines versus the law of cosines?

Reach for the law of cosines when you have three sides (SSS) or two sides and the included angle (SAS) — it works directly with those. Use the law of sines when you already have a complete angle-and-opposite-side pair, which is the case with two angles and a side (ASA, AAS).

How do you find an angle from three sides?

Rearrange the law of cosines. For the angle opposite side a: cos A = (b² + c² − a²) / (2bc), then take the inverse cosine. Repeat for a second angle and subtract both from 180° for the third.