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Walk-through

How to Use This Calculator

3 steps

1

Pick a mode based on what you know

Choose "Two Sides + Included Angle" (SAS) if you know two sides and the angle directly between them. Choose "Three Sides" (SSS) if you know all three side lengths and want to find an angle.

2

Enter the three known values

Fill in the fields for your chosen mode and leave the rest blank. The calculator solves the missing side or angle instantly using the law of cosines: c² = a² + b² − 2ab·cosC.

3

Check the Full Solve and Area tabs

The Solve tab gives you the one value you asked for. The Full Solve tab lays out every side and angle of the triangle plus a diagram, and the Area tab adds Heron's-formula area, perimeter, and circumradius.

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Reference

Formula & Methodology

3 formulas

The law of cosines (solving for a side)

c² = a² + b² − 2ab·cosC

Given two sides (a and b) and the angle between them (C, the included angle), this generalizes the Pythagorean theorem to any triangle — not just right triangles. When C = 90°, cosC = 0 and the formula collapses to c² = a² + b², exactly the Pythagorean theorem.

The law of cosines (solving for an angle)

cosC = (a² + b² − c²) / (2ab)

Rearranging the same identity to solve for the angle instead, given all three sides. This is the only case where the law of cosines is required to find any angle at all, since none of the three angles is directly known.

Heron's formula (area from three sides)

Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2

Once all three sides are known — either given directly (SSS) or solved via the law of cosines (SAS) — Heron's formula gives the triangle's area without needing any angle at all.

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Glossary

Key Terms Explained

7 terms

Law of cosines The identity c² = a² + b² − 2ab·cosC relating a triangle's three sides to the cosine of one angle. Works for any triangle and generalizes the Pythagorean theorem, which is the special case where that angle is 90°.

SAS Side-Angle-Side — two known side lengths plus the angle directly between them (the included angle). Always yields exactly one triangle; the law of cosines solves for the third side directly.

SSS Side-Side-Side — all three side lengths known, no angle given. The law of cosines is the only way to solve for an angle in this case, since there's no angle-side pair to use the law of sines with.

Included angle The angle formed directly between two known sides. SAS always uses the included angle — unlike SSA (side-side-angle), where the known angle is opposite one of the sides instead, which is why SSA needs the law of sines and can be ambiguous while SAS never is.

Oblique triangle Any triangle that isn't a right triangle — meaning none of its angles is exactly 90°. The law of cosines and law of sines are the standard tools for solving oblique triangles, since SOH-CAH-TOA only applies when a right angle is present.

Cosine rule Another common name for the law of cosines, used interchangeably in many textbooks and regions.

Heron's formula A formula for a triangle's area using only its three side lengths: Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter (a+b+c)/2. Named after Heron of Alexandria; works for any triangle without needing an angle or a height measurement.

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Scenarios

Real-World Examples

3 worked examples

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Two sides + included angle known (SAS)

Side a = 3, Side b = 4, Angle C = 90°

Side a 3 Side b 4 Angle C 90

c² = 3² + 4² − 2·3·4·cos(90°) = 9 + 16 − 0 = 25, so c = 5 — this is exactly the classic 3-4-5 right triangle, since a 90° included angle makes cosC = 0 and the law of cosines collapses to the Pythagorean theorem. The remaining angles are A ≈ 36.87° and B ≈ 53.13°, and the area (Heron's formula on sides 3, 4, 5) is exactly 6 square units.

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All three sides known (SSS)

Side a = 3, Side b = 4, Side c = 5

Side a 3 Side b 4 Side c 5

cosC = (3² + 4² − 5²) / (2·3·4) = (9 + 16 − 25) / 24 = 0/24 = 0, so C = 90°. The same 3-4-5 triangle, now solved the other direction — from all three sides to the angle. The other two angles work out to the same A ≈ 36.87° and B ≈ 53.13° as the SAS example, and the area is again 6 square units, confirming both directions of the law of cosines agree on the exact same triangle.

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Full-solve walkthrough

Side a = 7, Side b = 10, Angle C = 55°

Side a 7 Side b 10 Angle C 55

c² = 7² + 10² − 2·7·10·cos(55°) = 49 + 100 − 140·0.5736 ≈ 68.7, so c ≈ 8.29. From there, angle A = acos((b²+c²−a²)/(2bc)) ≈ 54.7° and angle B = 180 − C − A ≈ 70.3°. The Full Solve tab shows all of this at once — every side, every angle — and the Area tab adds the Heron's-formula area (≈ 28.7 sq units), perimeter (≈ 25.29 units), and circumradius (≈ 5.08 units).

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Deep Dive

Understanding the Law of Cosines Calculator

3 chapters · 4 min read

The law of cosines lets you solve a triangle when you know two sides and the angle between them (SAS), or all three sides (SSS) — the two cases the law of sines can't handle directly, since neither gives you a ready-made angle-side pair. This calculator solves for the missing side or angle, then chains the same identity to fill in the rest of the triangle and its area.

How the Law of Cosines Calculator works

The law of cosines, c² = a² + b² − 2ab·cosC, relates a triangle's three sides to the cosine of the angle opposite side c. In SAS mode, C is the known included angle between sides a and b, so plugging directly into the formula gives side c. In SSS mode, all three sides are known and the formula is rearranged to solve for the angle instead: cosC = (a² + b² − c²) / (2ab).

Once one unknown is solved, the calculator finds the remaining two angles using the law of cosines again (not the law of sines) — for example, A = acos((b²+c²−a²)/(2bc)), then B = 180° − A − C. Using the law of cosines for every angle avoids the ambiguous-case problem that the law of sines has with its two-valued sine function: cosine's inverse is single-valued over a triangle's 0°–180° angle range, so there's never a second candidate angle to rule out.

Inputs and what they mean

In SAS mode, angle C must be the included angle — the one directly between sides a and b, not opposite one of them. Entering an angle that's opposite a side instead of between two sides is the SSA case, which the law of cosines can't solve directly (that needs the law of sines, and carries the ambiguous-case risk this calculator's SAS/SSS design avoids entirely).

In SSS mode, all three sides must satisfy the triangle inequality — every pair of sides must add up to more than the third side (a+b>c, a+c>b, b+c>a). If they don't, no triangle exists with those measurements and the calculator flags it rather than returning a meaningless result.

Limits and edge cases

This calculator solves plane (Euclidean) triangles with exact input values — it doesn't model measurement uncertainty or spherical geometry (used in some navigation and astronomy problems). If you instead know an angle-side pair plus one more value (ASA, AAS, or SSA), the law of cosines isn't the right tool — use the law of sines, which may also need to flag the SSA ambiguous case.

A triangle that barely satisfies the SSS inequality, or a SAS angle very close to 0° or 180°, produces a valid but nearly degenerate triangle — one so "flat" its area is close to zero. The calculator still solves these correctly rather than rejecting them, but flags the result so you can double-check your inputs if that's unexpected.

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Questions

Frequently Asked Questions

6 questions

When should I use SAS versus SSS?+

Use SAS (two sides + included angle) when you know two side lengths and the angle directly between them. Use SSS (three sides) when you know all three side lengths but no angle. Both are solved with the law of cosines; the difference is just which values you start with.

How do I find a missing side?+

Use SAS mode: enter the two known sides and the included angle between them, and the calculator solves c² = a² + b² − 2ab·cosC directly for the third side.

How do I find a missing angle?+

Use SSS mode: enter all three side lengths, and the calculator rearranges the law of cosines to cosC = (a² + b² − c²) / (2ab) to solve for the angle.

How is this different from the Pythagorean theorem?+

The Pythagorean theorem (a² + b² = c²) only works for right triangles. The law of cosines generalizes it to any triangle by adding a correction term, −2ab·cosC — when angle C is exactly 90°, cosC = 0 and the two formulas become identical.

What units does this calculator use?+

Angle C is entered and displayed in degrees. Side lengths can be in any consistent unit of length — inches, feet, meters, etc. — as long as you use the same unit for every side, since the calculator works with the ratios and differences between them.

How is the area calculated?+

Once all three sides are known (either given directly in SSS mode, or solved via the law of cosines in SAS mode), the calculator uses Heron's formula: Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter (a+b+c)/2. This works for any triangle without needing a height measurement.

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Law of Cosines Calculator

Choose what you know

Solved Value

Full Triangle Solution

Triangle diagram

Area, Perimeter & Circumradius

How to Use This Calculator

Pick a mode based on what you know

Enter the three known values

Check the Full Solve and Area tabs

Formula & Methodology

Key Terms Explained

Real-World Examples

Two sides + included angle known (SAS)

All three sides known (SSS)

Full-solve walkthrough

Understanding the Law of Cosines Calculator

How the Law of Cosines Calculator works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

Law of Cosines Calculator

Choose what you know

Solved Value

Full Triangle Solution

Triangle diagram

Area, Perimeter & Circumradius

How to Use This Calculator

Pick a mode based on what you know

Enter the three known values

Check the Full Solve and Area tabs

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding the Law of Cosines Calculator

How the Law of Cosines Calculator works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

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