Law of Cosines Calculator

Three boxes, pre-filled with a = 10, b = 12, C = 30°. Click Calculate. This law of cosines calculator form takes the SAS setup: two sides and the angle between them. The law of cosines calculator is the SAS machine.

• Side a and side b are the two known sides.
• Angle C is the included angle, the one squeezed between sides a and b. In degrees.
• Seven answers come back: the third side c, both remaining angles A and B, plus the perimeter, the area, and the two circle radii — the triangle is fully solved.

The law of cosines calculator keeps full precision, so 10, 12, and 30 return c ≈ 6.013, A ≈ 56.3°, B ≈ 93.7°.

This page is more than a static law of cosines calculator — it is also an AI tutor you can ask questions:

• Ask in plain words. "Sides 5 and 9 with a 110° angle between them — find the third side." The AI fills the form and runs this law of cosines calculator for you.
• Take a photo of your homework. The AI reads the triangle and fills the law of cosines calculator boxes for you.
• Ask about the ideas, not just the answer. "Why does the 2ab cos C term shrink c?" The AI explains step by step — every number still comes from the law of cosines calculator, so nothing is guessed.
• Keep asking follow-ups. Ask "why?" as often as you need — follow-up questions are welcome.
• Three sides instead? That is the SSS case: the AI rearranges the same formula to hunt the angles with you.\n\nThe worked law of cosines examples below use the exact numbers this law of cosines calculator pre-fills, so you can follow along and check.

The Pythagorean theorem has one blind spot: it only works when the angle is 90°. The law of cosines removes that restriction — it is the Pythagorean theorem generalized to every triangle, and this law of cosines calculator runs it. Compare the two:

• Right angle: $c^2 = a^2 + b^2$
• Any angle: $c^2 = a^2 + b^2 - 2ab\cos C$

The new term $-2ab\cos C$ is the correction for the tilt. Point the angle wide (C near 180°) and cos C approaches −1, so the correction adds and c stretches. Squeeze it narrow (C near 0°) and cos C approaches 1, so c shrinks toward the gap between the sides. Set C = 90° and cos 90° = 0: the correction vanishes and the law of cosines calculator becomes the Pythagorean theorem exactly.

Read the picture first: two sides pinned at the angle C, third side waiting. Now the symbols the law of cosines calculator uses:

$c^2 = a^2 + b^2 - 2ab\cos\,C$

| $P = a + b + c$ | perimeter, once the third side lands || $K = \frac{1}{2}ab\sin C$ | area from the two known sides and their angle || $r = K/s$ | inradius; $s = P/2$ is the semi-perimeter — all four of these ride along in the law of cosines calculator || $R = c/(2\sin C)$ | circumradius, the circle through all three vertices || Symbol | Meaning | |---|---| | $a$, $b$, $c$ | sides opposite angles $A$, $B$, $C$ | | $C$ | the included angle between sides $a$ and $b$ |

Every number the law of cosines calculator returns is the first row plus two arccosines. Check any law of cosines calculator output by summing the three angles: they must total 180°. This is the law of cosines calculator's built-in lie detector.

Give the law of cosines calculator its home-game: a = 10, b = 12, and the angle between them C = 30°.

1. Third side: $c^2 = 100 + 144 - 2(10)(12)\cos 30\degree = 244 - 207.85 = 36.15$, so $c \approx 6.013$.
2. Angle A: $\cos A = (b^2 + c^2 - a^2)/(2bc) = 0.555$, so $A \approx 56.3\degree$.
3. Angle B: $180 - 30 - 56.3 = 93.7\degree$.

These exact numbers come pre-loaded in the law of cosines calculator above — click Calculate and the law of cosines calculator confirms all three instantly.

Use the law of cosines to find angle measures when all three sides are known. A triangle has sides 20, 25, and 18; find the angle opposite the side 20.

1. SSS case — rearrange: $\cos A = (b^2 + c^2 - a^2)/(2bc)$.
2. \cos A = (625 + 324 - 400)/(2 \. 25 \. 18) = 549/900 = 0.61.
3. $A \approx 52.4\degree$.

Three sides lock the triangle with no ambiguity — every angle comes out unique. The law of cosines calculator form above covers SAS. For this law of cosines SSS situation, ask the AI to run the rearranged version step by step.

Watch the correction term work with the law of cosines calculator: sides a = 3 and b = 4, first with C = 90°, then with C = 60°.

1. Right angle: $c^2 = 9 + 16 - 24\cos 90\degree = 25$, so $c = 5$ — the classic 3-4-5 triangle.
2. Narrower: $c^2 = 9 + 16 - 24\cos 60\degree = 25 - 12 = 13$, so $c = \sqrt{13} \approx 3.61$.
3. Smaller angle, smaller third side — the law of cosines calculator promised exactly this; the $-2ab\cos C$ term pulled it in.

The law of cosines calculator shows the sweep live: keep 3 and 4 fixed, walk C from 90° down to 60°. Watch the law of cosines calculator's side c shrink from 5 to 3.61.

Four slips the law of cosines calculator catches every day:

1. Using an angle that is not the included one. The formula wants the angle squeezed between the two known sides. Given angles A or B instead of C? That is a different setup — ask the AI on this law of cosines calculator page which law fits.
2. Dropping the minus sign. The correction is $-2ab\cos C$. Write $+2ab\cos C$ and every answer comes out wrong in the same subtle direction.
3. Feeding radians where degrees belong. The law of cosines calculator takes degrees in the angle box. Feed this law of cosines calculator 30, never 0.52.
4. Bringing an SSA problem here. Two sides and a non-included angle is the law of sines' ambiguous case — it can give two triangles. That case never reaches a law of cosines calculator: the law of cosines calculator's SAS setup is single-answer by design.