Law of Cosines Calculator

Law of Cosines calculator: the job and the audience

Searchers who type law of cosines calculator usually want side, angle, or area, not a page that restates the definition and hides the math. This URL puts the live Law of Cosines tool first, then the identity it uses: the triangle identity on this page.

On the shipped sample — side a of 3, side b of 4, side c of 5, angle of 60° — the engine shows 12. That figure is a walkthrough. It exists so you can see the model work before you overwrite the boxes with the numbers from your worksheet, exam review, or lab notebook.

A missing side, a missing angle, or an area from SAS/SSS. Mixing those formulas is how geometry tests get lost.

Related searches

People also search law of cosines formula, how to calculate law of cosines, side, angle, or area. Those queries still map to this model: the triangle identity on this page.

How to fill the Law of Cosines calculator

Each control below is a real variable in the triangle identity on this page. If a unit is already printed as ° or %, do not convert it again. The sample values are a walkthrough, not a suggestion for your homework set.

Side A

Side A uses the unit printed on the label. Match that unit; the engine will not guess centimeters versus meters. The sample uses 3. Overwrite it when your problem does not look like the demo.

Side B

Side B uses the unit printed on the label. Match that unit; the engine will not guess centimeters versus meters. The sample uses 4. Overwrite it when your problem does not look like the demo.

Side C

Side C uses the unit printed on the label. Match that unit; the engine will not guess centimeters versus meters. The sample uses 5. Overwrite it when your problem does not look like the demo.

Angle

Angle is in degrees on this form unless the label says radians. 90 means a right angle, not π/2. If your calculator is in radian mode, convert first. The sample uses 60°. Overwrite it when your problem does not look like the demo.

Step-by-step law of cosines example

The identity on this page is the triangle identity on this page. Walk the sample once, then change a single field.

Step 1. Enter side a as 3. That is the default shipped with this law of cosines calculator so you can see a finished headline before you touch anything.

Step 2. Enter side b as 4. That is the default shipped with this law of cosines calculator so you can see a finished headline before you touch anything.

Step 3. Enter side c as 5. That is the default shipped with this law of cosines calculator so you can see a finished headline before you touch anything.

Step 4. Enter angle as 60°. That is the default shipped with this law of cosines calculator so you can see a finished headline before you touch anything.

Result. The engine prints 12. That number is what the form at the top of this page is wired to show for those inputs.

If you change only one coefficient and the headline barely moves, that input is not doing the work on this model. If it jumps, it is. That is the useful part of a worked example — not the demo numbers themselves.

The identity behind this law of cosines calculator

On the Law of Cosines Calculator, the engine applies the triangle identity on this page to side a, side b, side c, angle. That is not a hidden solver and it is not a computer-algebra dump. It is the classroom identity that matches this slug (law-of-cosines-calculator).

Pythagoras needs a right angle. SSA is the ambiguous case. Do not feed an obtuse triangle into a right-triangle identity.

Two honest ways to break this formula: put the part in the whole box, or treat a percent as a decimal. Recalculate after you fix the unit. If the headline still looks absurd, this may simply be the wrong tool — a percentage page will not solve a quadratic, and a volume page will not do surface area.

Practical scenarios

This page is written for geometry students using Pythagoras, congruence, or triangle area who need a law of cosines number they can check.

Run the sample, then a second case that matches the problem in your book. If both headlines look reasonable, you probably have the right identity.

If you are checking homework, change only the field that differs from the demo. Changing three things at once is how people lose the thread.

Come back when one input changes — a new radius, a new data point, a new angle. Re-running the same demo every night teaches you nothing.

Mistakes that wreck a law of cosines estimate

• Feeding SSS data into an SAS formula.
• Treating this law of cosines calculator as a proof, an exam answer key, or a CAS. It is an educational estimate from the numbers you typed.

Continue on CalculatorWeb

Internal links here are editorial, not a dump of the whole Math library. They sit next to the same kind of problem as the Law of Cosines Calculator. Start with Law of Sines Calculator if you still have a missing piece.

• Law of Sines Calculator — keep the same numbers if you just finished the law of cosines run, so the two headlines can be compared.
• Direction Cosines Calculator — keep the same numbers if you just finished the law of cosines run, so the two headlines can be compared.
• Direction Cosines Calculator — keep the same numbers if you just finished the law of cosines run, so the two headlines can be compared.
• 30-60-90 Triangle Calculator — keep the same numbers if you just finished the law of cosines run, so the two headlines can be compared.

Citations

Start with Khan Academy triangles if you need the official or handbook definition that sits behind this law of cosines question. This article cites that page; it does not replace a textbook proof.

Also useful: NASA practical uses of math. Same rule — primary source over a content farm.

FAQ

What does this law of cosines calculator actually calculate?

It applies the triangle identity on this page to the fields on this page. The large number is the headline. It does not show every algebraic transformation a teacher might want on lined paper.

Is the Law of Cosines calculator free?

Yes. There is no signup wall on the tool. Use it whenever the problem changes — a new radius, a new data list, a new coefficient. The shortcode stays on this URL.

Why is the sample result 12?

Because those are the defaults shipped with the form. They exist so you can see a finished identity before you type. They are not “the” answer to your homework set.

How do I calculate law of cosines by hand?

Use the triangle identity on this page. Plug in the same units the labels use. If your hand calc disagrees with the headline, you usually converted a percent to a decimal twice, mixed degrees and radians, or swapped two fields.

Why would a graphing calculator or CAS show a different number?

Mode (deg/rad), exact versus floating output, sample versus population, or a different algebraic form of the same value. That is two models, not always one broken page.

Can I paste 12 into an exam as the official answer?

No. It is an educational estimate. Your teacher’s required form (exact, rounded, with units) still sits above it.

What should I change first?

Any field still sitting at the demo value that is not in your problem.

How is this different from the related tools on CalculatorWeb?

This slug is wired to one identity. If you need the neighboring question, open Law of Sines Calculator and reuse the same numbers so the two headlines can be compared.

Does this law of cosines calculator store my numbers?

The calculation runs in your browser from the fields on this page. Treat it like a notepad, not an account. If you need a record, print the inputs.

What search terms should I use if I want this page again?

The focus phrase is law of cosines calculator. People also search “how to calculate law of cosines” and “law of cosines formula.” Those queries should land here if the title and the H2s stay specific to this model.

Educational estimate only. This is not a substitute for a textbook proof, an exam rubric, or a teacher’s required form. Formula review: 19 August 2026.

Law of Cosines vs a handheld calculator

A handheld is faster when you already know the identity and just need a decimal. This law of cosines calculator is better when you want the identity in public, on a page you can send to a classmate who will not open your calculator history. The tradeoff is honesty about scope: the triangle identity on this page is what you get, not a CAS with hidden simplify() steps.

If you already live in Desmos or a TI-84, use this page as a checksum. Type the same side a and see whether your device and the shortcode agree. If they do not, one of you is in the wrong mode or the wrong unit.