Law of Cosines Calculator

The law of cosines (also called the cosine rule) links the three sides of any triangle to one of its angles: c² = a² + b² − 2ab cos C. It solves the two cases the sine rule cannot start on, SAS and SSS. Example: with sides 8 and 11 and a 37° angle between them, the third side is 6.6663 and the area is 26.4799 square units.

Two sister pages cover the neighboring questions. The Law of Sines Calculator handles AAS, ASA and the ambiguous SSA case, where you already know an angle and the side opposite it. The Triangle Calculator takes whatever mix of sides and angles you have and returns the full solution without you choosing a rule. Use this page when you want the cosine rule itself shown line by line, either to find a third side from SAS data or to turn three measured lengths into three angles.

How the law of cosines works

Every triangle obeys the same relationship between a side and the angle across from it:

Read it as the Pythagorean theorem plus a correction. If the angle C between sides a and b were exactly 90°, cos C would be 0, the last term would vanish, and you would be left with c² = a² + b². Anything other than a right angle needs the correction term 2ab cos C, and the sign of the cosine tells you which way it pushes. For an acute angle the cosine is positive, so you subtract something and the third side comes out shorter than the right-angle value. For an obtuse angle the cosine is negative, subtracting a negative adds length, and the third side comes out longer.

The three forms you actually use

The rule is symmetric, so it can be written once for each vertex, and it can be rearranged to give an angle instead of a side. Those are the four expressions this calculator applies:

The first three take SAS data and hand you the missing side. The fourth takes SSS data and hands you an angle, and because inverse cosine returns exactly one value between 0° and 180°, the answer is never ambiguous. That single-answer property is the reason many people reach for the cosine rule first and only then switch to the sine rule for the remaining angles.

Worked example 1: SAS, two sides and the angle between them

Suppose a = 8, b = 11, and the angle between those two sides is C = 37°. Work through it in five moves:

1. Square the known sides: a² = 64 and b² = 121, so a² + b² = 185.
2. Build the correction term: 2ab = 2 × 8 × 11 = 176, and cos 37° = 0.798636, so 2ab cos C = 140.5598.
3. Subtract: c² = 185 − 140.5598 = 44.4402, so c = 6.6663.
4. Find angle A from the rearranged rule: cos A = (121 + 44.4402 − 64) / (2 × 11 × 6.6663) = 0.691671, so A = 46.2375°.
5. Find angle B the same way: cos B = (64 + 44.4402 − 121) / (2 × 8 × 6.6663) = −0.117754, so B = 96.7625°.

Check the angle sum: 46.2375 + 96.7625 + 37 = 180.0000°. The area is 0.5 × 8 × 11 × sin 37° = 0.5 × 88 × 0.601815 = 26.4799 square units, and Heron's formula on the solved sides (s = 12.8332) returns the identical 26.4799, which is a clean cross-check. The perimeter is 25.6663 and the circumradius is 5.5385. Notice that angle B came out obtuse even though the only angle you were given was a modest 37°: the cosine rule found that on its own, with no ambiguity to resolve.

Worked example 2: SSS, three measured sides

Now take a triangle whose sides measure 7, 9 and 12, with no angle known at all. Use the rearranged form three times:

• cos C = (49 + 81 − 144) / (2 × 7 × 9) = −14 / 126 = −0.111111, so C = 96.3794°.
• cos A = (81 + 144 − 49) / (2 × 9 × 12) = 176 / 216 = 0.814815, so A = 35.4309°.
• cos B = (49 + 144 − 81) / (2 × 7 × 12) = 112 / 168 = 0.666667, so B = 48.1897°.

The three angles add to 180.0000°, confirming the arithmetic. For the area, the semiperimeter is s = 28 / 2 = 14, and Heron's formula gives the square root of 14 × 7 × 5 × 2 = the square root of 980 = 31.305 square units. The circumradius is 6.0374 and the inradius is 2.2361. The negative cosine on angle C is the shape test in action: 12² = 144 exceeds 7² + 9² = 130, so the triangle is obtuse.

How the third side changes with the included angle

Hold two sides fixed at a = 10 and b = 14 and swing the angle between them. The table shows the correction term, the resulting third side, and the area at each angle, all computed with the formula above.

Two things stand out. The 90° row is pure Pythagoras: 100 + 196 = 296, and the square root of 296 is 17.2047. And the area column is symmetric around 90° while the side column is not, because area depends on sin C, which is equal at 75° and 105°, while the side depends on cos C, which flips sign.

Shape test: what three sides tell you before you solve

In SSS work, comparing the longest side squared with the sum of the other two squares tells you the shape immediately, and the cosine rule then gives the exact angle. Each row below was computed with cos = (x² + y² − z²) / (2xy) for the largest angle, with Heron's formula for the area.

The three Pythagorean triples land on a cosine of exactly zero, which is what "right triangle" means in cosine-rule language. For a quick check of that special case on its own, the Pythagorean Theorem Calculator is faster.

How the angle grows as the opposite side grows

Fix a = 8 and b = 11 and let the third side c vary. The triangle inequality allows c anywhere strictly between 11 − 8 = 3 and 8 + 11 = 19; outside that range the shape cannot close. Inside it, every length of c produces exactly one angle C.

The area peaks near c = 14, which is close to the 90° row, exactly as expected: for two fixed sides the area 1/2 ab sin C is largest when the included angle is a right angle. The 6.6663 row is the SAS example from earlier read backwards, which is a good way to confirm that the two modes of this calculator agree.

How to use this calculator

1. Pick the case. Choose SAS if you know two sides and the angle between them. Choose SSS if you have measured all three sides.
2. Label consistently. Side a sits opposite angle A, side b opposite B, side c opposite C. In SAS mode, the two sides you enter are a and b, and the angle you enter is the one they form.
3. Type the values. Enter lengths in whatever unit you are working in, and the angle in degrees. The result recalculates as you type.
4. Read the headline. In SAS mode it shows the third side; in SSS mode it shows the largest angle. The tiles below carry all three angles and all three sides.
5. Check the derivation. The numbered steps reproduce the substitution, the correction term, the square root, and each inverse cosine, so you can copy the work into homework or a job sheet.
6. Set the display. Choose a unit label and one to four decimal places. Only the display rounds; the math always runs at full precision.

Who this calculator is for

• Trigonometry and precalculus students who need the cosine rule shown step by step, not just an answer.
• Carpenters and deck builders laying out a non-square corner from two run lengths and an angle.
• Surveyors and site technicians closing a traverse from three measured distances.
• Machinists and fabricators checking a bolt-circle chord or a mitered joint against a print.
• Navigators and pilots resolving a course-change leg where the turn angle is known but the closing distance is not.
• Anyone with three tape measurements who needs the angles between them without a protractor.

A real-world example: a deck corner

You are framing a deck where two runs leave the same corner post. One run is 12 ft, the other is 18 ft, and the angle between them is 110° rather than a square 90°. How long is the diagonal that closes the shape, and what angles do the runs make with it? Apply the cosine rule: cos 110° = −0.342020, so 2ab cos C = 2 × 12 × 18 × (−0.342020) = −147.7527. Then c² = 144 + 324 + 147.7527 = 615.7527, and the diagonal is 24.8144 ft, a little under 24 ft 9.8 in. The remaining angles follow from the rearranged rule: 42.9719° at the end of the 12 ft run and 27.0281° at the end of the 18 ft run, adding with 110° to a clean 180°. The enclosed area is 0.5 × 12 × 18 × sin 110° = 101.4868 square feet, which is what you would order decking for. Note how the obtuse angle stretched the diagonal well past the right-angle answer of 21.6333 ft.

Key terms explained

• Included angle: the angle formed between two named sides. In SAS data it is the angle you know, and it is opposite the side you are solving for.
• Oblique triangle: any triangle without a 90° angle. The cosine rule was built for these; right triangles are just the special case where the correction term is zero.
• Triangle inequality: the requirement that the two shorter sides add up to more than the longest side. It is the validity test for SSS input.
• Heron's formula: an area formula that needs only the three sides, Area = the square root of s(s − a)(s − b)(s − c) with s the semiperimeter.
• Circumradius (R): the radius of the circle through all three vertices, equal to abc divided by four times the area.
• Inradius (r): the radius of the largest circle that fits inside the triangle, equal to the area divided by the semiperimeter.
• Inverse cosine (arccos): the function that turns a cosine back into an angle. It returns a single value from 0° to 180°, which is why the cosine rule never produces two candidate triangles.

What changes the result the most

• The included angle. It is the most sensitive input in SAS mode. Moving from 60° to 120° on the same 10 and 14 sides stretches the third side from 12.49 to 20.8806, a 67% jump.
• The sign of the cosine. Crossing 90° flips the correction term from subtracting length to adding it, which is why obtuse triangles look so different from acute ones with the same two sides.
• Measurement error in the longest side. In SSS mode the largest angle is the one most affected by a tape that reads a fraction long, because it depends on the squared value of that side.
• Near-degenerate shapes. When the three sides barely satisfy the triangle inequality, the cosine is close to −1 and a tiny change in one length swings the angle by degrees.
• Rounding partway through. Rounding the cosine to two decimals before the inverse cosine can move the final angle by more than a tenth of a degree.

Tips for a clean, accurate answer

• Solve the largest angle first in SSS mode. It is the only one that can be obtuse, and the cosine rule handles it correctly, whereas finishing with the sine rule on a big angle can hide an obtuse answer.
• Cross-check the angle sum. Every solved triangle should total 180°. This calculator prints the sum for exactly that reason.
• Cross-check the area two ways. Heron's formula and 1/2 ab sin C must agree; if they do not, a side or angle got mislabeled.
• Keep full precision. Carry unrounded values through the derivation and round only the final display, which is what the calculator does internally.
• Use consistent units. All three lengths must be in the same unit. Mixing feet and inches is the single most common source of wrong answers in shop work.

Limitations and assumptions

• It assumes plane (Euclidean) geometry, so the angle sum is exactly 180°. Long-range geodesy on the curved surface of the earth needs spherical trigonometry instead.
• It takes SAS and SSS only. AAS, ASA and SSA data start with the law of sines, and the ambiguous SSA case needs the extra height test that page provides.
• Answers are decimal, not symbolic. A textbook answer of the square root of 44.4402 is shown as 6.6663 here.
• Angles are entered and displayed in degrees, not radians or gradians.
• Results are only as good as your measurements. A tape that is off by 1/8 in or a protractor read to the nearest degree limits the real accuracy far more than the arithmetic does.

How it compares to related calculators

This page answers "what are the missing parts of this triangle when I have SAS or SSS data?" For a different starting point, a sister tool fits better:

• If you know an angle and the side opposite it (AAS, ASA or SSA), use the Law of Sines Calculator, which also resolves the ambiguous case.
• If you would rather enter whatever you have and let the tool pick the rule, use the Triangle Calculator.
• If the triangle has a right angle, the Pythagorean Theorem Calculator is the shortcut.
• For sine, cosine and tangent values on their own, use the Trigonometry Calculator.
• For the area of other shapes once your triangle is solved, use the Area Calculator, and for the distance between two coordinate points use the Distance Calculator.

Sources

Everything on this page is deterministic plane geometry, so there are no agency figures to cite. The law of cosines, its rearranged form for an angle, Heron's formula, the triangle inequality and the 180° angle sum are exact theorems of Euclidean geometry, provable from first principles and independent of any measurement standard or jurisdiction. The only outside convention used here is the definition of the degree as 1/360 of a full turn and the units you attach to your own lengths; for the exact definitions of the inch (25.4 mm exactly) and the other US customary units you may be measuring in, the reference is NIST Special Publication 811, the National Institute of Standards and Technology guide to the SI and to US customary conversions. No non-primary sources are used, and no market, tax or health data appears on this page.