What is the Law of Cosines?
The Law of Cosines relates the three sides of a triangle to the cosine of one angle. It generalizes the Pythagorean theorem from right triangles to any valid triangle.
c² = a² + b² − 2ab cos(C)The same relationship can be written for the other two sides:
a² = b² + c² − 2bc cos(A) b² = a² + c² − 2ac cos(B)When should you use the Law of Cosines?
The Law of Cosines is the natural first tool in two important configurations:
- SAS: two sides and the included angle are known.
- SSS: all three sides are known.
When ASA, AAS or SSA data are given, the Law of Sines is usually more direct. SSA can also produce an ambiguous case with zero, one or two solutions.
SAS: finding a missing side
Suppose sides a and b and the included angle C are known. Solve:
c = √(a² + b² − 2ab cos(C))Example: a = 7, b = 10 and C = 60°.
c² = 7² + 10² − 2(7)(10)cos(60°) c² = 49 + 100 − 140(0.5) = 79 c = √79 ≈ 8.888Why the angle must be included in SAS
If you know sides a and b, the included angle is C because C lies between those two sides and is opposite c. Using a non-included angle in the same formula does not represent the stated SAS geometry.
This is one of the most common setup mistakes in oblique-triangle problems.
SSS: finding an angle
When all three sides are known, rearrange the Law of Cosines. To find angle C:
C = arccos((a² + b² − c²)/(2ab))For a=7, b=8 and c=9:
cos(C) = (49 + 64 − 81)/(112) = 32/112 ≈ 0.285714 C ≈ 73.398°Why arccos is convenient for SSS
For a valid triangle, the cosine-form expression for an angle lies between −1 and 1. Taking arccos returns the unique interior triangle angle between 0° and 180°.
This is different from the inverse-sine ambiguity that can occur in SSA problems. SSS determines a unique triangle up to reflection.
Triangle inequality matters
Three positive lengths form a nondegenerate triangle only if every pair sums to more than the remaining side:
a+b>c, a+c>b, b+c>aIf 2, 3 and 6 are entered, 2+3 is not greater than 6, so no ordinary triangle exists. A high-quality SSS calculator should reject those inputs before attempting inverse cosine.
Degenerate triangles
If two side lengths add exactly to the third, the three points lie on a straight line. The area is zero and the shape is degenerate rather than an ordinary triangle.
SonoCalculator requires strict triangle inequality for SSS so degenerate cases are not presented as valid solved triangles.
Law of Cosines and the Pythagorean theorem
If C = 90°, then cos(90°)=0. The cosine term disappears:
c² = a² + b²So the Pythagorean theorem is a special case of the Law of Cosines.
Acute and obtuse angles in the formula
The sign of cos(C) explains how the side opposite C behaves.
For an acute angle, cos(C) is positive, so the subtraction term reduces c². For an obtuse angle, cos(C) is negative, so subtracting a negative value increases c². The side opposite an obtuse angle is therefore relatively long.
Largest side and largest angle
In every triangle, the largest side is opposite the largest angle. This provides a quick reasonableness check after solving SSS.
If c is clearly the longest side but the computed C is not the largest angle, something is wrong with the side-angle pairing or arithmetic.
Solving the whole triangle from SAS
A robust SAS workflow is:
- Use the Law of Cosines to find the third side.
- Use the Law of Cosines again, or carefully use the Law of Sines, to determine another angle.
- Use the 180° angle sum for the final angle.
This calculator computes all three angles from the solved side lengths, which avoids accidentally choosing a supplementary sine angle.
Solving the whole triangle from SSS
With three valid sides, compute angles with:
A = arccos((b²+c²−a²)/(2bc)) B = arccos((a²+c²−b²)/(2ac)) C = arccos((a²+b²−c²)/(2ab))The sum should be 180° within numerical rounding.
Why calculators clamp arccos inputs
Floating-point arithmetic can occasionally produce a value like 1.0000000000000002 when the exact mathematical value should be 1. Passing that directly to arccos would fail.
A numerically careful calculator clamps tiny rounding excursions back into the valid interval [−1,1] before evaluating arccos. This protects against machine-rounding noise without turning genuinely impossible triangles into valid ones.
Area from three sides: Heron's formula
After a full triangle is solved, its area can be found from side lengths using the semiperimeter:
s = (a+b+c)/2 Area = √[s(s−a)(s−b)(s−c)]Heron's formula is convenient because it works directly from SSS data and also works after SAS has produced the missing side.
Area from two sides and included angle
For SAS, the same area can be found before the third side is known:
Area = ½ab sin(C)The two area formulas should agree, apart from rounding, once the whole triangle is solved.
Perimeter and semiperimeter
The perimeter is:
P = a+b+cThe semiperimeter is half of it:
s = P/2Semiperimeter is useful in Heron's formula and several other triangle identities.
Circumradius
The circumradius R is the radius of the circle through the three vertices. Once area K is known:
R = abc/(4K)It also follows from the extended Law of Sines:
a/sin(A) = b/sin(B) = c/sin(C) = 2RInradius
The inradius r is the radius of the inscribed circle tangent to all three sides:
r = K/swhere K is the area and s is the semiperimeter.
Altitudes
Each altitude is the perpendicular height to its corresponding side:
hₐ = 2K/a, hᵦ = 2K/b, h𝚌 = 2K/cThese follow directly from the area formula K = ½(base)(height).
Triangle classification by sides
A triangle is equilateral when all three sides are equal, isosceles when at least two sides are equal, and scalene when all three are different.
Because computed values can differ by tiny floating-point amounts, a calculator should use a small numerical tolerance rather than exact string equality.
Triangle classification by angles
A triangle is acute if all angles are less than 90°, right if one angle is 90°, and obtuse if one angle exceeds 90°.
The Law of Cosines gives a useful side-only classification too. For the largest side c:
c² < a²+b² → acute c² = a²+b² → right c² > a²+b² → obtuseDegrees versus radians
The formulas are valid in either angle unit, provided the trigonometric function uses the same convention. This calculator uses degrees because most school geometry and triangle-solving problems are presented that way.
Law of Cosines versus Law of Sines
| Given information | Best first method | Why |
|---|---|---|
| SSS | Law of Cosines | No known opposite side-angle pair |
| SAS | Law of Cosines | Finds the third side directly |
| ASA / AAS | Law of Sines | Known angle data and side-angle pair |
| SSA | Law of Sines | May involve the ambiguous case |
Worked example: 5, 8 and included 110°
Given a=5, b=8 and C=110°:
c² = 25 + 64 − 80cos(110°)Since cos(110°) is negative, the final subtraction adds to c². This is consistent with C being obtuse and c being the longest side.
After c is found, all three side lengths are available and the remaining angles can be computed from the cosine formulas.
Worked example: a 3-4-5 triangle
For a=3, b=4 and c=5, angle C is:
C = arccos((3²+4²−5²)/(2·3·4)) C = arccos(0) = 90°This directly recovers the familiar right triangle.
Near-degenerate triangles and sensitivity
When one side is almost equal to the sum of the other two, the triangle becomes extremely thin. The area approaches zero and one angle approaches 180°.
In such cases, small measurement errors can create relatively large changes in angles or area. Numerical output may be mathematically valid while still being sensitive to input precision.
Measurement error in real-world triangles
Surveying, construction and physical measurement data are rarely exact. If side lengths are rounded, computed angles inherit that uncertainty.
For practical work, retain realistic precision and avoid reporting many decimal places that the original measurements cannot support.
Law of Cosines in vectors
The cosine rule is closely related to the dot product. If vectors u and v form angle θ, then:
|u−v|² = |u|² + |v|² − 2|u||v|cos(θ)This has the same structure as the Law of Cosines and connects triangle geometry with vector algebra.
Law of Cosines in surveying and navigation
If two distances from a known point and the included angle are known, the Law of Cosines can calculate the distance between the two remote points in a planar model.
For large geographic distances, Earth curvature and map projection effects matter, so spherical or geodesic methods may be required instead of ordinary Euclidean triangle geometry.
Common Law of Cosines mistakes
- Using the wrong angle in SAS. The angle must be included between the two known sides.
- Mismatching opposite labels. Side c is opposite C, and similarly for a/A and b/B.
- Forgetting the cosine term's minus sign.
- Using degrees while a calculator is set to radians.
- Skipping the triangle inequality check in SSS.
- Taking arccos before isolating the cosine expression correctly.
- Rounding too early. Preserve extra digits until the final displayed result.
- Using Law of Sines first for SSS. There is no known side-angle pair yet.
- Assuming a diagram is drawn exactly to scale. Use the computed values, not the sketch, for conclusions.
How to verify a solved triangle
Check the angle sum:
A+B+C = 180°Check the triangle inequality, then substitute the solved values back into at least one Law of Cosines equation. You can also verify the area independently using both Heron's formula and ½ab sin(C).
For a right triangle, the cosine rule should reduce to the Pythagorean theorem.
Frequently asked questions
What is the Law of Cosines formula?
For side c opposite angle C: c² = a² + b² − 2ab cos(C).
When should I use the Law of Cosines?
Use it especially for SAS and SSS triangles.
Can the Law of Cosines solve SSS?
Yes. Rearrange the formula and use arccos to find each angle.
Can it solve SAS?
Yes. Two sides and their included angle determine the third side uniquely.
Can it solve SSA?
SSA is generally a Law of Sines situation and can have zero, one or two solutions.
Why does the triangle inequality matter?
Without it, three side lengths may not be able to close into a nondegenerate triangle.
How is the Pythagorean theorem related?
It is the special case when the included angle is 90° and cos(90°)=0.
Why use arccos instead of arcsin for SSS?
The cosine form directly returns the unique interior angle associated with the three sides and avoids the supplementary ambiguity of inverse sine.
Does this calculator find area too?
Yes. It reports area, perimeter, semiperimeter, circumradius, inradius and altitudes after solving the triangle.
Why might my real-world result need fewer decimals?
Computed precision should not exceed the meaningful precision of the measurements you entered.
Final note: use the Law of Cosines when SSS or SAS gives you no immediate opposite side-angle pair. Match each side to its opposite angle, verify triangle inequality for SSS, and keep the included angle straight in SAS. This calculator solves the full triangle and exposes area and circle-radius checks so the result can be validated from several independent relationships.