When do I use the law of cosines?

Use it when you know two sides and the included angle, or when you know all three sides and want an angle.

Is the law of cosines always unique?

For the cases it is designed for, it gives a unique triangle. It is especially helpful when the law of sines would be ambiguous.

Can it solve any triangle?

It solves the common SAS and SSS triangle cases and is a standard companion to the law of sines.

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Law of Cosines Calculator

Solve a triangle from sides and the included angle.

Side a Side b Included Angle C (degrees)

Solved Side c

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Angle A

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Angle B

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Comprehensive Guide

The Law of Cosines is a triangle formula that connects sides and angles when the triangle is not a simple right triangle. It is the go-to tool for side-angle-side and side-side-side situations. If you know two sides and the angle between them, you can solve the third side. If you know all three sides, you can solve an angle.

That makes the calculator useful in geometry, surveying, engineering, and navigation. The law of sines is great when the opposite side and angle are already paired up, but the law of cosines is more reliable when the triangle is missing that clean pairing. It removes ambiguity and gives a direct answer.

Because triangles come up in many practical settings, having a quick solver is valuable. It turns raw lengths and angles into a complete shape that can be measured, cut, or drawn with confidence.

Real-World Examples

If side a is 7, side b is 9, and the included angle C is 60°, the calculator finds side c directly. This is useful when you know two edges of a roof triangle and the angle between them, or when you are laying out a part in the field. A single angle change will shift the solved side.

If all three sides are known, the formula can reverse to find an angle. That is helpful for checking a measured triangle against a plan or for validating a geometry problem. The output changes if any side changes, which is why precision matters.

Reference Table

Inputs	Solved	Use Case
a, b, C	c	SAS triangle
a, b, c	A, B, or C	SSS triangle
Two sides + included angle	Third side	Construction and geometry

Frequently Asked Questions

When should I use the law of cosines?

Use it when you have two sides and the included angle, or all three sides and need an angle. It is the standard triangle solver for those cases. It is often the easiest direct route.

Why not always use the law of sines?

Because the law of sines can be ambiguous in some triangle setups. The law of cosines avoids that issue in SAS and SSS problems. It is the more stable choice when the opposite-side pair is not obvious.

Can this calculate area too?

The calculator is focused on sides and angles, but once a triangle is solved you can use the side and angle values to compute area separately. That makes the result a foundation for more geometry work. The law itself is about shape, not direct area.

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