Law of Sines & Cosines Calculator NZ

This calculator solves a triangle using the law of cosines and the law of sines, the two rules that let you find unknown sides and angles in any triangle, not just right-angled ones. While basic trigonometry handles right triangles, real problems in surveying, navigation, engineering and physics involve triangles of every shape, and that is where these two laws come in. The law of cosines relates the three sides and one angle, and is the go-to when you know two sides and the angle between them, or all three sides. The law of sines relates each side to the sine of its opposite angle, and finishes off the remaining angles once you have a side-angle pair. This tool takes the common and reliable case where you know two sides and the angle between them, often called side-angle-side. You enter the two sides and the included angle in degrees, and the calculator uses the law of cosines to find the third side, then the law of sines to find the other two angles, and also returns the area of the triangle. The results update as you type, so you can explore how changing the angle or a side reshapes the triangle. Use it for geometry and trigonometry homework, for practical layout and surveying problems, or to check a hand calculation. The side-angle-side case always gives a single valid triangle, avoiding the ambiguous situations that can arise from other combinations. The calculations are exact for your inputs, with sides, angles and area rounded for display.

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6.24
third side (a)
Angle B43.9°
Angle C76.1°
Area15.16

Uses side-angle-side: two sides and the angle between them. Angle A is opposite side a. Rounded for display.

How it works

The law of cosines finds the third side: a squared equals b squared plus c squared minus twice bc times the cosine of the included angle A. The law of sines then finds angle B from the ratio of side b to side a and the sine of A, and angle C is what remains to make 180 degrees. The area is half of bc times the sine of A.

Worked example

With sides b = 5 and c = 7 and an included angle A of 60 degrees, the third side a is the square root of 25 plus 49 minus 2 times 35 times cosine 60, which is about 6.24. The law of sines then gives angle B about 43.9 degrees and angle C about 76.1 degrees, and the area is half of 5 times 7 times sine 60, about 15.16.

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Frequently asked questions

When do I use the law of cosines versus the law of sines?

Use the law of cosines when you know two sides and the angle between them, or all three sides. Use the law of sines when you have a side and its opposite angle, plus one more piece. This calculator combines both to solve a side-angle-side triangle.

What is side-angle-side?

It means you know two sides of a triangle and the angle between them. This always defines a single, unique triangle, so it is a reliable case to solve, avoiding the ambiguous results that can come from knowing two sides and a non-included angle.

How do I find the area of a non-right triangle?

When you know two sides and the included angle, the area is half the product of those two sides times the sine of the angle between them. This calculator computes it automatically along with the missing side and angles.