Law of Cosines, Law of Sines, and triangle area
MathematicsPlane GeometryAges 15–16
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Plane Geometry
Triangle ABC is built from two sides a, b and the included angle C; drag any of the three vertices freely. Every side and angle updates live while you verify: c² = a² + b² − 2ab·cosC, a/sinA = b/sinB = c/sinC = 2R, and area S = ½ab·sinC = √(p(p−a)(p−b)(p−c)) = abc/4R = pr. You can also show the circumscribed circle (radius R) and the altitude.
Lesson: Law of Cosines, Law of Sines, and triangle area formulas
What it shows
In any triangle ABC, the Law of Cosines c² = a² + b² − 2ab·cosC links three sides and one angle. It reduces to Pythagoras' theorem when C = 90°, and for an obtuse angle cosC < 0, so c² > a² + b². The Law of Sines says that each side divided by the sine of the opposite angle equals 2R, the diameter of the circumscribed circle, so the longest side faces the largest angle. The area can be found in several equivalent ways, ½ab·sinC, Heron's formula, abc/4R and pr, where r is the radius of the inscribed circle and p the semi-perimeter.
How to use
Set Side a = BC, Side b = CA and Angle C, or drag the vertices A, B, C directly, and check that c², the three ratios and the four area formulas agree. Move Angle C through 90° to see the Law of Cosines turn into Pythagoras' theorem. Turn on Show circumscribed circle (R) to connect the ratio a/sinA with the radius, and Show altitude AH to link ½ab·sinC with ½ × base × height.
Parameters you can change
- Side a = BC 1–12 cm
- Side b = CA 1–12 cm
- Angle C 10–170 °
- Show circumscribed circle (R)
- Show altitude AH
Questions to explore
- For which angle C does c² = a² + b², and what kind of triangle is it then?
- With a and b fixed, which angle C gives the largest area, and why?
- Why is the longest side of a triangle always opposite its largest angle?
