Twelfth Grade · Math · week 28

The Law of Cosines and the area of a triangle

Use the Law of Cosines to solve triangles given SAS or SSS, and find a triangle's area using the sine area formula and Heron's formula.

The Law of Cosines handles the two cases the Law of Sines cannot start with directly -- SAS (two sides and the included angle) and SSS (three sides, no angle at all): c^2 = a^2+b^2-2ab*cos(C), where C is the angle between sides a and b. Notice this is the Pythagorean theorem with one extra correction term -- when C=90 degrees, cos(C)=0 and the formula collapses exactly back to a^2+b^2=c^2.

Once you know two sides and the INCLUDED angle, the area of a triangle does not need a base and height at all: Area = (1/2)*a*b*sin(C). And when you know all three sides but no angle, Heron's formula finds area directly: with s = (a+b+c)/2 (the semi-perimeter), Area = sqrt(s(s-a)(s-b)(s-c)).

📋 For the teacher
Teach it (10–15 min):
  • Point out the Pythagorean theorem hiding inside the Law of Cosines explicitly -- setting C=90 degrees should feel like coming home to a familiar formula.
  • Practice recognizing SAS and SSS on sight, and matching each to its correct tool (Law of Cosines for both; Law of Sines only after an angle is known).
  • Use the sine area formula whenever an included angle is available -- it is almost always faster than finding a height first.
  • Re-verify the 13-14-15 triangle's area of exactly 84 as a memorable, clean checkpoint for Heron's formula done correctly.
Talk about it:
  • Why does the Law of Cosines reduce to the Pythagorean theorem when the included angle is 90 degrees?
  • Why can the sine area formula skip finding a triangle's height entirely?
  • Why does Heron's formula need only the three side lengths, with no angle at all?

Mastery looks like: They apply the Law of Cosines to SAS and SSS triangles, and correctly choose between the sine area formula and Heron's formula depending on what is given.

Common stumbles: Forgetting the -2ab*cos(C) correction term and applying the plain Pythagorean theorem to a non-right triangle; mismatching which side is opposite which angle in the Law of Cosines setup.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

Unit CircleAngles, radians, and sine & cosine come alive.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
← Back to todayAll coursesYear planner