Trigonometry

Triangle Area Calculator

The calculator finds the area of a plane triangle. You can enter two sides and the included angle, or all three sides. The result is the area, with either (1/2)ab sin(C) or Heron’s formula written using the numbers you typed.

Known parts

Needed for Heron’s formula.
Needed for the sine formula. C sits between sides a and b.

Try a worked example

Lesson: two routes to the same area

The SAS formula comes from the usual (1/2) × base × height. If two sides enclose angle C, the height to one of those sides is the other side times sin(C). Multiplying gives (1/2)ab sin(C). When C is 90°, sin(C) is 1 and the formula collapses to the right-triangle version (1/2)ab.

Heron’s formula is the SSS route. You do not need an angle. The semi-perimeter s is half the perimeter. The product s(s − a)(s − b)(s − c) is positive only when the three lengths satisfy the triangle inequality. A side of 0, a negative length, or three lengths that cannot close produces an error instead of an area.

The two formulas agree on a legal triangle. If you already solved for the sides on another page of this site, Heron’s formula is the fastest check. If you have SAS data and have not found the third side yet, the sine formula is enough.

Practice problems

1. Included angle

Sides a = 7 and b = 10 meet at 60°. Find the area.

2. Three sides

The sides are 5, 5, and 6. Find the area with Heron’s formula.

3. Impossible sides

What happens if the sides are 2, 3, and 9?

Show answers

Problem 1. Area = (1/2) · 7 · 10 · sin(60°) = 35 · √3 / 2, about 30.3109.

Problem 2. s = 8. Area = √[8 · 3 · 3 · 2] = √144 = 12.

Problem 3. 2 + 3 is not greater than 9, so no triangle exists.