Laws of Sines & Cosines

Trigonometric Equations

Overview The Laws Law of Sines Law of Cosines Triangle Area

Triangle Area

At this point, you should know two formulas, \(A = \frac{1}{2}bh\) and \(A = \frac{1}{2}ab\sin(\theta)\), for finding the area of a triangle. The first formula requires knowing the length of the base and height of the triangle while the second formula requires knowing the length of two sides and the measure of the angle between those sides. A third method involves using Heron's formula, which is named after the Greek mathmetician and engineer Heron of Alexandria (around 60-70 CE).

A scalene triangle ABC with angles A, B, and C. The side opposite angle A is side a the side opposite angle B is side b, and the side opposite angle C is side c.

Heron's Formula: Given triangle ABC with opposite sides a, b, and c, respectively, the semi-perimeter of the triangle is \(s = \frac{1}{2}(a+b+c)\) and the area of the triangle can be computed using the following formula.

\[A = \sqrt{s(s-a)(s-b)(s-c)}\]

This means that Heron's formula is the best choice for finding the triangle area if we know all 3 sides of the triangle.

Heron's Formula - Triangle Area
[https://youtu.be/JeQWGYh1dDM]

Self-Check #5: Use Heron's formula to compute the area of a triangle having sides with length 14, 23, and 27. (Enter your answer rounded to the nearest whole number.)

You must enter an answer.

(Answer: 161) -- We must first compute the semi-perimeter using the formula \(s = \frac{1}{2}\left(14+23+27\right) = 32\). Then we can find the area.

\[A = \sqrt{32(32-14)(32-23)(32-27)} = \sqrt{32(18)(9)(5)} = 72\sqrt{5} \approx 161.\]