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Exercises Trigonometry

Example 5 from Algebra and Trigonometry, 10.2 Non-right Triangles: Law of Cosines

Using Heron’s Formula to Find the Area of a Given Triangle

Find the area of the triangle in Figure 9 using Heron’s formula.

A triangle with angles A, B, and C and opposite sides a, b, and c, respectively. Side a = 10, side b - 15, and side c = 7.

Figure 9

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
First, we calculate $s .$

$\begin{array}{l} s = \frac{(a + b + c)}{2} \\ s = \frac{(10 + 15 + 7)}{2} = 16 \end{array}$
Then we apply the formula.

$\begin{array}{l} Area = \sqrt{s (s - a) (s - b) (s - c)} \\ Area = \sqrt{16 (16 - 10) (16 - 15) (16 - 7)} \\ Area \approx 29.4 \end{array}$
The area is approximately 29.4 square units.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Use Heron’s formula to find the area of a triangle with sides of lengths $a = 29.7 ft, b = 42.3 ft,$ and $c = 38.4 ft .$

How did it go?

PreviousCalculating Distance Traveled Using a SAS Triangle Next Applying Heron’s Formula to a Real-World Problem

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 10.2, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.