Topic: Oblique Triangles and the Laws

Subtopic: Area of a Triangle

**Overview**

Remember the area of a triangle formula `A=1/2bh`? This formula works on both right and oblique triangles, but you have to know the length of the altitude (h) of the triangle. In the triangles we have been working with lately, we aren't given the altitude, just sides and/or angles. So, new area formulas would be useful.

The common `A=1/2bh` formula is actually a special case of the area of a triangle formula `A=1/2ab*sin gamma` where gamma is the included angle between sides of length a and b. This formula can be used to find the area of an oblique triangle given SAS.

To find the area of an oblique triangle given SSS, use __Heron's Area Formula__, `A=sqrt(s(s-a)(s-b)(s-c))` where s is the semiperimeter of the triangle. This formula has an interesting history and extends into higher dimensions. See the supplementary resources below for more information.

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 7.4.1 Find the area of an oblique triangle given SSS via Heron's Area Formula
- 7.4.2 Find the area of an oblique triangle given SAS

**Terminology**

Define: semiperimeter **`s=1/2(a+b+c)`**

Heron's Area Formula **`A=sqrt(s(s-a)(s-b)(s-c))`**

Area formula **`A=1/2ab*sin gamma`**

**Supplementary Resources**

- Read a biography of Heron of Alexandria.
- Learn more about the history of Heron and his formula at Math Forum 's Ask Dr. Math - Heron's Formula.
- Check-out some proofs of Heron's Formula: Joe Wilson's Algebraic Proof, Joe Wilson's Geometric Proof, Joe Wilson's Trigonometric Proof, Cut-The-Knot's Geometric Proof (with Java-based interactive figure), Univ. of Virginia's Geometrical Proof.
- Watch Wild Trig's Heron's Formula Viewed Rationally for an explanation of finding a triangle's area. Note: Historical context is interesting, ie. Archimedes's precursor to Heron's formula, but details are complex.
- Explore the area of an SSS triangle with Math Open References's Heron's Formula Java applet.
- Extend your understanding of the derivation of Heron's area of a triangle formula and its extension to the volume of a tetrahedron at Wikipedia's Heron's Formula. (I know Wiki sites are not acceptable academic sites but at least on 2019-02-10 when I last reviewed this page the material was accurate and included a pretty cool extension of the formula from 2D to 3D.)