8.3.1: Resources and Key Concepts

Prerequisite Topics

This page lists the prerequisite skills (all of which can be reviewed in CRC's Corequisite Codex) needed for this section that have not already been mentioned in any previous section. If you are enrolled in a course with a Support section, some (but definitely not all) of these topics might be covered or reviewed in the Support section of your course.

• Geometry
  • Common Areas: The section builds upon the basic formula for the area of a triangle, \(A = \frac{1}{2}bh\), to derive the trigonometric formula for the area. This basic formula is also needed for the "Skills Refresher" exercises.

Videos

• Area of a Triangle
  • Trigonometric Functions of Angles - Areas of Triangles
  • Non-Right Triangles - Area of a Triangle (Part 1)
• Heron's Formula
  • The Law of Cosines - Heron's Formula: Theorem, Proof, and Examples

Definitions

• Semiperimeter: Half the perimeter of a triangle. For a triangle with side lengths \(a\), \(b\), and \(c\), the semiperimeter \(s\) is calculated as \(s = \frac{1}{2}(a+b+c)\).

Theorems

• Area of a Triangle: If a triangle has two sides of length \(a\) and \(b\), and the angle between those two sides is \(\theta\), then the area of the triangle is given by \(A = \frac{1}{2}ab\sin(\theta)\).
• Heron's Formula: If a triangle has sides of length \(a\), \(b\), and \(c\), and the semiperimeter is \(s\), then the area of the triangle is given by \(A = \sqrt{s(s-a)(s-b)(s-c)}\).

Common Mistakes

• Using a Non-Included Angle: The formula \(A = \frac{1}{2}ab\sin(\theta)\) is only valid if the angle \(\theta\) is the angle included between the sides of length \(a\) and \(b\). Using an angle that is not between the two given sides will result in an incorrect area.
• Confusing Perimeter and Semiperimeter: Heron's formula requires the semiperimeter (\(s\)), which is half the perimeter. Using the full perimeter in the formula will produce an incorrect result.