Students will learn how to recognize and solve 30-60-90 and 45-45-90 right triangles. They will also learn how to evaluate the six basic trig functions for the angles 30, 45, and 60. After that, ask your students to find the area of a regular hexagon with side length of \(2\sqrt{3}.\) To see this problem solved, watch this. Next, give your students this challenge: Find \(\overline{AB}\) and \(\overline{AC}\) in the figure below.

Here's the solution: Drop a perpendicular from the apex to the base. Call the length of this perpendicular h, the height of the triangle. Because the perpendicular cuts the triangle into a 45-45-90 triangle, and a 30-60-90 triangle, the side of length 8 has been cut into lengths \(h\) and \(\sqrt{3}h\) respectively. Then it's just a matter of solving \(h + \sqrt{3}h = 8.\) Before attempting this puzzle, students should know about special right triangles, and how to rationalize the denominator. Conclude by giving your students this challenge:

Each circle has radius 1. The triangle is a 45-45-90 triangle. The circles and triangle have been drawn to show you how the holly leaf has been constructed, but are not part of the shape itself. What is the perimeter of the holly leaf, and what's its area? Here's the solution.

I improved the question and graphic provided by NRICH.

Conclude by giving your students these challenges:

- Cubes Within Cubes by NRICH
- Terminating or Not by NRICH
- Bendy Quad by NRICH
- 2016 AMC 8, Problem 22
- 2003 AMC 8, Problem 21