The purpose of this Warm-up is to ensure all students are able to find the area and perimeter of a triangle. In subsequent activities students will be decomposing polygons into triangles to find the areas and perimeters of the shapes.

The goal of this discussion is for students to consider multiple strategies for finding the height of the equilateral triangle and how this value helps them find the area and perimeter.

Display the image, and invite students to share their strategies for calculating . There is no need to emphasize the exact value of because decimal approximations will be more useful for subsequent activities.

Remind students of the area formula for a triangle and the definition of perimeter as needed. These ideas will continue to be relevant in subsequent activities.

During this activity students are asked to compute the perimeter of inscribed regular polygons given only the radius. They will need to figure out that drawing in the altitude will form a right triangle. Drawing this auxiliary segment shows that students recognize the important structure of right triangles to calculate unknown information (MP7). In this activity students are building towards an equation to approximate the value of , which they will complete in the next lesson.

The digital version of this activity includes an applet that may be helpful to display for the class during the Synthesis.

This activity requires more interpretation than a basic right triangle problem and requires students to use skills that will help them in mathematical modeling (MP4). First, students need to draw a diagram and figure out how to label it. Second, students need to recognize that the units do not match and figure out how to convert to a common unit. At this point the problem looks as if there are several basic right triangles, and students will be ready to use their usual techniques of trigonometry and the Pythagorean Theorem.