Final Exam Part 1: Open Response

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Students will be able to demonstrate what they know and understand about foundational geometry ideas, constructions, and proof, on the open response portion of the final exam.

Big Idea

In this part of the exam, students have the opportunity to show what they know and to explain their thinking and reasoning about some of the most important ideas from the entire year.

Final Exam Part 1: Open Response

90 minutes

In the first part of the final exam, which is all open response, students work through several problems that target their understanding of foundational geometry topics in novel ways.  Several of the problems, like problems 4, 5, and 6, require students to synthesize several tools and geometric concepts in order to solve.  Some of the problems are my own, but some have been adapted from an NCTM calendar of problems.


  • Problem 1: compare the areas of circles and annuluses
  • Problem 2: find the perimeter of a heptagon made by right triangles
  • Problem 3: determine the number of sides of a regular polygon using angle relationships
  • Problem 4: use square properties, the Pythagorean Theorem, and trigonometry to find the area of a slanted rectangle
  • Problem 5: apply triangle similarity, trigonometry, proportional reasoning to solve a problem
  • Problem 6: construct a regular hexagon and solve using areas of sectors
  • Problem 7: prove intersecting chords in a circle create similar triangles
  • Problem 8: prove a regular hexagon can be divided into a rectangle and two congruent isosceles triangles such that the rectangle’s area is two-thirds the area of the regular hexagon