K Khaemenes GeometryOpen-Age · Printable Visual Lesson
Unit 07 Course Home Student Lesson

Unit 07 · Lesson 06 · Polygons & Quadrilaterals

Regular Polygons, Apothem & Central Angles · Teacher Key

Use apothem, perimeter, central angle, and decomposition into congruent triangles.

Open-AgePrintableFoundation · Core · ExtendedMastery target 80%

Learning intention

Use accurate diagrams, definitions, measurements, and reasoning to use apothem, perimeter, central angle, and decomposition into congruent triangles.

Success evidence

  • Label the figure and units before calculating.
  • Select and justify the governing definition, theorem, relationship, or construction.
  • Show enough work that another learner can audit the conclusion.
  • Check the result against the diagram, scale, and stated constraints.

Visual model

Polygons geometry diagram A clear labeled diagram supporting polygons reasoning. a

Classify polygons and quadrilaterals using angle sums, side relationships, diagonals, symmetry, and coordinate evidence.

S=(n-2)180^\circE=360^\circA=\frac12 aP

Worked reasoning

Process: List the properties required by the classification, calculate or prove each one, then state the most specific valid name.

  1. Read the figure and rewrite all given information with correct notation.
  2. Identify the exact relationship that connects the known and unknown quantities.
  3. Substitute, construct, or prove one justified step at a time.
  4. State the result with units and a sentence explaining what it means.

Misconception check: A square belongs to both the rectangle and rhombus families; classifications are hierarchical rather than mutually exclusive.

Khaemenes Academy · Open-Age Geometry

Regular Polygons, Apothem & Central Angles

Find, construct, prove, or explain each result. Show reasoning and label units.

U07 · L06
Name
Date
1

Find the sum of the interior angles of a 9-gon.

Polygons geometry diagram A clear labeled diagram supporting polygons reasoning. a
Answer: 1260
Reasoning: (n−2)180°=(9−2)180°.
2

Find each exterior angle of a regular 9-gon.

Polygons geometry diagram A clear labeled diagram supporting polygons reasoning. a
Answer: 40
Reasoning: Regular exterior angles equal 360°/9.
3

Which quadrilateral has four congruent sides and four right angles?

Polygons geometry diagram A clear labeled diagram supporting polygons reasoning. a
Answer: Square
Reasoning: A square is both a rhombus and a rectangle.
4

What do the diagonals of a parallelogram do?

Polygons geometry diagram A clear labeled diagram supporting polygons reasoning. a
Answer: Bisect each other
Reasoning: This property also provides a converse test.
5

What additional property turns a parallelogram into a rectangle?

Polygons geometry diagram A clear labeled diagram supporting polygons reasoning. a
Answer: Congruent diagonals or one right angle
Reasoning: Either condition is sufficient.
6

A regular polygon has perimeter 48 and apothem 7. Find its area.

Polygons geometry diagram A clear labeled diagram supporting polygons reasoning. a
Answer: 168
Reasoning: A=1/2·aP=1/2·7·48=168.