K Khaemenes GeometryOpen-Age · Printable Visual Lesson
Unit 08 Course Home Student Lesson

Unit 08 · Lesson 07 · Circles, Arcs & Tangency

Sector Area, Segment Area & Annuli · Teacher Key

Connect radii, chords, arcs, angles, tangents, secants, sectors, and coordinate equations in a unified circle model.

Open-AgePrintableFoundation · Core · ExtendedMastery target 80%

Learning intention

Use accurate diagrams, definitions, measurements, and reasoning to connect radii, chords, arcs, angles, tangents, secants, sectors, and coordinate equations in a unified circle model.

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

Circles geometry diagram A clear labeled diagram supporting circles reasoning. O

Connect radii, chords, arcs, angles, tangents, secants, sectors, and coordinate equations in a unified circle model.

C=2\pi rA=\pi r^2s=\frac{\theta}{360^\circ}2\pi r(x-h)^2+(y-k)^2=r^2

Worked reasoning

Process: Identify the angle or segment type, locate its intercepted arc or related products, apply the correct theorem, and confirm the result fits the diagram.

  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: An inscribed angle is half its intercepted arc, while a central angle has the same measure as its intercepted arc.

Khaemenes Academy · Open-Age Geometry

Sector Area, Segment Area & Annuli

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

U08 · L07
Name
Date
1

A circle has radius 6. Find its circumference in exact form.

Circles geometry diagram A clear labeled diagram supporting circles reasoning. O
Answer: 12π
Reasoning: C=2πr=12π.
2

A circle has radius 6. Find its area in exact form.

Circles geometry diagram A clear labeled diagram supporting circles reasoning. O
Answer: 36π
Reasoning: A=πr²=36π.
3

In a circle of radius 6, find the arc length for a 45° central angle in exact form.

Circles geometry diagram A clear labeled diagram supporting circles reasoning. O
Answer: 1.5π
Reasoning: s=(45/360)·2π·6.
4

An intercepted arc measures 45°. Find the inscribed angle.

Circles geometry diagram A clear labeled diagram supporting circles reasoning. O
Answer: 22.5°
Reasoning: An inscribed angle is half its intercepted arc.
5

What angle is formed by a radius and a tangent at the point of tangency?

Circles geometry diagram A clear labeled diagram supporting circles reasoning. O
Answer: 90°
Reasoning: The radius is perpendicular to the tangent.
6

Write the equation of a circle centered at (3,−2) with radius 5.

Circles geometry diagram A clear labeled diagram supporting circles reasoning. O
Answer: (x−3)²+(y+2)²=25
Reasoning: Use (x−h)²+(y−k)²=r².