A circle has radius 6. Find its circumference in exact form.
Unit 08 · Lesson 08 · Circles, Arcs & Tangency
Equation of a Circle & Coordinate Applications
Identify center and radius from standard form and build an equation from geometric information.
Learning intention
Use accurate diagrams, definitions, measurements, and reasoning to identify center and radius from standard form and build an equation from geometric information.
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
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.
- Read the figure and rewrite all given information with correct notation.
- Identify the exact relationship that connects the known and unknown quantities.
- Substitute, construct, or prove one justified step at a time.
- 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
Equation of a Circle & Coordinate Applications
Find, construct, prove, or explain each result. Show reasoning and label units.
Name
Date
A circle has radius 6. Find its area in exact form.
In a circle of radius 6, find the arc length for a 120° central angle in exact form.
An intercepted arc measures 120°. Find the inscribed angle.
What angle is formed by a radius and a tangent at the point of tangency?
Write the equation of a circle centered at (3,−2) with radius 5.