Circle questions connect radius, diameter, circumference, area, arcs, central angles, and coordinate equations. Most errors come from confusing radius with diameter or using the wrong fraction of the full circle.
Essential concepts
\[C=2pi r qquad A=pi r^2 qquad (x-h)^2+(y-k)^2=r^2\]
The coordinate-form center is (h, k) and the radius is r.
SAT strategy and application
Find a circle’s center and radius
Rewrite \(x^2+y^2-6x+4y-12=0\) in standard form.
- Group and complete squares
\((x-3)^2-9+(y+2)^2-4=12\).
- Move constants
\((x-3)^2+(y+2)^2=25\).
Center: \((3,-2)\); radius: \(5\).
Mistakes and traps
Check your understanding
What is the circumference of a circle with radius \(4\)?
- \(4pi\)
- \(8pi\)
- \(16pi\)
- \(32pi\)
Show answer and explanation
Answer: \(8pi\)
\(C=2pi r=2pi(4)=8pi\).
Key takeaways
What to remember
- Convert diameter to radius before using circle formulas.
- Use central-angle fractions for arc length and sector area.
- Relate inscribed angles to half their intercepted arcs.
- Complete the square to reveal a circle’s center and radius.
Continue learning
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.