Circles show up in both the algebra and geometry portions of the SAT. You'll need the equation of a circle, plus formulas for arc length and sector area.
Standard Equation of a Circle
- Center:
- Radius:
Example 1:
Center: (note the sign change for )
Radius:
Converting from General Form
Sometimes the equation is expanded:
Complete the square for both and .
Example 2: Find the center and radius of
Group:
Complete the square:
Center: , Radius:
Writing a Circle Equation
Example 3: Write the equation of a circle with center and radius .
Circumference and Area
Example 4: A circle has radius 6. Find its circumference and area.
Arc Length
Arc length is a fraction of the circumference:
where is the central angle in degrees.
Example 5: Find the arc length of a 90° arc in a circle with radius 8.
Sector Area
A sector is a "pizza slice" of a circle:
Example 6: Find the area of a sector with central angle 60° and radius 9.
Central Angles and Inscribed Angles
- A central angle equals the arc it intercepts
- An inscribed angle equals half the arc it intercepts
Example 7: An inscribed angle intercepts an arc of 100°. What is the inscribed angle?
Practice Problems
Problem 1: What are the center and radius of ?
Solution
Center: , Radius:
Problem 2: Find the arc length of a 120° arc in a circle with radius 6.
Solution
Problem 3: A circle has the equation . What is the radius?
Solution
. Radius = .
Key Takeaways
- Standard form: . The radius is , not !
- Complete the square to convert from general to standard form
- Arc length and sector area use the same fraction:
- Inscribed angle = half the central angle (for the same arc)
- Always check signs when reading center coordinates from the equation
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