Circles

Circles on the SAT focus on arc length, sector area, equation of a circle, and angle properties (central vs. inscribed).

Equation of a Circle

The standard form of a circle in the $xy$-plane with center $(h, k)$ and radius $r$ is:

• Center: $(h, k)$ — note the signs inside the parentheses are flipped relative to the coordinates.

• Radius: $r = \sqrt{r^2}$.

• General Form to Standard Form: If given $x^2 + y^2 + Ax + By + C = 0$, complete the square for both $x$ and $y$ terms to rewrite it into standard form.

Radians, Arcs, and Sectors

A full circle covers $360^\circ$ or $2\pi\text{ radians}$.

• Degree-to-Radian Conversion:

  $$\text{Radians} = \text{Degrees} \times \frac{\pi}{180^\circ}$$

• Arc Length ($s$): The distance along a fraction of the circumference.

  • In degrees: $s = \dfrac{\theta}{360^\circ} \times 2\pi r$

  • In radians: $s = r\theta$

• Sector Area ($A$): The area of a "slice" of the circle.

  • In degrees: $A = \dfrac{\theta}{360^\circ} \times \pi r^2$

  • In radians: $A = \frac{1}{2} r^2 \theta$

Angle Properties in Circles

• Central Angle: An angle with its vertex at the center of the circle. Its measure equals the measure of its intercepted arc.

• Inscribed Angle: An angle with its vertex on the edge of the circle. Its measure is half the measure of its intercepted arc (and half the central angle subtending the same arc). 

  
  

• Tangent Line Property: A tangent line touches a circle at exactly one point and is always perpendicular ($90^\circ$) to the radius drawn to that point of tangency.

1. Complete the Square to Find Center and Radius. When given an expanded circle equation, group terms, halve the linear coefficients, square them, and add to both sides. 

2. Use Proportions for Arc Length and Sector Area. Think of arcs and sectors as a simple fraction of the total circle: $\dfrac{\text{Part}}{\text{Whole}} = \dfrac{\theta}{360^\circ}$. 

   Example: A circle with radius $9$ has a central angle of $40^\circ$. What is the area of the sector? 

   
   

   • Total Area: $\pi r^2 = \pi (9)^2 = 81\pi$

   • Sector Fraction: $\dfrac{40^\circ}{360^\circ} = \dfrac{1}{9}$

   • Sector Area: $\dfrac{1}{9} \times 81\pi = 9\pi$

3. Apply the Inscribed Angle Theorem for Quick Mental Math, An inscribed angle subtending the same arc as a central angle is always half the size. 

   Example: Points $A, B, C$ lie on a circle centered at $O$. If central angle $\angle AOB = 80^\circ$, what is inscribed angle $\angle ACB$? 

   
   

   • Inscribed Angle Rule: $\angle ACB = \dfrac{1}{2}(\angle AOB)$

   • Calculation: $\angle ACB = \dfrac{80^\circ}{2} = 40^\circ$