Circles

1) Radius & Isosceles Triangle

• Radius: distance from center \(C\) to any point \(P\) on the circle: \(\;CP=r\).
• Any two radii \(CA\) and \(CB\) with chord \(\overline{AB}\) form \(\triangle CAB\) with \(CA=CB\) (isosceles). ⇒ Base angles at \(A\) and \(B\) are equal.
• When a central angle \(\angle ACB\) is known, you instantly know the arc \(\widehat{AB}\) and often the base angles in \(\triangle CAB\).

2) Radians \(\leftrightarrow\) Degrees

• \(\theta_{\deg}=\theta_{\text{rad}}\cdot \dfrac{180^\circ}{\pi}\),   \(\theta_{\text{rad}}=\theta_{\deg}\cdot \dfrac{\pi}{180^\circ}\).
• Use degrees when a problem gives a percentage of the circle (e.g., \(45^\circ\) is \(\tfrac18\) of the circle). Use radians when applying \(S=r\theta\) directly.

3) Circumference

• \(C=2\pi r\). If diameter \(d\) is given, \(C=\pi d\).
• Arc-length questions often reduce to finding the fraction of the full circumference.

4) Area

• \(A=\pi r^2\). Less common on SAT circle items, but it appears in mixed-figure or sector problems.

5) Arc Length

• Radians: \(\displaystyle S=r\theta\) where \(\theta\) is the central angle in radians.
• Degrees: \(\displaystyle \frac{S}{C}=\frac{\theta_{\deg}}{360^\circ}\) so \(S=\dfrac{\theta_{\deg}}{360^\circ}\cdot 2\pi r\).
• The central angle \(\theta\) always measures the arc \(\widehat{AB}\) it intercepts.

6) Circle Equation

• Standard form: \((x-h)^2+(y-k)^2=r^2\) with center \((h,k)\) and radius \(r\).
• General form: \(x^2+y^2+Dx+Ey+F=0\). Complete the square in \(x\) and \(y\) to convert to standard: \[ (x+\tfrac{D}{2})^2+(y+\tfrac{E}{2})^2=\tfrac{D^2+E^2}{4}-F, \] so \(\;h=-\tfrac{D}{2},\;k=-\tfrac{E}{2},\;r^2=\tfrac{D^2+E^2}{4}-F.\)

7) Tangents

• The tangent at \(P\) is perpendicular to radius \(\overline{CP}\): \(\overline{CP}\ \perp\ \text{tangent at }P\).
• In slope terms (when defined): if \(m_{\text{rad}}\) is the slope of \(\overline{CP}\) then \(m_{\text{tan}}=-\dfrac{1}{m_{\text{rad}}}\). Vertical radius \(\Rightarrow\) horizontal tangent; horizontal radius \(\Rightarrow\) vertical tangent.

Prerequisites frequently used: perpendicular/parallel slopes, triangle angle sums, 30–60–90 and 45–45–90 triangles, and SOH–CAH–TOA.