Circle Properties and Formulas

Cheatsheet Content

Circle Definition A circle is the set of all points in a plane that are equidistant from a given point (the center). The distance from the center to any point on the circle is called the radius ($r$). Standard Equation of a Circle Center at Origin $(0,0)$ $x^2 + y^2 = r^2$ Center at $(h,k)$ $(x-h)^2 + (y-k)^2 = r^2$ General Form of a Circle $Ax^2 + Ay^2 + Bx + Cy + D = 0$ To convert to standard form, complete the square for $x$ and $y$ terms. Example: $x^2 + y^2 + 4x - 6y - 12 = 0$ $(x^2 + 4x) + (y^2 - 6y) = 12$ $(x^2 + 4x + 4) + (y^2 - 6y + 9) = 12 + 4 + 9$ $(x+2)^2 + (y-3)^2 = 25$ Center: $(-2, 3)$, Radius: $r = 5$ Key Properties and Formulas Radius ($r$) Distance from the center to any point on the circle. Diameter ($d$) Distance across the circle through the center. $d = 2r$ Circumference ($C$) The distance around the circle. $C = 2\pi r$ $C = \pi d$ Area ($A$) The space enclosed by the circle. $A = \pi r^2$ $A = \frac{\pi d^2}{4}$ Arc Length and Sector Area Arc Length ($L$) Length of a portion of the circumference. In radians: $L = r\theta$ (where $\theta$ is the central angle in radians) In degrees: $L = \frac{\theta}{360^\circ} \cdot 2\pi r$ (where $\theta$ is the central angle in degrees) Sector Area ($A_{sector}$) Area of a portion of the circle bounded by two radii and an arc. In radians: $A_{sector} = \frac{1}{2}r^2\theta$ (where $\theta$ is the central angle in radians) In degrees: $A_{sector} = \frac{\theta}{360^\circ} \cdot \pi r^2$ (where $\theta$ is the central angle in degrees) Tangent and Normal Lines Tangent Line A line that touches the circle at exactly one point. The tangent line is perpendicular to the radius at the point of tangency. Slope of radius from center $(h,k)$ to point $(x_1, y_1)$ is $m_r = \frac{y_1-k}{x_1-h}$. Slope of tangent line $m_t = -\frac{1}{m_r}$ (if $m_r \ne 0$). Normal Line A line perpendicular to the tangent line at the point of tangency. The normal line always passes through the center of the circle. Important Theorems Thales' Theorem If A, B, and C are distinct points on a circle where the line AC is a diameter, then the angle $\angle ABC$ is a right angle ($90^\circ$). Intersecting Chords Theorem If two chords intersect inside a circle, then the product of the segments of one chord is equal to the product of the segments of the other chord. If chords AB and CD intersect at P, then $AP \cdot PB = CP \cdot PD$. Tangent-Secant Theorem If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external segment. If tangent PT and secant PAB are drawn from P, then $(PT)^2 = PA \cdot PB$.