SAT Math 15: Circles

Circle — definition

A circle is the set of all points in a plane at a fixed distance from a fixed point called the center.

Circle versus disk

A circle is technically only the boundary curve. The disk is the entire planar region enclosed by that circle.

Radius and diameter

A radius is a segment from the center to a point on the circle. A diameter passes through the center, has endpoints on the circle, and has length $d=2r$.

Chord

A chord is a line segment whose endpoints lie on a circle. A diameter is the longest possible chord.

Secant

A secant is a line in the plane of a circle that intersects the circle at two points; it is an extended chord.

Tangent

A tangent is a line that intersects a circle at exactly one point, called the point of tangency.

Arc, sector, and segment

An arc is a connected portion of a circle. A sector is bounded by two radii and an arc, while a segment is bounded by a chord and an arc.

Circumference

The circumference is the distance around a circle. Its formulas are $C=2\pi r=\pi d$.

Area of a circle

The area enclosed by a circle of radius $r$ is $A=\pi r^2$, or equivalently $A=\frac{\pi d^2}{4}$.

Scaling relationships for circles

Circumference is proportional to radius, $C\propto r$, while area is proportional to the square of the radius, $A\propto r^2$. Thus, doubling the radius doubles the circumference but quadruples the area.

Central angle and radian measure

For a central angle intercepting arc length $s$ in a circle of radius $r$, its radian measure is $\theta=\frac{s}{r}$. One radian intercepts an arc whose length equals the radius.

Full-circle angle conversions

A complete revolution measures $2\pi$ radians or $360^\circ$.

Arc length in radians

For a circle of radius $r$ and central angle $\theta$ measured in radians, the arc length is $s=r\theta$.

Arc length in degrees

For a central angle of $\theta^\circ$, the arc length is $s=\frac{\theta}{360^\circ}(2\pi r)$.

Area of a circular sector

A sector with radius $r$ and central angle $\theta$ in radians has area $A=\frac{1}{2}r^2\theta$.

Sector area in degrees

For a central angle of $\theta^\circ$, the sector area is $A=\frac{\theta}{360^\circ}\pi r^2$.

Equation of a circle in Cartesian coordinates

A circle with center $(a,b)$ and radius $r$ satisfies $(x-a)^2+(y-b)^2=r^2$. This follows from the Pythagorean theorem.

How does the Cartesian equation simplify for a circle centered at the origin?

For center $(0,0)$, the equation becomes $x^2+y^2=r^2$.

Upper and lower semicircle functions

For center $(x_0,y_0)$ and radius $r$, the upper and lower halves are $y=y_0\pm\sqrt{r^2-(x-x_0)^2}$, with $x_0-r\le x\le x_0+r$.

Tangent line at a point on a circle

For a circle centered at $(a,b)$ and a point $(x_1,y_1)$ on it, the tangent line is $(x_1-a)(x-a)+(y_1-b)(y-b)=r^2$. The tangent is perpendicular to the radius through the point of tangency.

Tangent line to a circle centered at the origin

At $(x_1,y_1)$ on $x^2+y^2=r^2$, the tangent line is $x_1x+y_1y=r^2$.

Slope of a tangent to a circle

For a circle centered at $(a,b)$, the tangent slope at $(x_1,y_1)$ is $-\frac{x_1-a}{y_1-b}$ when $y_1\ne b$.

Chord perpendicular-bisector theorem

The perpendicular bisector of any chord passes through the circle's center. Conversely, a perpendicular from the center to a chord bisects that chord.

Relationship between equal chords and distance from the center

Two chords of the same circle are equal in length if and only if they are equally distant from the center.

Chord length from a central angle

A chord subtending central angle $\theta$ in a circle of radius $r$ has length $\ell=2r\sin\left(\frac{\theta}{2}\right)$. For a $90^\circ$ central angle, $\ell=r\sqrt{2}$.

Equal tangents from an external point

The two tangent segments drawn from the same point outside a circle have equal lengths.

Intersecting chords theorem

If two chords intersect inside a circle, the products of the two segment lengths on each chord are equal: $AC\cdot AD=AB\cdot AE$.

Tangent-secant theorem

If a tangent and a secant are drawn from the same external point, the square of the tangent length equals the product of the secant's external and entire lengths: $AF^2=AC\cdot AD$.

Angle between a tangent and a chord

The angle formed by a tangent and a chord equals half the measure of the arc intercepted by the chord, measured on the opposite side.

Angle formed by two secants outside a circle

The measure of an exterior angle formed by two secants equals half the difference of the intercepted arcs.

Inscribed angle theorem

An inscribed angle equals half the measure of the central angle subtending the same arc. Therefore, inscribed angles intercepting the same arc are equal.

Inscribed angles on the same side of a chord

Two inscribed angles that subtend the same chord on the same side of that chord have equal measures.

Thales' theorem for circles

An inscribed angle that subtends a diameter is a right angle because the corresponding central angle is $180^\circ$.

Inscribed angles on opposite sides of a chord

Two inscribed angles subtending the same chord on opposite sides of that chord are supplementary.

Cyclic quadrilateral exterior-angle theorem

For a quadrilateral inscribed in a circle, an exterior angle equals the measure of the interior angle opposite it.

Sagitta

The sagitta is the perpendicular distance from the midpoint of a chord to the arc. For chord length $y$ and sagitta $x$, the radius is $r=\frac{y^2}{8x}+\frac{x}{2}$.