Circles

A circle is the set of all points that are the same distance from one point, the center. We name a circle by its center: the circle below is “circle $O$,” written $\odot O$.

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• Radius: a segment from the center to a point on the circle.
• Diameter: a chord that passes through the center. It is the longest chord, and $d = 2r$.
• Chord: a segment whose two endpoints are on the circle.
• Secant: a line that cuts the circle at two points.
• Tangent: a line that touches the circle at exactly one point.

A sector cut by a central angle.

An arc is a curved piece of the circle between two points.

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• Minor arc: the shorter way around (less than $180^\circ$). Named with two letters, like $\overset{\frown}{AB}$.
• Major arc: the longer way around (more than $180^\circ$). Named with three letters to show which way you travel.
• Semicircle: exactly half the circle ($180^\circ$); its endpoints are the ends of a diameter.

The circumference $C$ is the perimeter of a circle:

The number $\pi$ (“pi”) is about $3.14$. We leave answers in terms of $\pi$ unless a decimal is asked for; then we use $\pi \approx 3.14$.

A sector cut by a central angle.

An arc is a fraction of the whole circle. If its central angle is $n^\circ$, then

The fraction $\dfrac{n}{360}$ is “how much of the full $360^\circ$” the arc covers.

A central angle has its vertex at the center of the circle. The measure of a minor arc equals the measure of its central angle. A full trip around the circle is $360^\circ$, so all the central angles around the center add to $360^\circ$.

A sector cut by a central angle.

An inscribed angle has its vertex on the circle, and its two sides are chords. The

So the intercepted arc is twice the inscribed angle. (Notice a central angle equals its arc, but an inscribed angle is only half of it.)

A sector cut by a central angle.

A tangent touches a circle at exactly one point. At that point of tangency, the tangent line is perpendicular to the radius. That right angle is the key to almost every tangent problem, because it makes a right triangle you can use with the Pythagorean Theorem.

A tangent touches at exactly one point.

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• Equal chords: In the same circle, chords that are the same distance from the center are congruent (equal in length).
• Perpendicular from the center: A radius (or diameter) that is perpendicular to a chord bisects it --- it cuts the chord into two equal halves.

A chord joins two points on the circle.

The area inside a circle is

Remember it is $r$ squared, then times $\pi$. Area is measured in square units.

A sector cut by a central angle.

A sector is a “pizza slice” bounded by two radii and an arc. If its central angle is $n^\circ$,

It is the same fraction $\dfrac{n}{360}$ of the whole circle that we used for arc length.

A circle with center $(h,k)$ and radius $r$ has the equation

Read it carefully: the numbers subtracted from $x$ and $y$ give the center, and the right side is $r^2$ (so take the square root to get the radius).

Everything follows from the radius.

Why is an inscribed angle exactly half its arc? Look at the easy case where one side of the angle is a diameter. Draw radius $\overline{OB}$. Since $OB = OC = r$, triangle $OBC$ is isosceles, so its base angles are equal: $\angle OCB = \angle OBC$. Now $\angle AOB$ is an exterior angle of that triangle, so it equals the sum of the two remote interior angles:

The central angle $\angle AOB$ equals arc $\overset{\frown}{AB}$, so the inscribed angle $\angle ACB$ is half that arc. (Any inscribed angle can be split into cases like this by drawing the diameter through $C$.)

A chord joins two points on the circle.

Section on chords stated it; here is why. Drop $\overline{OM}\perp\overline{AB}$ and draw radii $\overline{OA}$ and $\overline{OB}$. Then right triangles $OMA$ and $OMB$ share leg $\overline{OM}$ and have equal hypotenuses $OA = OB = r$. By Hypotenuse--Leg, the triangles are congruent, so $AM = MB$. The perpendicular from the center must split the chord in half.

When a tangent and a chord meet at the point of tangency, the angle they form is half the intercepted arc --- the same “half the arc” rule as an inscribed angle:

In the picture, chord $\overline{PB}$ and tangent $t$ meet at $P$; the marked angle is half of arc $\overset{\frown}{PB}$.

A cyclic quadrilateral has all four vertices on a circle. Each angle is inscribed, so it equals half of the arc across from it. The two arcs opposite a pair of angles together make the whole circle ($360^\circ$), so each pair of opposite angles adds to half of that:

If two lines through a point $P$ each meet the circle, the products of the two distances along each line are equal. There are three versions of this one idea:

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• Two chords crossing inside at $P$: $\ PA\cdot PB = PC\cdot PD$.
• Two secants from an outside point $P$: $\ PA\cdot PB = PC\cdot PD$ (each product is near point $\times$ far point).
• Secant and tangent from an outside point, with tangent length $PT$: $\ PT^2 = PA\cdot PB$.

A segment is the region between a chord and its arc --- what is left of a sector after you remove the triangle formed by the two radii. So

An equation like $x^2 + y^2 - 6x + 4y - 12 = 0$ is a circle in disguise. To find its center and radius, complete the square on the $x$-terms and the $y$-terms separately, adding the same amounts to both sides. Each time, take half the middle coefficient and square it.