Circle Geometry

1. Introduction

A circle is one of the most perfectly symmetric objects in mathematics: the set of all points in a plane that lie at a fixed distance, the radius $r$, from a single fixed point, the center. That one-sentence definition hides an astonishing amount of structure. Because every point on a circle is "the same" as every other point (you can rotate the circle onto itself), circles obey a rich collection of theorems relating angles, arcs, chords, tangents, and secants.

Circle geometry matters far beyond the classroom. Gears, wheels, satellite orbits, lenses, radar sweeps, and the design of arches and domes all depend on circular relationships. In mathematics itself, the circle is the gateway to trigonometry (via the unit circle), to radian measure, and to the study of conic sections.

In this article we build the subject from the ground up. We start with the basic vocabulary, then derive the central and inscribed angle theorems, work through the chord, secant, and tangent power relationships, study circles in the coordinate plane, and finish with arc length and sector area. Throughout, since we cannot show pictures, we describe each figure carefully in words — practice sketching them yourself as you read.

When you encounter a circle problem, the first question is always geometric: what is fixed (center, radius) and what is moving (a point on the circle, a tangent line, an inscribed angle)? The second question is positional: is the angle vertex on the circle, inside it, or outside? Those two answers route you to the correct theorem almost every time. Circle geometry rewards systematic labeling — name every arc, mark every right angle from tangents, and draw radii liberally.

2. Core Concepts

2.1 Anatomy of a Circle

Imagine a circle drawn on paper with center $O$. The following terms describe its parts:

• Radius: a segment from the center $O$ to any point on the circle; all radii have the same length $r$.
• Diameter: a chord passing through the center; it equals $2r$ and is the longest possible chord.
• Chord: any segment whose two endpoints both lie on the circle.
• Secant: a line that passes through the circle, intersecting it at two points (a chord is the portion of a secant inside the circle).
• Tangent: a line that touches the circle at exactly one point, called the point of tangency.
• Arc: a connected portion of the circle itself. A minor arc is shorter than a semicircle; a major arc is longer. We measure an arc by the central angle that subtends it.
• Sector: the "pie slice" region bounded by two radii and the arc between them.
• Segment (of a circle): the region between a chord and the arc it cuts off.

A key intuition: because all radii are equal, any triangle you form using two radii is isosceles. This single fact powers many proofs.

2.2 Central Angles and Arc Measure

A central angle has its vertex at the center $O$, with two radii as its sides. We define the measure of the intercepted arc to equal the measure of its central angle. So a central angle of $70^\circ$ cuts off an arc of $70^\circ$. The entire circle is $360^\circ$.

Do not confuse arc measure (an angle, in degrees) with arc length (an actual distance, in the same units as $r$). Two circles of different sizes can have arcs of the same measure ($60^\circ$) but very different lengths.

2.3 The Inscribed Angle Theorem

An inscribed angle has its vertex on the circle, with two chords as its sides. The central result:

$\text{inscribed angle} = \tfrac{1}{2}\,(\text{intercepted arc}).$

Why is this true? Consider an inscribed angle $\angle ABC$ where one side $BA$ happens to pass through the center $O$ (the simplest case). Draw radius $OC$. Triangle $OBC$ is isosceles because $OB = OC = r$, so its base angles are equal: $\angle OBC = \angle OCB = \beta$. The central angle $\angle AOC$ is an exterior angle of triangle $OBC$, so it equals the sum of the two remote interior angles: $\angle AOC = \beta + \beta = 2\beta$. But $\angle AOC$ is the central angle for arc $AC$, and $\angle ABC = \beta$ is the inscribed angle for the same arc. Hence the inscribed angle is half the arc. The general case (where the center is inside or outside the angle) follows by adding or subtracting two such configurations.

Two famous corollaries fall out immediately:

• Angles in the same segment are equal: any two inscribed angles subtending the same arc are equal, because each is half of that arc.
• Thales' Theorem: an angle inscribed in a semicircle is a right angle, because the intercepted arc is $180^\circ$ and $\tfrac{1}{2}(180^\circ) = 90^\circ$.

2.4 Tangents and the Radius

A tangent line touches the circle at one point $P$. The defining property is:

$\text{a tangent is perpendicular to the radius drawn to the point of tangency.}$

Why? Of all points on the tangent line, $P$ is the closest one to the center $O$ (every other point lies outside the circle, so it is farther than $r$). The shortest segment from a point to a line is the perpendicular, so $OP$ must be perpendicular to the tangent.

A useful consequence is the Two-Tangent Theorem: from an external point $E$, the two tangent segments to a circle have equal length. The two right triangles formed (each with the radius, the tangent segment, and the shared hypotenuse $OE$) are congruent by HL.

2.5 Angles Formed by Chords, Secants, and Tangents

Where lines cross relative to the circle changes the angle formula:

• Two chords intersecting inside the circle: the angle equals half the sum of the two intercepted arcs.
• Two secants/tangents meeting outside the circle: the angle equals half the difference of the intercepted arcs (far arc minus near arc).
• A tangent and a chord meeting at the point of tangency: the angle equals half the intercepted arc.

The pattern: vertex on the circle gives $\tfrac{1}{2}(\text{arc})$, vertex inside gives $\tfrac{1}{2}(\text{sum})$, vertex outside gives $\tfrac{1}{2}(\text{difference})$.

2.6 Power of a Point

When lines through a point $P$ cut a circle, the products of the segment lengths are equal — this constant is called the power of the point:

• Two chords crossing inside at $P$, splitting into $(a, b)$ and $(c, d)$: $\;a\cdot b = c\cdot d$.
• Two secants from an external point $P$: $\;(\text{whole}_1)(\text{near}_1) = (\text{whole}_2)(\text{near}_2)$.
• Tangent and secant from an external point: $\;t^2 = (\text{whole})(\text{near})$, where $t$ is the tangent length.

These all follow from similar triangles created by the inscribed angle theorem.

2.7 Chord Length and Distance from the Center

A chord of length $c$ sits at perpendicular distance $d$ from the center. Half the chord, the radius, and $d$ form a right triangle, so

$\left(\frac{c}{2}\right)^2 + d^2 = r^2 \quad \Longrightarrow \quad c = 2\sqrt{r^2 - d^2}.$

A chord closer to the center is longer; the diameter ($d = 0$) is the longest chord. Two chords equidistant from the center are equal in length — a fact often used in proofs about symmetry.

2.8 Cyclic Quadrilaterals

A cyclic quadrilateral has all four vertices on one circle. Its opposite angles are supplementary (they sum to $180^\circ$), because each pair intercepts arcs that together make the full circle. Conversely, if a quadrilateral has opposite angles summing to $180^\circ$, it can be inscribed in a circle. This links angle chasing on circles to polygon problems.

2.9 The Circle in the Coordinate Plane

A circle with center $(h, k)$ and radius $r$ consists of all points $(x, y)$ whose distance from $(h, k)$ equals $r$. By the distance formula,

$(x - h)^2 + (y - k)^2 = r^2.$

The special case $(x)^2 + (y)^2 = r^2$ is a circle centered at the origin. If an equation is given in expanded form $x^2 + y^2 + Dx + Ey + F = 0$, complete the square in $x$ and $y$ separately to read off the center and radius. The circle exists as a real figure only when the completed-square form has a nonnegative $r^2$.

2.10 Arc Length, Sector Area, and Radian Measure

On a circle of radius $r$, an arc whose central angle is $\theta$ (in degrees) has length

$s = \frac{\theta}{360^\circ}\cdot 2\pi r.$

In radians, where a full turn is $2\pi$, the formula simplifies to $s = r\theta$. Sector area follows the same fraction of the whole disk: $A = \tfrac{\theta}{360^\circ}\pi r^2$ in degrees, or $A = \tfrac{1}{2}r^2\theta$ in radians. Radians are natural here because arc length equals radius times angle with no conversion factor.

2.11 Inscribed Angle and Arc Notation

When we say inscribed angle $\angle ABC$ intercepts arc $AC$, we mean the arc that lies opposite the vertex $B$ — the arc from $A$ to $C$ that does not pass through $B$. If both arcs from $A$ to $C$ are mentioned, the minor arc is typically used unless the problem specifies the major arc. Central angle $\angle AOC$ for the same arc uses the center $O$ and equals the full arc measure, not half. Keeping arc notation consistent prevents the most common inscribed-angle error.

2.12 Common Tangents and Concentric Circles

Two concentric circles share a center but have different radii $r_1 < r_2$. The ring between them is an annulus. A line tangent to both circles (a common external tangent) is perpendicular to the radius at each point of tangency; the segment joining the centers and the two radii form a right trapezoid whose vertical leg has length $r_2 - r_1$, which helps find tangent length from geometry.