Circles

You have been surrounded by circles your entire life. The wheels on your car, the face of your watch, the pizza you order for dinner, the roundabout you drive through on your commute - circles are everywhere. There is something deeply satisfying about a circle’s perfect symmetry. Unlike a rectangle with its different lengths and widths, or a triangle with its varying angles, a circle is the same in every direction. This elegant simplicity hides a wealth of fascinating properties that humans have been exploring for thousands of years.

The ancient Greeks were captivated by circles. They discovered relationships between a circle’s parts that hold true for every single circle, no matter how large or small. These same relationships help engineers design gears that mesh perfectly, architects create domes that stand for centuries, and pizza makers cut slices that are actually fair. Let us explore this perfectly round shape together.

Core Concepts

Parts of a Circle: The Basic Vocabulary

A circle is the set of all points that are exactly the same distance from a single point called the center. This might sound like a complicated definition, but think about it: if you stick a pin in a piece of paper and tie a string to it, then pull the string tight and draw while keeping it tight, you will draw a circle. Every point on that circle is exactly one string-length away from the pin.

The center is the point right in the middle of the circle. We often name a circle by its center. A circle with center $O$ is called “circle $O$” or written as $\odot O$.

The radius (plural: radii) is the distance from the center to any point on the circle - like that string you used to draw. We typically use $r$ to represent the radius. Every radius of a circle has the same length (that is what makes it a circle!).

The diameter is the distance across the circle through the center. It is like drawing a line from one edge of the circle, through the center, to the opposite edge. The diameter is always exactly twice the radius:

$$d = 2r$$

Or equivalently:

$$r = \frac{d}{2}$$

A chord is any line segment with both endpoints on the circle. Think of it as a “shortcut” across the circle that does not necessarily go through the center. The diameter is a special chord - it is the longest possible chord because it passes through the center.

Secants and Tangents: Lines Meeting Circles

Now let us talk about lines that interact with circles.

A secant is a line that intersects a circle at exactly two points. Imagine extending a chord in both directions forever - that line is a secant. The word comes from the Latin “secare,” meaning “to cut,” and that is exactly what a secant does: it cuts through the circle.

A tangent is a line that touches the circle at exactly one point. It just barely grazes the edge of the circle without cutting through it. That single point where the tangent touches the circle is called the point of tangency.

Here is a crucial fact about tangents: a tangent line is always perpendicular to the radius drawn to the point of tangency. If you draw a radius to the point where a tangent touches the circle, those two lines form a $90°$ angle. Always. This property is incredibly useful for solving problems.

Arcs: Pieces of the Circle

An arc is a portion of the circle’s edge - a curved piece between two points on the circle. When you cut a pizza, each slice has a curved crust edge; that curved edge is an arc.

Two points on a circle actually create two arcs: a shorter one and a longer one (unless the points are exactly opposite each other, in which case both arcs are equal). We distinguish between them:

• Minor arc: The shorter arc (less than half the circle)
• Major arc: The longer arc (more than half the circle)
• Semicircle: Exactly half the circle (when the endpoints are the ends of a diameter)

We name arcs using their endpoints. The minor arc from point $A$ to point $B$ is written $\stackrel{\frown}{AB}$. For major arcs, we include a third point on the arc to make it clear which arc we mean: $\stackrel{\frown}{ACB}$ (the arc from $A$ to $B$ passing through $C$).

Arc measure is measured in degrees, just like angles. A full circle is $360°$, so a semicircle is $180°$, and a quarter of a circle is $90°$.

Central Angles: Angles from the Center

A central angle is an angle whose vertex is at the center of the circle. The two rays of the angle intersect the circle at two points, creating an arc.

Here is the beautiful simplicity of central angles: the measure of a central angle equals the measure of its intercepted arc. If a central angle measures $75°$, the arc it “cuts off” (called the intercepted arc) also measures $75°$.

This makes intuitive sense when you think about it. The central angle and its arc are measuring the same thing: how much of the full $360°$ rotation you are capturing.

Arc Length: The Actual Distance Along the Arc

Arc measure tells you what fraction of the circle an arc represents (in degrees). But what about the actual length of the curved path along the arc?

Arc length is the distance along the arc - if you straightened out that curved piece, how long would it be?

To find arc length, you need to know what fraction of the circle the arc represents, then take that fraction of the circumference. If an arc has a measure of $\theta$ degrees:

$$\text{Arc Length} = \frac{\theta}{360°} \times 2\pi r = \frac{\theta}{360°} \times \pi d$$

Or, if you prefer working with the arc measure as a fraction:

$$\text{Arc Length} = \frac{\theta}{360°} \times C$$

where $C$ is the circumference of the circle.

Inscribed Angles: Angles from the Edge

An inscribed angle is an angle whose vertex is on the circle (not at the center) and whose sides are chords of the circle. Picture sitting on the edge of a circular stadium and looking at two points on the opposite side - the angle of your view is an inscribed angle.

Here is where things get exciting. Inscribed angles have a special relationship with their intercepted arcs.

The Inscribed Angle Theorem

The Inscribed Angle Theorem states: An inscribed angle is half the measure of its intercepted arc.

$$\text{Inscribed Angle} = \frac{1}{2} \times \text{Intercepted Arc}$$

Or equivalently, the intercepted arc is twice the inscribed angle:

$$\text{Intercepted Arc} = 2 \times \text{Inscribed Angle}$$

Compare this to central angles, where the angle equals the arc. Inscribed angles are more “distant” from the arc (the vertex is on the edge rather than at the center), so they “see” the arc as smaller - exactly half the size.

This theorem has powerful consequences:

Corollary 1: All inscribed angles that intercept the same arc are congruent. No matter where you sit on the circle, if you are looking at the same arc, you see the same angle.

Corollary 2: An inscribed angle that intercepts a semicircle is a right angle ($90°$). Since a semicircle measures $180°$, the inscribed angle is $\frac{180°}{2} = 90°$. This means if you inscribe a triangle in a circle where one side is a diameter, the angle opposite that diameter is always $90°$.

Angles Formed by Chords, Secants, and Tangents

Beyond central and inscribed angles, there are other angles formed when chords, secants, and tangents interact with circles.

Angles formed by two chords intersecting inside the circle: When two chords intersect inside a circle, they form two pairs of vertical angles. The measure of each angle is half the sum of the two intercepted arcs:

$$\text{Angle} = \frac{1}{2}(\text{arc}_1 + \text{arc}_2)$$

Angles formed by two secants, two tangents, or a secant and a tangent meeting outside the circle: When these lines meet outside the circle, the angle formed is half the difference of the intercepted arcs:

$$\text{Angle} = \frac{1}{2}(\text{larger arc} - \text{smaller arc})$$

Angle formed by a tangent and a chord at the point of tangency: This angle equals half the intercepted arc (similar to an inscribed angle):

$$\text{Angle} = \frac{1}{2} \times \text{Intercepted Arc}$$

Segment Relationships in Circles

When chords, secants, and tangents intersect, the segments they create have special length relationships.

Two chords intersecting inside a circle: If two chords intersect inside a circle, the products of their segments are equal:

$$a \cdot b = c \cdot d$$

where one chord is divided into segments of length $a$ and $b$, and the other into segments of length $c$ and $d$.

Two secants from the same external point: If two secants are drawn from the same point outside a circle, then:

$$(whole_1) \times (external_1) = (whole_2) \times (external_2)$$

where “whole” means the entire length from the external point through the circle, and “external” means just the part outside the circle.

A secant and a tangent from the same external point: If a secant and a tangent are drawn from the same external point, then:

$$(\text{tangent})^2 = (\text{whole secant}) \times (\text{external part of secant})$$

Two tangents from the same external point: If two tangent segments are drawn to a circle from the same external point, they are congruent (equal in length).

Circumference and Area: A Formal Review

You have likely seen these formulas before, but let us state them with proper notation.

Circumference (the distance around the circle):

$$C = \pi d = 2\pi r$$

Area (the space inside the circle):

$$A = \pi r^2$$

The number $\pi$ (pi) is the ratio of any circle’s circumference to its diameter. It is approximately $3.14159…$, but it is an irrational number that goes on forever without repeating. For calculations, we often use $\pi \approx 3.14$ or leave answers in terms of $\pi$ for exactness.

Notation and Terminology

Term	Meaning	Example
Circle	Set of points equidistant from center	Circle $O$ or $\odot O$
Radius	Distance from center to edge	$r$
Diameter	Distance across through center	$d = 2r$
Chord	Segment with endpoints on circle	$\overline{AB}$ where $A$, $B$ are on the circle
Secant	Line intersecting circle twice	Extends through two points
Tangent	Line touching circle at one point	Perpendicular to radius at point of tangency
Arc	Part of a circle	$\stackrel{\frown}{AB}$
Minor arc	Arc less than $180°$	Shorter path between two points
Major arc	Arc greater than $180°$	Longer path between two points
Semicircle	Arc of exactly $180°$	Half the circle
Central angle	Vertex at center	Equals intercepted arc measure
Inscribed angle	Vertex on circle	Half of intercepted arc measure
Intercepted arc	Arc “cut off” by an angle	Between the sides of the angle
Pi ($\pi$)	Ratio of circumference to diameter	$\approx 3.14159$

Examples

Key Properties and Rules

Central and Inscribed Angle Relationships

Central Angle: $$\text{Central angle measure} = \text{Intercepted arc measure}$$

Inscribed Angle: $$\text{Inscribed angle measure} = \frac{1}{2} \times \text{Intercepted arc measure}$$

Inscribed angle intercepting a semicircle: $$\text{Angle} = \frac{180°}{2} = 90°$$

Angles Formed by Chords, Secants, and Tangents

Two chords intersecting inside the circle: $$\text{Angle} = \frac{1}{2}(\text{arc}_1 + \text{arc}_2)$$

Two secants, two tangents, or a secant and tangent meeting outside: $$\text{Angle} = \frac{1}{2}|\text{larger arc} - \text{smaller arc}|$$

Tangent and chord at point of tangency: $$\text{Angle} = \frac{1}{2} \times \text{Intercepted arc}$$

Segment Relationships

Intersecting chords (inside circle): $$a \cdot b = c \cdot d$$

Two secants from external point: $$(\text{whole}_1)(\text{external}_1) = (\text{whole}_2)(\text{external}_2)$$

Secant and tangent from external point: $$(\text{tangent})^2 = (\text{whole secant})(\text{external secant})$$

Two tangents from external point: $$\text{tangent}_1 = \text{tangent}_2$$

Arc Length and Circumference

Circumference: $$C = 2\pi r = \pi d$$

Arc length: $$\text{Arc Length} = \frac{\theta}{360°} \times 2\pi r$$

where $\theta$ is the central angle in degrees.

Area

Area of a circle: $$A = \pi r^2$$

Area of a sector (pie slice): $$A_{\text{sector}} = \frac{\theta}{360°} \times \pi r^2$$

Tangent Properties

• A tangent is perpendicular to the radius at the point of tangency: $\text{tangent} \perp \text{radius}$
• Two tangent segments from the same external point are congruent

Real-World Applications

Wheel and Gear Design

Every wheel is a circle, and the properties of circles are essential to making wheels work. The circumference of a wheel determines how far a vehicle travels in one rotation. A car tire with a diameter of 26 inches has a circumference of $\pi \times 26 \approx 81.7$ inches. Each time the wheel rotates once, the car moves about 82 inches forward.

Gears use circles and their properties to transfer motion. When two gears mesh, the relationship between their radii determines the gear ratio - how many times one gear turns for each rotation of the other. This is how bicycles give you mechanical advantage when climbing hills and speed when riding on flat ground.

Pizza and Pie Slicing

When you cut a pizza into slices, you are creating sectors (pie-shaped pieces). Each cut from the center creates a central angle, and fair slices mean equal central angles. For 8 slices, each central angle is $\frac{360°}{8} = 45°$.

But here is something interesting: the crust on each slice (the arc) depends on both the angle AND the size of the pizza. A 45-degree slice from a large pizza has more crust than a 45-degree slice from a small pizza, even though the angles are the same. That is the difference between arc measure (in degrees) and arc length (in inches or centimeters).

Roundabouts and Circular Tracks

Traffic roundabouts are designed using circle properties. The inscribed angle theorem helps engineers understand sight lines - what drivers can see as they enter and navigate the circle. The radius of the roundabout determines both the circumference (how far you drive to go around) and the curvature (how sharply you need to turn).

Running tracks have circular ends. A standard outdoor track is 400 meters around, but the inside lane has a tighter curve (smaller radius) than the outside lanes. That is why runners in different lanes start at staggered positions - to compensate for the different distances around the curves.

Clock Faces and Angles

A clock face is a perfect example of central angles in action. The minute hand sweeps $360°$ every hour, or $6°$ every minute. The hour hand moves $30°$ every hour (since $\frac{360°}{12} = 30°$), or $0.5°$ every minute.

At 3:00, the hands form a $90°$ angle. At 6:00, they form a $180°$ angle. But what about 3:30? The minute hand is at the 6 (pointing down), and the hour hand has moved halfway between the 3 and 4. The angle is not quite $90°$ anymore - it is $90° - 15° = 75°$ (since the hour hand moved $15°$ in that half hour).

Architecture: Domes and Arches

Circular arcs are structurally strong, which is why domes and arches have been used in architecture for thousands of years. The Pantheon in Rome has a circular floor plan and a hemispherical dome - essentially half a sphere, which is a circle rotated in three dimensions.

When architects design arched doorways or windows, they use the inscribed angle theorem to ensure that the arch looks proportional from different viewing angles. The relationship between the chord (the width of the opening) and the arc (the curved top) determines the arch’s appearance and structural integrity.

Self-Test Problems

Summary

• A circle is the set of all points equidistant from a center point. The radius is the distance from center to edge; the diameter is twice the radius and passes through the center.

• Chords are segments with endpoints on the circle. Secants are lines that cut through the circle at two points. Tangents touch the circle at exactly one point and are perpendicular to the radius at that point.

• Arcs are portions of the circle. Minor arcs are less than $180°$; major arcs are greater than $180°$; semicircles are exactly $180°$.

• Central angles have their vertex at the center. A central angle equals its intercepted arc.

• Inscribed angles have their vertex on the circle. An inscribed angle is half its intercepted arc (the Inscribed Angle Theorem). All inscribed angles intercepting the same arc are congruent.

• Arc length is the actual distance along an arc: $\text{Arc Length} = \frac{\theta}{360°} \times 2\pi r$

• Angles formed by intersecting chords inside a circle equal half the sum of the intercepted arcs. Angles formed by secants or tangents meeting outside the circle equal half the difference of the intercepted arcs.

• Segment relationships: Intersecting chords follow $a \cdot b = c \cdot d$. Secants from an external point follow $(\text{whole}_1)(\text{external}_1) = (\text{whole}_2)(\text{external}_2)$. A tangent and secant from the same point follow $(\text{tangent})^2 = (\text{whole})(\text{external})$.

• Circumference $C = 2\pi r = \pi d$ and Area $A = \pi r^2$ are fundamental measurements for any circle.

• Circle properties appear everywhere in real life: wheels, gears, clocks, pizza, roundabouts, running tracks, and architectural domes all rely on the elegant geometry of circles.