Circles

A circle is the simplest curve you can draw and one of the deepest ideas in mathematics. It is nothing more than the set of all points that sit exactly the same distance from a single center point — yet from that one sentence flow the wheel, the orbit, the gear, the lens, radio waves, and the mysterious number $\pi$ that refuses to be written as a fraction. Almost every rotating, oscillating, or "smoothly curved" thing in the physical world hides a circle inside it.

What makes circles worth real study is that they force us to confront curvature. A straight line is easy to measure with a ruler. But how long is a curved boundary, and how much area does it enclose? Answering that honestly took humanity thousands of years and led to some of the most beautiful reasoning in the history of thought. This page teaches you the vocabulary, the two headline formulas, the angle theorems that show up on every geometry exam, and the human story of measuring the circle.

Learning Objectives

By the end of this page, you should be able to:

• Define and identify the key parts of a circle: center, radius, diameter, chord, arc, sector, tangent, and secant.
• Use $C = 2\pi r$ and $A = \pi r^2$ correctly, and explain why each is true rather than just plugging in.
• Compute arc length and sector area for a given central angle.
• Apply the central angle and inscribed angle theorems, including Thales' theorem.
• Explain what $\pi$ actually is and how Archimedes bounded it between $3\tfrac{10}{71}$ and $3\tfrac{1}{7}$.

Quick Answer

A circle is all points a fixed distance (the radius $r$) from a center. The diameter is $d = 2r$, the width straight through the center. The distance around the circle, its circumference, is $C = 2\pi r = \pi d$, and the area it encloses is $A = \pi r^2$. The number $\pi \approx 3.14159$ is the same ratio of circumference to diameter for every circle in existence — that universality is why it deserves its own symbol. A central angle at the center cuts off an arc, and an inscribed angle on the circle that looks at the same arc is always exactly half as large. Master those few facts and most circle problems fall apart quickly.

Where It Came From

The need to measure circles is as old as the wheel and the sky. Ancient builders, potters, and surveyors needed to know how much material wrapped around a round cistern, and early astronomers watched the Sun, Moon, and stars trace circular paths overhead. Everyone could see that a bigger circle needed more rope to fence it and more clay to fill it — but by exactly how much?

Early civilizations found that the circumference was always "a bit more than three times" the diameter, no matter the size of the circle. The Babylonians used $3$ or $3\tfrac{1}{8} = 3.125$. The Egyptian Rhind Papyrus (around 1650 BCE) used a rule equivalent to $\pi \approx \left(\tfrac{16}{9}\right)^2 \approx 3.16$. These were practical estimates, not proofs — and nobody yet knew whether the ratio was truly constant or merely close.

The decisive breakthrough came from Archimedes of Syracuse (around 250 BCE). Instead of measuring a circle directly — impossible with a curved edge — he trapped it between two shapes he could measure: regular polygons drawn just inside and just outside the circle. A hexagon, then a 12-gon, 24-gon, 48-gon, and finally a 96-gon squeezed the circle from both sides. His conclusion was a rigorous inequality:

$3\frac{10}{71} < \pi < 3\frac{1}{7}$

That is roughly $3.1408 < \pi < 3.1429$ — astonishingly tight for a proof done entirely by hand with geometry and no decimals. This "method of exhaustion" is a direct ancestor of integral calculus, invented nearly two thousand years early.

One question haunted geometers for the next two millennia: squaring the circle — using only compass and straightedge, construct a square with exactly the same area as a given circle. Countless people tried and failed. The mystery was only settled in 1882, when Ferdinand von Lindemann proved that $\pi$ is transcendental (not the root of any polynomial with whole-number coefficients). That single fact makes the construction provably impossible — a rare case where mathematics proves that something cannot be done, ending a 2000-year quest.

The Anatomy of a Circle

Before any formula, you need the vocabulary. Every term below describes a line, region, or angle related to the circle.

• Center: the fixed point everything is measured from.
• Radius ($r$): any segment from the center to the circle; also its length.
• Diameter ($d$): a chord through the center; $d = 2r$. It is the longest possible chord.
• Chord: any segment joining two points on the circle.
• Arc: a portion of the circle itself (the curved part). A minor arc is the shorter piece, a major arc the longer.
• Sector: a "pizza slice" — the region between two radii and the arc between them.
• Segment: the region between a chord and its arc (a sector with the triangle sliced off).
• Tangent: a line touching the circle at exactly one point; it is always perpendicular to the radius at that point.
• Secant: a line cutting the circle at two points.

Worked example: reading a diagram

A circle has a diameter of 26 cm. A chord of length 24 cm is drawn. How far is the chord from the center?

The radius is $r = 26/2 = 13$ cm. Drop a perpendicular from the center to the chord; it bisects the chord, giving two halves of $24/2 = 12$ cm. This makes a right triangle with hypotenuse $r = 13$ (a radius to the chord's endpoint) and one leg $12$. By the Pythagorean theorem the distance $x$ from center to chord satisfies:

$x^2 + 12^2 = 13^2 \quad\Rightarrow\quad x^2 = 169 - 144 = 25 \quad\Rightarrow\quad x = 5 \text{ cm}.$

Notice how a circle problem quietly became a right-triangle problem — that happens constantly.

Circumference, Area, and the Meaning of Pi

The single most important fact about circles is that the ratio of circumference to diameter is the same for every circle. We name that ratio $\pi$:

$\pi = \frac{C}{d} \approx 3.14159265\ldots$

Rearranging gives the circumference formula, and since $d = 2r$:

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

The area formula is $A = \pi r^2$. Here is an intuition for why, not just the formula. Cut the circle into many thin pie slices and lay them side by side, alternating point-up and point-down. They interlock into a shape close to a rectangle. Its height is the radius $r$, and its width is half the circumference (all the arc-tips from the top), which is $\tfrac{1}{2}(2\pi r) = \pi r$. The more slices you use, the closer this gets to a true rectangle, so:

$A = \text{height} \times \text{width} = r \times \pi r = \pi r^2.$

Worked example: circumference and area

A circular running track has radius $r = 50$ m. Find its circumference and enclosed area (use $\pi \approx 3.14159$).

$C = 2\pi r = 2 \times 3.14159 \times 50 \approx 314.16 \text{ m}.$

$A = \pi r^2 = 3.14159 \times 50^2 = 3.14159 \times 2500 \approx 7853.98 \text{ m}^2.$

So one lap is about 314 m, and the field inside covers nearly 7854 square metres. Watch the units: circumference is in metres, area in metres squared, because area multiplies two lengths.

Worked example: arc length and sector area

A pizza of radius 15 cm is cut into a slice with a central angle of $60°$. Find the length of the crust (arc) and the area of the slice (sector).

A full circle is $360°$, so this slice is $\tfrac{60}{360} = \tfrac{1}{6}$ of the whole. Take that same fraction of the circumference and of the area:

$\text{Arc} = \frac{60}{360}\times 2\pi r = \frac{1}{6}\times 2\pi(15) = 5\pi \approx 15.71 \text{ cm}.$

$\text{Sector area} = \frac{60}{360}\times \pi r^2 = \frac{1}{6}\times \pi (15)^2 = 37.5\pi \approx 117.81 \text{ cm}^2.$

The general formulas, for a central angle $\theta$ in degrees, are $\text{Arc} = \tfrac{\theta}{360}\cdot 2\pi r$ and $\text{Sector} = \tfrac{\theta}{360}\cdot \pi r^2$. (In radians, they simplify beautifully to $r\theta$ and $\tfrac{1}{2}r^2\theta$.)

Central and Inscribed Angles

This is the topic that separates students who "know the formulas" from those who truly understand circles.

A central angle has its vertex at the center. Its measure equals the measure of the arc it cuts off — this is essentially the definition of arc measure.

An inscribed angle has its vertex on the circle, with its two sides being chords. The Inscribed Angle Theorem states:

An inscribed angle is exactly half the central angle that subtends the same arc.

So if two people look at the same arc — one standing at the center, one standing on the far side of the circle — the person on the circle always measures half the angle. A powerful special case is Thales' Theorem: any inscribed angle standing on a diameter subtends a semicircle (a $180°$ arc), so it must be $\tfrac{1}{2}\times 180° = 90°$. Every triangle inscribed in a circle with one side as the diameter is right-angled.

Worked example: inscribed angle

Points $A$, $B$, and $C$ lie on a circle with center $O$. The central angle $\angle AOC = 110°$. What is the inscribed angle $\angle ABC$ (with $B$ on the major arc)?

Both angles subtend arc $AC$. By the inscribed angle theorem:

$\angle ABC = \frac{1}{2}\angle AOC = \frac{1}{2}(110°) = 55°.$

A neat corollary: any other point on the same major arc also sees arc $AC$ at exactly $55°$. All these viewing angles are equal — which is why a fixed goalpost looks the "same width" from anywhere along a certain arc.

Real-World Applications

• Engineering and machines: gears, wheels, pulleys, turbines, and bearings all rely on constant radius; $C = 2\pi r$ tells you how far a wheel of radius $r$ travels in one rotation.
• Astronomy and navigation: planetary orbits are near-circular; the whole system of latitude/longitude divides Earth's circular cross-sections into $360°$.
• Medicine: CT and MRI scanners rotate detectors on a circle; pupil and blood-vessel areas are estimated with $\pi r^2$; stent sizing uses circumference.
• Architecture and construction: pipes, tanks, domes, and arches — the volume of a cylindrical water tank is its circular area times its height, $\pi r^2 h$.
• Everyday life: a 12-inch and a 16-inch pizza differ in area by $\pi(8^2 - 6^2) = 28\pi \approx 88$ square inches — the larger one gives far more food than its diameter suggests, because area grows with the square of the radius.

Common Mistakes

Mistake 1: Confusing radius and diameter in the formulas. Students plug the diameter into $A = \pi r^2$. Because $r$ is squared, using $d$ instead of $r$ gives an area four times too large. Always ask first, "Is the number I was given the radius or the diameter?" If it is the diameter, halve it before using $r$.

Mistake 2: Thinking area and circumference scale the same way. Doubling the radius doubles the circumference but quadruples the area (since $A \propto r^2$). A circle of twice the radius does not hold twice the area — it holds four times as much. This trips people up constantly with pizzas, ponds, and pipes.

Mistake 3: Treating an inscribed angle as equal to the central angle. They subtend the same arc, but the inscribed angle is half, not equal. Many students set them equal and get double the correct answer. Remember Thales: the $90°$ angle in a semicircle comes from halving the $180°$ diameter arc.

Comparison and Connections

Idea	What it measures	Formula	Common confusion
Circumference	Distance around	$2\pi r$	Mixed up with area
Area	Region inside	$\pi r^2$	Forgetting to square $r$
Arc length	Part of the boundary	$\tfrac{\theta}{360}\cdot 2\pi r$	Using area fraction
Sector area	Part of the inside	$\tfrac{\theta}{360}\cdot \pi r^2$	Using arc formula
Chord	Straight line inside	$2r\sin(\tfrac{\theta}{2})$	Confused with arc

A circle is also a special ellipse (both axes equal) and the boundary of a disk. In coordinate geometry it becomes the equation $(x-h)^2 + (y-k)^2 = r^2$ — the Pythagorean theorem in disguise, connecting this page directly to Coordinate Geometry. The unit circle then becomes the foundation of all of Trigonometry.

Practice Questions

Recall

State the formulas for the circumference and area of a circle of radius $r$, and give the value of $\pi$ to two decimal places.

Answer: $C = 2\pi r$, $A = \pi r^2$, and $\pi \approx 3.14$.

Understanding

Explain why doubling a circle's radius multiplies its area by 4 but its circumference by only 2.

Answer: Circumference is linear in $r$ ($2\pi r$), so scaling $r$ by 2 scales $C$ by 2. Area depends on $r^2$ ($\pi r^2$), so scaling $r$ by 2 scales area by $2^2 = 4$.

Application

A circular garden of radius 7 m is surrounded by a path. Find the length of fencing needed to enclose the garden and the area to be planted (use $\pi \approx \tfrac{22}{7}$).

Answer: $C = 2\times\tfrac{22}{7}\times 7 = 44$ m of fencing. $A = \tfrac{22}{7}\times 7^2 = \tfrac{22}{7}\times 49 = 154$ m².

Analysis

A triangle is inscribed in a circle so that one side is a diameter of length 10, and one other side has length 6. Find the third side.

Answer: By Thales' theorem the angle opposite the diameter is $90°$, so the triangle is right-angled with hypotenuse 10 (the diameter). The third side is $\sqrt{10^2 - 6^2} = \sqrt{100-36} = \sqrt{64} = 8$.

FAQ

Why does $\pi$ never end or repeat? Because $\pi$ is irrational — it cannot be written as a fraction of whole numbers, proven in 1761. Its decimal expansion therefore goes on forever without a repeating pattern. It is also transcendental, an even stronger property.

Is $\pi$ exactly $\tfrac{22}{7}$? No. $\tfrac{22}{7} \approx 3.142857$ is a handy approximation, slightly larger than $\pi \approx 3.14159$. It is Archimedes' upper bound. For rough work it is fine, but it is not exact — nothing rational is.

What is the difference between an arc and a chord? A chord is the straight segment between two points on the circle; the arc is the curved path along the circle between those same two points. The arc is always longer than the chord.

Why is a tangent perpendicular to the radius? The radius to the point of contact is the shortest distance from the center to the tangent line, and the shortest distance from a point to a line is always the perpendicular. So the radius meets the tangent at exactly $90°$.

Do all inscribed angles on the same arc really equal each other? Yes. Every inscribed angle subtending a given arc equals half that arc's central angle, so they are all equal to each other regardless of where the vertex sits on the remaining arc. This is the key to many "cyclic quadrilateral" proofs.

Can you ever square the circle? Not with compass and straightedge alone — Lindemann proved this impossible in 1882 by showing $\pi$ is transcendental. You can do it with other tools, but the classical Greek challenge is provably unsolvable.

Quick Revision

• Circle: all points at distance $r$ from a center. Diameter $d = 2r$.
• Circumference: $C = 2\pi r = \pi d$. Area: $A = \pi r^2$.
• $\pi = C/d \approx 3.14159$ — same for every circle; irrational and transcendental.
• Arc length $= \tfrac{\theta}{360}\cdot 2\pi r$; Sector area $= \tfrac{\theta}{360}\cdot \pi r^2$ (degrees).
• Central angle $=$ arc measure; Inscribed angle $=$ half the central angle on the same arc.
• Thales: angle in a semicircle is $90°$.
• Tangent $\perp$ radius at the point of contact.
• Area scales with $r^2$; circumference scales with $r$.
• Archimedes: $3\tfrac{10}{71} < \pi < 3\tfrac{1}{7}$ using inscribed and circumscribed 96-gons.

Related Topics

Prerequisites

• Lines and Angles
• Triangles

Related Topics

• Coordinate Geometry
• Area and Volume

Next Topics

• The Unit Circle
• Graphs and Periodic Motion