Circle

A circle is a special shape. It is made up of all the points that are the same distance from one central point. This distance from the centre to any point on the circle is called the radius. If you draw a line through the centre connecting two points on the circle, that line is called the diameter. The area inside a circle is called a disc.

Circles have been known since long before people started writing history. We see natural circles all around us, like the full moon or a round piece of fruit. The circle is also the basis for the wheel, and together with inventions like gears, it helps make many modern machines work. Studying circles has been very important in mathematics, helping to develop areas like geometry, astronomy, and calculus.

Terminology

Here are some important words and ideas about circles:

An annulus is a ring-shaped area between two circles that share the same centre.
An arc is any connected piece of a circle. If you pick two points on a circle, you can draw two arcs between them that together make the whole circle.
The centre of a circle is the point that is the same distance from every point on the circle.
A chord is a straight line that connects two points on a circle, splitting the circle into two parts.
The circumference is the distance around the circle.
The diameter is a straight line that goes through the centre of the circle and connects two points on the circle. It is the longest distance between any two points on the circle and is twice as long as the radius.
A disc is the area inside a circle. Sometimes people use the word "circle" to mean both the line around the circle and the area inside it.
A lens is the shape you get when two circle areas overlap.
The radius is a line from the centre of the circle to any point on the circle. Its length is half of the diameter. We usually write radius as r.
A sector is the area between two lines that start from the centre of the circle and end on the circle.
A segment is the area between a chord and the curve of the circle.
A secant is a straight line that cuts through a circle at two points.
A semicircle is half of a circle, made by cutting the circle in half through its diameter.
A tangent is a straight line that touches a circle at just one point.

All of these shapes can be thought of with or without their outer edges.

Etymology

The word circle comes from an old Greek word, kirkos or kuklos. These words mean "hoop" or "ring." The words circus and circuit also come from similar words.

History

People long ago made shapes like stone circles and timber circles. Circles also appear in old petroglyphs and cave paintings. Well-known examples are the Nebra sky disc and special jade discs called Bi.

Later, the Rhind papyrus from around 1700 BCE showed a way to find the area of a circle, using a number close to π. Euclid's Elements described circles and how they work. In Plato's writings, circles were talked about as perfect shapes. During the middle ages, many thought circles were special and perfect. In 1880, Ferdinand von Lindemann showed that π is a special kind of number, proving an old problem called squaring the circle could not be solved with simple tools.

With modern abstract art in the early 1900s, circles became important in paintings. Artists like Wassily Kandinsky used them often.

Symbolism and religious use

Circles have been important symbols for thousands of years in many cultures. They can represent ideas like unity, infinity, balance, and perfection. Different cultures use circles in art and religion in many ways, such as in magic circles and other special symbols like the Dharma wheel.

Analytic results

Circumference

Main article: Circumference

The distance around a circle is called its circumference. It is related to the circle's width, called its diameter, by a special number called π (pi). This number is about 3.14. If you know the circumference and the diameter, you can find π by dividing the circumference by the diameter. You can also find the circumference by multiplying the diameter by π.

Area enclosed

Main article: Area of a circle

A smart man named Archimedes found that the space inside a circle, called its area, can be calculated using a special rule. The area of the circle is found by multiplying π by the radius squared.

Radian

Main article: Radian

Imagine drawing an angle with the point where the two lines meet at the center of a circle. If the angle cuts out a piece of the circle’s edge, called an arc, the size of the angle in radians is the length of that arc divided by the radius of the circle. A full circle makes an angle of 2π radians, which is the same as 360 degrees.

Equations

Cartesian coordinates

Equation of a circle

In a special grid called a Cartesian coordinate system, a circle can be described by a simple rule. If the center of the circle is at point (a, b) and the radius is r, then every point (x, y) on the circle follows this rule: (x - a)² + (y - b)² = r². If the circle is centered at the origin (0, 0), the rule becomes x² + y² = r².

One coordinate as a function of the other

A circle can also be described by two rules, one for the top half and one for the bottom half. These rules tell you the y-value for any x-value on the circle.

Parametric form

The circle can also be described using two special math functions called sine and cosine. This gives a way to find any point on the circle using a single number called a parameter.

3-point form

There is also a way to find the equation of a circle if you know three points on the circle that are not in a straight line.

Homogeneous form

In a more advanced type of coordinates, circles follow a specific pattern that can be written as an equation.

Polar coordinates

In another way of describing places called polar coordinates, a circle follows a special rule that looks different depending on where the center of the circle is.

Complex plane

In a special area of math called the complex plane, a circle with center c and radius r follows a simple rule: |z - c| = r.

Tangent lines

Main article: Tangent lines to circles

A line that just touches a circle at one point without crossing it is called a tangent line. This line is always perpendicular to the radius at the point where it touches the circle.

Properties

A circle is a special shape where every point is the same distance from the middle point, called the center. This distance is named the radius.

Circles are very balanced shapes. Every line passing through the center can be flipped over and look the same, and the circle looks the same after turning around the center at any angle. All circles are the same shape, just different sizes. The distance around a circle (its circumference) and its radius have a special relationship, as do the space inside the circle (its area) and the radius squared. The constants in these relationships are 2π and π. A circle with radius 1, centered at the starting point, is called the unit circle.

Chords in a circle are lines that connect two points on the circle. Chords that are the same length are the same distance from the center. A line that cuts a chord into two equal parts always passes through the center. The angle at the center of the circle is twice any angle on the circle that uses the same chord. Angles on the same chord and the same side of the chord are equal, while angles on the same chord but opposite sides add up to 180 degrees. An angle on the circle that uses the whole width of the circle (a diameter) is always a right angle, which measures 90 degrees. The diameter is the longest chord in a circle.

A tangent is a line that just touches the circle at one point. A line drawn from the center to where the tangent touches is perpendicular to the tangent. From any point outside the circle, two tangents can be drawn, and they are the same length.

Compass and straightedge constructions

There are many ways to make circles using a compass and straightedge. The easiest way is when you know the centre of the circle and a point on the circle. You place one leg of the compass on the centre point and the other leg on the point of the circle, then rotate the compass.

You can also make a circle if you know the diameter. First, find the midpoint of the diameter. Then, make a circle with that midpoint as the centre, using one end of the diameter as a point on the circle—it will also go through the other end.

You can even make a circle through three points that are not in a straight line. Find the midpoint line between two of the points, then find the midpoint line between another pair of points. Where these two lines cross is the centre of your circle. Use this centre and one of the three points to draw the circle, and it will go through all three points.

Circle of Apollonius

Apollonius of Perga showed that a circle can be described as a group of points that have a fixed relationship in distance to two special points, called foci. This special circle is sometimes said to be drawn around these two points.

The idea has two parts. First, if you pick two points and a special distance relationship, any point that fits this relationship will always lie on one special circle. Second, every point on this circle will fit the special distance relationship that was chosen.

Cross-ratios

A related idea about circles uses a special math tool called the cross-ratio. For three points, the circle of Apollonius is made of all points where the size of this cross-ratio is exactly one.

Generalised circles

Inscription in or circumscription about other figures

In every triangle, there is a special circle called the incircle that fits inside and touches each side.

We can also draw a circle called the circumcircle around any triangle so that it passes through all three vertices of the triangle.

Some shapes, like certain four-sided figures and all regular shapes, can have a circle drawn inside them touching each side. These shapes are called tangential polygons. Other shapes can have a circle drawn around them passing through all their corners, called cyclic polygons. Shapes that can do both are called bicentric polygons.

A hypocycloid is a special curve made by tracing a point on a smaller circle as it rolls inside a larger circle.

Limiting case of other figures

A circle is a special shape. It can look like the end result of many other shapes. For example, if you keep adding more sides to a shape like a pentagon or hexagon, it begins to look more like a circle.

A circle is also a special type of ellipse where the two points that define the shape are in the same place. It can even be a special case of more complex shapes, like a Cartesian oval or a superellipse, under certain conditions.

Locus of constant sum

Imagine you have a few points on a flat surface. If you look for all the places where the total of the squared distances to those points stays the same, you will find a circle. The middle point of this circle is right at the center of all the points you started with.

If you use the corners of a regular shape, like a triangle or a square, and look for places where a certain power of the distances to those corners adds up to a constant number, you will also find circles. The middle of each of these circles is the center of the shape you started with.

Squaring the circle

Main article: Squaring the circle

Squaring the circle is a problem that ancient geometers wanted to solve. They tried to make a square with the same size as a circle, using only a compass and straightedge in a few steps.

In 1882, mathematicians showed this could not be done. This is because pi (π) is a special number that makes this problem unsolvable. Even so, many people still find the idea fascinating.

Generalisations

In other p-norms

A circle is usually thought of as all points that are the same distance from a center point. But if we change how we measure distance, we can get different shapes that still act like circles.

In normal geometry, we use the "Euclidean" way of measuring distance. But there are other ways, like the "taxicab" way, where distance is measured like how a car drives on a grid of streets. In this way, circles look like squares tilted at a 45-degree angle.

Topological definition

In topology, a circle is a special kind of shape that can be stretched and bent but not torn. All circles, no matter how stretched, are considered the same in this study.

Orientation

An oriented circle is a circle that has a direction — either clockwise or anti-clockwise. This direction is shown by a number, +1 or -1, attached to the circle.