Circles: Circumference, Area, Angles and Equations

What is a circle?

A circle is the set of all points in a plane that are the same distance, called the radius, from one fixed point, called the center. The circle is only the curve; the curve together with its inside is the disk, and in everyday language "the area of a circle" means the area of the disk.

A circle is perfectly symmetric: every line through the center is a line of symmetry, and turning the circle by any angle about the center leaves it unchanged. It is also similar to every other circle, since one is a scaled copy of the other (the same idea as similar triangles). That is the reason the ratio of circumference to diameter is one number for all circles.

What is π, and how do you find the circumference and area of a circle?

The number π≈3.14159 is the ratio of the circumference of any circle to its diameter, so the circumference is C=π d=2π r, and the area of the disk is A=π r^2.

Why π r^2? Cut the disk into many thin sectors and lay them side by side, alternately pointing up and down. They form a shape that looks more and more like a rectangle as the slices get thinner. Its height is r and its width is half the circumference, π r, so the area is π r· r=π r^2.

For r=7 the circumference is 14π≈43.98 and the area is 49π≈153.94. Keep π as a symbol as long as you can: the answer "49π" is exact, while 153.94 is rounded. A ring (annulus) between radii R and r has area π R^2-π r^2=π(R-r)(R+r); for R=10 and r=6 that is 64π≈201.06.

What kind of number is π? It is irrational, so no fraction equals it, and transcendental, so it is not the solution of any polynomial equation with whole-number coefficients, a fact that ends the old puzzle of squaring the circle; see pi, e and transcendental numbers. Its first digits are 3.14159 26535 89793. Useful fractions come from rational approximations: 3.14 is off by 0.05\%, (22)/(7)≈3.142857 by 0.04\%, and (355)/(113)≈3.1415929 by less than 0.00001\%.

A word problem: a round pond has a diameter of 14 m and is to be fenced, with π≈(22)/(7). The fence is C=(22)/(7)·14=44 m long, and at 12 dollars per meter it costs 44·12=528 dollars.

Archimedes found bounds for π by squeezing the circle between two polygons. The tool below repeats his doubling, from the hexagon to a polygon with thousands of sides.

How do you find arc length, sector area and segment area?

A central angle of θ cuts off the fraction (θ)/(360) of the circle, so the arc is that fraction of the circumference and the sector is that fraction of the disk; the segment is the sector minus the triangle of the two radii and the chord.

Radians. One radian is the central angle whose arc is exactly as long as the radius. A full turn has (2π r)/(r)=2π radians, so 180^\circ=π rad and 1 rad≈57.2958^\circ. In radians the arc is simply rθ, which is why mathematics prefers them.

Examples. For r=6 and θ=120^\circ the fraction is \frac13, so the arc is \frac13·12π=4π≈12.57 and the sector is \frac13·36π=12π≈37.70. This is the multiplication of fractions with π carried along. For r=10 and θ=90^\circ: arc 5π≈15.71, sector 25π≈78.54, chord 10√2≈14.14 and segment 25π-50≈28.54, since the triangle has area \frac12·10·10=50.

Choose a radius (or any of the other three quantities) and slide the angle below.

What are the angle theorems of a circle?

An inscribed angle (a vertex on the circle) is half the central angle on the same arc, and from this one theorem come the right angle in a semicircle, equal angles in the same segment and the 180^\circ of a cyclic quadrilateral.

• Central angle. An angle at the center measures the same, in degrees, as the arc it cuts off.
• Inscribed angle theorem. An angle \angle APB with P on the circle is half the central angle \angle AOB on the same arc: central 120^\circ gives inscribed 60^\circ.
• Same segment. Inscribed angles on the same arc are equal, wherever P moves on the other part of the circle.
• Thales. If AB is a diameter, the arc is 180^\circ, so every inscribed angle \angle APB is 90^\circ. The converse is the fact that the circumcenter of a right triangle is the midpoint of its hypotenuse, as in the centers of a triangle.
• Cyclic quadrilateral. If all four vertices lie on a circle, opposite angles add up to 180^\circ, because they are half of two arcs that together make 360^\circ. With \angle A=75^\circ the opposite angle is \angle C=105^\circ. Ptolemy's theorem and Brahmagupta's area formula in the history of quadrilaterals are about these shapes.
• Tangent and radius. A tangent is perpendicular to the radius at the point where it touches.
• Tangent and chord. The angle between a tangent and a chord equals the inscribed angle on the other side of the chord (the alternate segment).

Why one half? Take the case where one side of the angle passes through the center. The triangle OPA is isosceles (OP=OA=r), so its base angles are equal, and the central angle is an exterior angle of that triangle, equal to the sum of the two base angles, as in the angle sum and exterior angle. So the central angle is twice the inscribed one. The general case adds or subtracts two such cases.

Slide the four points below: the inscribed angles on the same side of AB stay equal, and on opposite sides they add up to 180^\circ.

What are the rules for chords, tangents and secants?

The perpendicular from the center to a chord cuts the chord in half, a tangent is perpendicular to the radius, and the products of the pieces of crossing chords or secants are equal; all three come from the right triangles that the radius makes.

• Chord and center. If a chord is at distance d from the center, its length is 2√(r^2-d^2). With r=5 and d=3 it is 2√(25-9)=8, the 3–4–5 triangle of the Pythagorean theorem. Equal chords are equally far from the center, and the closer a chord is to the center the longer it is; the diameter (d=0) is the longest.
• Two tangents. From a point P outside the circle there are two tangents, and the two tangent segments have equal length √(OP^2-r^2). With r=5 and OP=13 the length is √(169-25)=12, the 5–12–13 triangle.
• Crossing chords. If two chords AB and CD cross at P inside the circle, then PA· PB=PC· PD. For PA=3, PB=8 and PC=4 we get PD=6, since 3·8=4·6=24.
• Secants and tangents. From a point P outside, two secants give PA· PB=PC· PD, and a tangent PT and a secant give PT^2=PA· PB. For the tangent 12 above and the secant through the center, PA=13-5=8 and PB=13+5=18, and 8·18=144=12^2.

All of these say one thing: the number |PO|^2-r^2, the power of the point P, is negative inside the circle, zero on it and positive outside, and for every line through P that meets the circle at A and B the product PA· PB equals its absolute value. Radicals like 2√(r^2-d^2) are often best left in simplified form.

What is the equation of a circle?

The circle with center (h,k) and radius r is the set of points (x,y) with (x-h)^2+(y-k)^2=r^2, because that says the squared distance to the center is r^2. For the center (3,-2) and r=5 it is (x-3)^2+(y+2)^2=25.

Expanding (see expanding brackets) gives the general form x^2+y^2+Dx+Ey+F=0 with D=-2h, E=-2k and F=h^2+k^2-r^2. Our circle is x^2+y^2-6x+4y-12=0. To go back, complete the square in x and in y: x^2-6x=(x-3)^2-9 and y^2+4y=(y+2)^2-4, so (x-3)^2+(y+2)^2=12+9+4=25. The general form describes a real circle only when r^2=(D^2)/(4)+(E^2)/(4)-F is positive.

Where is a point? Compare the squared distance (x-h)^2+(y-k)^2 with r^2: smaller means inside, equal means on the circle, larger means outside. For the circle above, (0,0) gives 13<25 (inside), (8,-2) gives 25 (on it) and (9,5) gives 85>25 (outside).

Through three points. Three points that are not on one line lie on exactly one circle, the circumcircle of the triangle they form. The circle through (0,0), (6,0) and (0,8) has center (3,4), the midpoint of the hypotenuse, and r=5: (x-3)^2+(y-4)^2=25.

The unit circle. For r=1 and the center at the origin, x^2+y^2=1, and the points are (\cos t,\sin t). Rational points also exist: (\frac35,\frac45) is on it because 3^2+4^2=5^2, the same Pythagorean triple in disguise.

How can a line, a point or another circle meet a circle?

Compare a distance with the radius: a line that is farther than r from the center misses the circle, one at distance exactly r is a tangent, and one closer than r is a secant; two circles are decided the same way by the distance d between their centers.

A line and a circle. Let d be the distance from the center to the line. If d>r there is no common point, if d=r there is exactly one (the line is a tangent, and the point of contact is where the perpendicular from the center lands), and if d<r there are two. Algebraically, substituting the line into (x-h)^2+(y-k)^2=r^2 gives a quadratic equation, and these are its cases: no solution, one double solution, two solutions. For the circle x^2+y^2=25 the line y=3 is at distance 3 and meets it at (±4,3), the line x=5 at distance 5 touches it at (5,0), and the line x=6 misses it.

Two circles with radii r_1,r_2 and center distance d lie in one of these ways, where \delta=|r_1-r_2| is the difference of the radii:

With radii 5 and 3 these are the distances d=9, 8, 6, 2 and 1. The tool uses exact fractions, so a tangent is recognized as exactly a tangent. It gives the intersection points in the exact form a+b√ m, and the tangent points from an outside point.

How do you work with circles in code?

Keep the squared radius, not the radius, and every question about a circle with whole-number or fraction data becomes an exact comparison of squared distances; use math.pi only for lengths and areas at the end.

from fractions import Fraction as F
from math import pi, sin, cos, radians

def power(c, p):                     # |PC|² − r²: negative inside, 0 on the circle, positive outside
    (h, k, r2), (x, y) = c, p
    return (x - h) ** 2 + (y - k) ** 2 - r2

def meets(c, a, b, k):               # the line a·x + b·y = k against the circle: none / tangent / secant
    h, kk, r2 = c
    d2 = F(a * h + b * kk - k) ** 2 / (a * a + b * b)   # squared distance from the center to the line
    return "none" if d2 > r2 else "tangent" if d2 == r2 else "secant"

def sector(r, deg):                  # arc length, sector area, segment area
    t = radians(deg)
    return r * t, r * r * t / 2, r * r * (t - sin(t)) / 2

>>> c = (3, -2, 25)                  # center (3, -2) and the square of the radius, 25
>>> power(c, (0, 0)), power(c, (8, -2)), power(c, (9, 5))
(-12, 0, 60)
>>> meets((0, 0, 25), 0, 1, 3), meets((0, 0, 25), 1, 0, 5), meets((0, 0, 25), 1, 0, 6)
('secant', 'tangent', 'none')
>>> round(2 * pi * 7, 4), round(pi * 7 ** 2, 4)
(43.9823, 153.938)
>>> 22 / 7, 355 / 113, pi
(3.142857142857143, 3.1415929203539825, 3.141592653589793)
>>> sector(10, 90)
(15.707963267948966, 78.53981633974483, 28.539816339744828)
>>> sin(pi)                          # should be 0 ...
1.2246467991473532e-16
>>> cos(radians(3)) ** 2 + sin(radians(3)) ** 2 == 1   # ... and this should be exactly 1
False

const r = 7
2 * Math.PI * r                          // 43.982297150257104
Math.PI * r ** 2                         // 153.93804002589985
Math.sin(Math.PI)                        // 1.2246467991473532e-16, not 0
const power = ([h, k, r2], [x, y]) => (x - h) ** 2 + (y - k) ** 2 - r2
power([3, -2, 25], [9, 5])               // 60: outside, exact for whole numbers

#include <iostream>
#include <cstdio>
#include <cmath>
using namespace std;

const double PI = acos(-1.0);

// |PC|^2 - r^2: negative inside, 0 on the circle, positive outside
int power(int h, int k, int r2, int x, int y) {
  return (x - h) * (x - h) + (y - k) * (y - k) - r2;
}

// the line a x + b y = c against the circle with center (h, k) and radius^2 = r2:
// compare d^2 = (a h + b k - c)^2 / (a^2 + b^2) with r2, in whole numbers
const char* meets(int h, int k, int r2, int a, int b, int c) {
  int num = (a * h + b * k - c) * (a * h + b * k - c);
  int rhs = r2 * (a * a + b * b);
  if (num > rhs) return "none";
  if (num == rhs) return "tangent";
  return "secant";
}

int main() {
  cout << power(3, -2, 25, 0, 0) << " " << power(3, -2, 25, 8, -2) << " " << power(3, -2, 25, 9, 5) << endl;
  cout << meets(0, 0, 25, 0, 1, 3) << " " << meets(0, 0, 25, 1, 0, 5) << " " << meets(0, 0, 25, 1, 0, 6) << endl;
  printf("%.4f %.4f\n", 2 * PI * 7, PI * 7 * 7);                      // circumference and area, r = 7
  printf("%.10f %.10f %.10f\n", 22.0 / 7, 355.0 / 113, PI);
  double r = 10, t = 90 * PI / 180;                                    // a sector of 90 degrees
  printf("%.4f %.4f %.4f\n", r * t, r * r * t / 2, r * r * (t - sin(t)) / 2);   // arc, sector, segment
  printf("%d\n", fabs(sin(PI)) < 1e-12);                              // sin(pi) is 1.2e-16, not 0
  return 0;
}

/* output:
-12 0 60
secant tangent none
43.9823 153.9380
3.1428571429 3.1415929204 3.1415926536
15.7080 78.5398 28.5398
1
*/

The pitfalls, in order of how often they bite:

Where does the study of circles come from?

The circle is the oldest curve people studied, and the effort to find π is among the longest in mathematics.

• c. 1800 BCE. Babylonian texts take the circumference to be three times the diameter, and later 3\frac18.
• c. 1650 BCE. Problem 50 of the Rhind Papyrus says that a round field of diameter 9 has the area of a square of side 8, which is π≈\left((16)/(9)\right)^2≈3.16.
• c. 300 BCE. Euclid's Elements, Book III, is the theory of circles: chords, tangents, the inscribed angle (III.20), the right angle in a semicircle (III.31) and the crossing chords (III.35, III.36). Book XII shows that the areas of circles are in the ratio of the squares of their diameters.
• c. 250 BCE. Archimedes, in Measurement of a Circle, squeezes the circle between polygons with 96 sides and proves 3(10)/(71)<π<3\frac17; he also shows that the area of a circle equals that of a right triangle with legs r and C.
• c. 480 CE. Zu Chongzhi in China finds (355)/(113), correct to six decimals, a record that lasted nearly a thousand years.
• c. 1400. Madhava of Sangamagrama finds an infinite series for π, the first of its kind.
• 1706. William Jones uses the Greek letter π for the ratio, and Euler makes the symbol standard in the 1730s.
• 1761 and 1882. Lambert proves that π is irrational and Lindemann that it is transcendental, so the circle cannot be squared with compass and straightedge.
• 1873. James Thomson coins the word radian for an idea Roger Cotes had described in 1714.

What are the common mistakes with circles?

The most common mistakes with circles are the nine below, each with the correct statement.

Practice: test your understanding

These questions mix everything above. A wrong answer costs nothing here: read the hint and try again.

Summary: circles at a glance

• Definition: all points at distance r from the center O; diameter d=2r is the longest chord; all circles are similar.
• Measures: C=2π r=π d, A=π r^2; π is irrational and transcendental.
• Arcs and sectors: arc =(θ)/(360)·2π r=rθ, sector =\frac12r^2θ, segment =\frac12r^2(θ-\sinθ); 180^\circ=π rad.
• Angles: inscribed =\frac12 central; 90^\circ in a semicircle; opposite angles of a cyclic quadrilateral sum to 180^\circ; a tangent is perpendicular to the radius.
• Lengths: chord =2√(r^2-d^2); tangent length =√(OP^2-r^2); crossing chords PA· PB=PC· PD; PT^2=PA· PB.
• Equation: (x-h)^2+(y-k)^2=r^2, or x^2+y^2+Dx+Ey+F=0 with r^2=(D^2+E^2)/(4)-F.
• Position: compare a distance with r (point, line) or with r_1± r_2 (two circles).
• Code: keep r^2 and compare squared distances exactly; math.pi and radians only at the end.

Glossary: key terms

Circle

The set of all points in a plane that are the same distance from a fixed point called the center.

Radius

A segment from the center to a point on the circle, or its length, the distance that defines the circle.

Diameter

A chord that passes through the center, with length twice the radius, the longest chord of the circle.

Chord

A segment that joins two points on a circle.

Arc

The part of a circle between two of its points.

Sector

The region of a disk between two radii and the arc between their endpoints.

Segment

The region of a disk between a chord and the arc it cuts off.

Tangent

A line that touches a circle at exactly one point and is perpendicular to the radius there.

Secant

A line that meets a circle in two points.

Circumference

The length of the circle itself, equal to pi times the diameter.

Pi

The ratio of the circumference of any circle to its diameter, about 3.14159, an irrational and transcendental number.

Central angle

An angle whose vertex is the center of the circle, equal in degrees to the arc it cuts off.

Inscribed angle

An angle whose vertex is on the circle and whose sides are chords, equal to half the central angle on the same arc.

Radian

The central angle whose arc is as long as the radius, so that 180 degrees equals pi radians.

Cyclic quadrilateral

A quadrilateral whose four vertices lie on one circle, so that opposite angles add up to 180 degrees.

Frequently asked questions

References

1. Thomas L. Heath (translator) (1908). The Thirteen Books of Euclid's Elements (2nd ed.), Book III (circles) and Book XII, proposition 2. Cambridge University Press.
2. Thomas L. Heath (translator) (1897). The Works of Archimedes, Measurement of a Circle. Cambridge University Press.
3. Petr Beckmann (1971). A History of Pi. St. Martin's Press.
4. H. S. M. Coxeter (1969). Introduction to Geometry (2nd ed.), chapter 2 (circles and angles). Wiley.
5. H. S. M. Coxeter and Samuel L. Greitzer (1967). Geometry Revisited, chapter 2 (the circle). Mathematical Association of America.
6. Python Software Foundation. The Python Standard Library: math, mathematical functions. docs.python.org. https://docs.python.org/3/library/math.html.

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