Angles: Measuring, Types, Pairs and Parallel Lines

What is an angle?

An angle is the figure made by two rays, called its sides or arms, that start at the same point, called its vertex; its size says how far you must turn one ray to land on the other. Open a door, and the angle between the door and the frame grows; swing a clock hand, and it sweeps an angle. The size does not depend on how long the rays are drawn, only on the turn between them.

Naming. An angle is written with three letters and the angle sign, with the vertex in the middle: \angle ABC has its vertex at B and its sides along BA and BC. A single letter (\angle B) is enough when no other angle shares the vertex, and a Greek letter (θ, α) is often used for the size itself.

Direction. In mathematics an angle is measured from a starting ray, the initial side, to a terminal side, and a turn counterclockwise is positive and clockwise is negative. So -90^\circ is a quarter turn clockwise, and 450^\circ is a full turn plus a quarter: it ends in the same place as 90^\circ. Euclid, in the Elements, described an angle as the inclination of two lines to one another in a plane, and for centuries geometry used only angles below a straight angle; the idea of an angle as an amount of turning, which can be added, negative or larger than a full turn, is what trigonometry, rotations and navigation need.

Why we measure. Two angles can be compared and added: placing them side by side along a common ray gives the sum. To say how big they are we need a unit, and the next sections describe the units, the names of angles by size, and the rules that relate angles to each other and to lines, polygons, circles, clocks and vectors.

How are angles measured in degrees, radians and turns?

A full turn is 360 degrees, 2π radians, 400 gradians or 1 turn, and every unit is a way of cutting the same turn into equal parts.

Degrees. The degree is (1)/(360) of a turn. The number 360 comes from the Babylonians, who counted in sixties and whose year was close to 360 days; it is divisible by 24 numbers, which makes halves, thirds, quarters, fifths, sixths and eighths of a turn whole numbers. A degree is cut into 60 minutes (1^\circ=60') and a minute into 60 seconds (1'=60''), as an hour is: 12^\circ30'15''=12+(30)/(60)+(15)/(3600)=12.504(16 repeating)^\circ. Latitude and longitude on maps are written this way.

Radians. One radian is the angle at the center of a circle that cuts an arc as long as the radius. Since the circumference is 2π r, a full turn is 2π radians, and an angle of θ radians cuts an arc of length rθ (see arcs and sectors for the circle). Radians make the formulas of calculus clean (the derivative of \sin x is \cos x only when x is in radians), so mathematicians and programming languages use them by default.

Converting. Since 180^\circ=π radians, radians=degrees·(π)/(180),\qquaddegrees=radians·(180)/(π). Common angles: 30^\circ=(π)/(6), 45^\circ=(π)/(4), 60^\circ=(π)/(3), 90^\circ=(π)/(2), 120^\circ=(2π)/(3), 135^\circ=(3π)/(4), 180^\circ=π, 270^\circ=(3π)/(2), 360^\circ=2π. One radian is (180)/(π)≈57.2958^\circ, and one degree is about 0.017453 radians.

Type an angle in any unit below, or pick one of the common angles, and read it in all of them, with its name and its complement, supplement and explement (the angle that completes it to 360^\circ).

What are the types of angles and the pairs of angles?

Angles are named by their size, and two angles are named as a pair when their sizes are tied by a simple rule.

Pairs. - Complementary angles add up to 90^\circ: the complement of 35^\circ is 55^\circ. Only acute angles have a complement. - Supplementary angles add up to 180^\circ: the supplement of 112^\circ is 68^\circ. The word is supplement, not "complement", and the memory aid is C before S as 90 before 180. - Explementary (or conjugate) angles add up to 360^\circ. - Adjacent angles share a vertex and one side and do not overlap. If their outer sides form a straight line they are a linear pair, and a linear pair is supplementary. - Vertical angles are the opposite angles where two lines cross, and vertical angles are equal. Proof: if the angles in order around the crossing are a, b, a', b', then a+b=180^\circ and b+a'=180^\circ (both are linear pairs), so a=a'; the same argument shows b=b'.

Two lines that cross at a right angle are perpendicular. Around a point the angles add up to 360^\circ, and along a line to 180^\circ; most of the "find the missing angle" exercises are these two facts and the pairs above, used over and over.

Which angles are equal when a line crosses two parallel lines?

A line, called a transversal, that crosses two parallel lines makes eight angles, and they take only two values: an angle α and its supplement 180^\circ-α. Number them 1 to 8, the first four at the upper line and the other four at the lower line, each time in the order upper left, upper right, lower left, lower right. The interior angles are the four between the parallel lines (3, 4, 5, 6) and the exterior angles the other four.

The theorem and its converse. If the two lines are parallel, corresponding angles are equal, and if a transversal makes equal corresponding angles, the two lines are parallel. This is the form in which Euclid's parallel postulate is used (Elements I.27 and I.29), and it is a way to prove lines parallel. The other pairs follow from the corresponding angles and the vertical angles: angle 3 is vertical to 2, and 2 corresponds to 6, so 3=6, the alternate interior pair. The same chain gives the others.

Using it. Given one angle, all eight are known: if \angle2=65^\circ then \angle1=\angle4=\angle5=\angle8=115^\circ and \angle3=\angle6=\angle7=65^\circ. The widget below lets you move the transversal and highlights the pairs of each kind with their values. The same reasoning proves that the angles of a triangle add up to 180^\circ: draw a parallel to one side through the opposite vertex (see the angle sum of a triangle).

What is the sum of the angles of any polygon?

The interior angles of a polygon with n sides add up to (n-2)·180^\circ, and its exterior angles add up to 360^\circ, whatever the shape.

Why (n-2)·180^\circ. Choose one vertex and draw all the diagonals from it. They cut the polygon into n-2 triangles (a quadrilateral into 2, a pentagon into 3), and the angles of the triangles together make up exactly the interior angles of the polygon. Each triangle contributes 180^\circ. So sum=(n-2)·180^\circ: 180^\circ, 360^\circ, 540^\circ, 720^\circ, … for n=3,4,5,6,… (The triangle gives 180^\circ and the quadrilateral 360^\circ, which are the facts of the triangle and the quadrilateral.) This works for convex polygons and also for concave ones.

Exterior angles. At each vertex the exterior angle is the turn you make when you walk around the polygon: it is 180^\circ minus the interior angle. After going once around you have turned a full turn, so the exterior angles add up to 360^\circ, for any convex polygon. Check with the interior sum: n·180^\circ-(n-2)·180^\circ=360^\circ.

Regular polygons. If all the angles are equal, each interior angle is ((n-2)·180^\circ)/(n) and each exterior angle is (360^\circ)/(n): 60^\circ for a triangle, 90^\circ for a square, 108^\circ for a pentagon, 120^\circ for a hexagon, 135^\circ for an octagon. Given an angle you can find n: a regular polygon with exterior angle 24^\circ has (360)/(24)=15 sides. A polygon has (n(n-3))/(2) diagonals.

Tiling. Copies of a regular polygon fit around a point without a gap exactly when the interior angle divides 360^\circ. That happens for the triangle (6 around a point), the square (4) and the hexagon (3) only, which is why bathroom tiles are squares, triangles or hexagons, and why a honeycomb is hexagonal. Slide the number of sides below.

How is an angle related to an arc of a circle?

An angle at the center of a circle is the fraction of the turn that its arc takes, and every length and area of the slice is that fraction of the whole. If the central angle is θ radians, the arc has length rθ, the sector has area \frac12r^2θ and the chord joining the ends has length 2r\sin\fracθ2. That the arc divided by the radius gives the angle is the definition of the radian.

An angle with its vertex on the circle, between two chords, is an inscribed angle. For the chord AB with a central angle of θ, the inscribed angle at a point P outside the arc from A to B is always \fracθ2, wherever P moves, and at a point on that arc it is 180^\circ-\fracθ2. When AB is a diameter (θ=180^\circ) the inscribed angle is always 90^\circ (Thales). The proofs and more theorems are in the angle theorems of a circle.

The widget below measures the angle at P from the coordinates of the points, with no theorem, and compares it with half the central angle: they agree wherever you drag P. It also gives the arc, sector and chord for a radius you type.

How do the hands of a clock make angles?

The minute hand turns 6^\circ every minute and the hour hand 0.5^\circ every minute, so at h hours and m minutes the angle between them is |30h-5.5m|, and if that is more than 180^\circ the angle is 360^\circ minus it. Measure each hand clockwise from the 12. The hour hand is at 30h+0.5m degrees: 30^\circ for each hour, plus the share of the next hour that has passed. The minute hand is at 6m degrees. Their difference is |30h+0.5m-6m|=|30h-5.5m|. Examples: at 3:00 it is |90-0|=90^\circ; at 2:30 it is |60-165|=105^\circ; at 12:30 it is |0-165|=165^\circ; at 6:00 it is 180^\circ.

How often. The minute hand gains 5.5^\circ a minute on the hour hand, so it laps it every (360)/(5.5)=65(5)/(11) minutes: the hands overlap 11 times in twelve hours, first at 1:05:27\frac311 (and not at 1:05 exactly), then at 2:10:54\frac611, and so on, and the eleventh overlap is at 12:00 again. The relative angle runs through all values from 0^\circ to 360^\circ eleven times in twelve hours, so any angle strictly between 0^\circ and 180^\circ occurs 22 times (twice per lap, once with each hand ahead): the hands are at right angles 22 times in twelve hours, 44 times a day, and exactly opposite 11 times. The right angles are at 3:00 and 9:00 and at twenty other, awkward times such as 12:16:21(9)/(11).

Move the hands below, and choose an angle to see all the times at which the hands make it, to the exact fraction of a second (click a time to set the clock).

How do compass bearings relate to the angles of mathematics?

A compass bearing is an angle measured clockwise from north, while a mathematical angle is measured counterclockwise from east, so the two are related by bearing =90^\circ-θ (modulo 360^\circ). Mathematics puts angle 0^\circ along the positive x-axis (east) and turns toward y (north). Navigators and surveyors start at north and turn the way a compass needle's markings run, clockwise. The conversion is a reflection and a shift:

The sixteen points of the compass are 22.5^\circ apart: N, NNE, NE, ENE, E, ESE, SE, SSE, S, SSW, SW, WSW, W, WNW, NW, NNW. A bearing is usually written with three digits (045^\circ), and a back bearing (the reverse direction) is the bearing plus or minus 180^\circ. Programs that draw on screens add a third convention: screen y grows downward, so a counterclockwise angle needs its sine negated.

The second mode of the widget at the end of the next section converts between the two systems and gives the compass name.

How do you find the angle between two lines or two vectors?

Between two vectors the angle comes from the dot product, \cosθ=(\mathbf a·\mathbf b)/(|\mathbf a||\mathbf b|), and between two lines from their slopes, \tanθ=\left|(m_1-m_2)/(1+m_1m_2)\right|.

Vectors. For \mathbf a=(a_1,a_2) and \mathbf b=(b_1,b_2) the dot product is a_1b_1+a_2b_2=|\mathbf a||\mathbf b|\cosθ and the cross product a_1b_2-a_2b_1=|\mathbf a||\mathbf b|\sinθ. The sign of the dot product tells the kind of angle: positive for acute, zero for a right angle, negative for obtuse. Dividing gives \tanθ, which is why the angle can be computed as θ=\operatornameatan2\bigl(|a_1b_2-a_2b_1|, a_1b_1+a_2b_2\bigr) with no division and no square roots. Many programs instead write θ=\arccos(\mathbf a·\mathbf b)/(|\mathbf a||\mathbf b|), which is right on paper but loses precision in floating point: the cosine of a tiny angle is 1 to sixteen digits, so for θ=10^-8 the arc cosine gets \cosθ=1 exactly and answers 0, and near 180^\circ it answers exactly 180^\circ (see floating-point numbers). Without the absolute value, the same expression is the directed angle in (-180^\circ,180^\circ], positive counterclockwise.

Lines. A line with slope m makes the angle \arctan m with the x-axis, so the angle between two lines is θ_1-θ_2, and the formula for the tangent of a difference gives \tanθ=(m_1-m_2)/(1+m_1m_2). The lines are perpendicular exactly when m_1m_2=-1, and parallel when m_1=m_2. For example the lines with slopes 1 and -1 meet at 90^\circ, and slopes 0 and √3 at 60^\circ. A vertical line has no slope: use its direction vector instead.

Try two vectors below: the presets include an angle of 10^-8 radians, where the two formulas differ.

How do you work with angles in Python, JavaScript and C++?

Programs measure angles in radians, so convert degrees first, normalize results into a fixed range, and use atan2 rather than acos for the angle between vectors. The trigonometric functions of every language (sin, cos, tan) take radians, so sin(30) is the sine of 30 radians, not of 30^\circ. The remainder operator needs care: Python's % always gives a result with the sign of the divisor, so -30 % 360 is 330; JavaScript's and C++'s keep the sign of the dividend, so -30 % 360 is -30 and you add 360 and take the remainder again (C++ uses fmod for decimals). And since a radian measure is a floating-point number, sin(π) is not 0 but about 1.2×10^-16. The Python program below normalizes angles, finds the angle between vectors both ways, the angle between clock hands, the sum of the angles of polygons and a compass bearing. The JavaScript and C++ programs do the same in their way.

import math
from fractions import Fraction


def normalize(deg):
    """The same direction as an angle in [0, 360): Python's % already gives a non-negative result."""
    return deg % 360


def angle_between(a, b):
    """The angle between two 2D vectors in degrees, from atan2: accurate for every angle."""
    cross = a[0] * b[1] - a[1] * b[0]
    dot = a[0] * b[0] + a[1] * b[1]
    return math.degrees(math.atan2(abs(cross), dot))


def angle_between_acos(a, b):
    c = (a[0] * b[0] + a[1] * b[1]) / (math.hypot(*a) * math.hypot(*b))
    return math.degrees(math.acos(max(-1.0, min(1.0, c))))


def clock_angle(h, m):
    """The angle between the hands at h:m, from 0 to 180 degrees."""
    d = abs(30 * (h % 12) - 5.5 * m) % 360
    return min(d, 360 - d)


def interior_sum(n):
    return (n - 2) * 180


def bearing(math_angle):
    """Compass bearing (clockwise from north) of a direction given counterclockwise from east."""
    return (90 - math_angle) % 360


print(normalize(-30), normalize(725), normalize(360))
print(angle_between((1, 0), (1, 1)), angle_between((1, 0), (0, 2)), angle_between((1, 0), (-1, 0)))
print(angle_between((1, 0), (1, 1e-8)), angle_between_acos((1, 0), (1, 1e-8)))
print(clock_angle(3, 0), clock_angle(2, 30), clock_angle(12, 30), clock_angle(6, 0))
print([interior_sum(n) for n in (3, 4, 5, 6, 10)], [(n - 2) * 180 / n for n in (3, 4, 5, 6, 8, 12)])
print(math.degrees(math.pi / 6), math.radians(180), Fraction(150, 180))
print(bearing(0), bearing(90), bearing(135), bearing(-90))
print(math.sin(math.radians(30)), math.cos(math.radians(60)), math.sin(math.pi))
# 330 5 0
# 45.0 90.0 180.0
# 5.729577951308232e-07 0.0
# 90.0 105.0 165.0 180.0
# [180, 360, 540, 720, 1440] [60.0, 90.0, 108.0, 120.0, 135.0, 150.0]
# 29.999999999999996 3.141592653589793 5/6
# 90 0 315 180
# 0.49999999999999994 0.5000000000000001 1.2246467991473532e-16

const toDeg = (r) => (r * 180) / Math.PI
const normalize = (d) => ((d % 360) + 360) % 360          // JavaScript's % keeps the sign of d, so fix it
const angleBetween = (a, b) => toDeg(Math.atan2(Math.abs(a[0] * b[1] - a[1] * b[0]), a[0] * b[0] + a[1] * b[1]))
const angleBetweenAcos = (a, b) => toDeg(Math.acos((a[0] * b[0] + a[1] * b[1]) / (Math.hypot(...a) * Math.hypot(...b))))
;[-30 % 360, normalize(-30), normalize(725), normalize(360)]       // [-30, 330, 5, 0]
;[angleBetween([1, 0], [1, 1]), angleBetween([1, 0], [0, 2])]      // [45, 90]
angleBetween([1, 0], [1, 1e-8])                                    // 5.729577951308232e-7
angleBetweenAcos([1, 0], [1, 1e-8])                                // 0: the arc cosine lost the angle
angleBetween([1, 0], [-1, 1e-8])                                   // 179.9999994270422
angleBetweenAcos([1, 0], [-1, 1e-8])                               // 180
const clockAngle = (h, m) => { const d = Math.abs(30 * (h % 12) - 5.5 * m) % 360; return Math.min(d, 360 - d) }
;[clockAngle(3, 0), clockAngle(2, 30), clockAngle(12, 30)]         // [90, 105, 165]
const bearing = (mathAngle) => normalize(90 - mathAngle)
;[bearing(0), bearing(90), bearing(135)]                           // [90, 0, 315]
;[Math.sin(Math.PI / 6), Math.sin(30)]                             // [0.49999999999999994, -0.9880316240928618]: sin takes radians
;[Math.atan2(1, 1), Math.atan2(-1, -1)].map(toDeg)                 // [45, -135]: atan2 knows the quadrant, atan does not

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

const double PI = 3.14159265358979323846;

double degrees(double rad) { return rad * 180.0 / PI; }

double angleBetween(double ax, double ay, double bx, double by) {
  double cross = ax * by - ay * bx;
  double dot = ax * bx + ay * by;
  return degrees(atan2(fabs(cross), dot));
}

double clockAngle(int h, int m) {
  double d = fabs(30.0 * (h % 12) - 5.5 * m);
  while (d >= 360) d -= 360;
  return d > 180 ? 360 - d : d;
}

int main() {
  printf("%.1f\n", angleBetween(1, 0, 1, 1));
  printf("%.1f\n", angleBetween(1, 0, 0, 2));
  printf("%.1f %.1f %.1f\n", clockAngle(3, 0), clockAngle(2, 30), clockAngle(12, 30));
  printf("%d\n", (5 - 2) * 180);                               // the angles of a pentagon
  double a = -30;
  printf("%.1f\n", a - 360.0 * floor(a / 360.0));              // for decimals: subtract whole turns, rounding down
  return 0;
}

/* output:
45.0
90.0
90.0 105.0 165.0
540
330.0
*/

Where do the degree and the angle come from?

The 360^\circ circle is Babylonian, the theory of angles is Greek, and the radian is a modern idea.

• By the first millennium BC. Babylonian astronomers, counting in base 60, divided the circle into 360 parts, close to the number of days in the year, and their sixtieths are the minutes and seconds we still use for angles and for time.
• About 300 BC. Euclid's Elements defines the plane angle and the right angle, proves that vertical angles are equal, states the parallel postulate and proves that a transversal of parallel lines makes equal alternate angles (Book I).
• About 150 BC. Hipparchus compiled the first table of chords, the ancestor of the sine table, dividing the circle into 360^\circ in the Greek world; Ptolemy used it in the Almagest (about 150 AD).
• About 500 AD. Indian astronomers, such as Aryabhata, replaced chords by half-chords, the sine, which led to the trigonometry used today.
• 1714. Roger Cotes described measuring an angle by the arc it cuts on a circle of radius 1, the idea of the radian, without giving it a name.
• 1790s. The French metric system proposed dividing the right angle into 100 grades (gradians, now gons), still used in surveying.
• 1873. The name radian appeared in print, credited to the Irish engineer James Thomson, the brother of Lord Kelvin.
• 1900s–today. Angles became rotations: the unit of calculus, of complex numbers (e^iθ), of computer graphics and robotics, and of the turn used for frequency.

What are the common mistakes with angles?

The most common mistakes with angles 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: angles at a glance

• Angle: the turn between two rays at a vertex; counterclockwise is positive; 360^\circ=2π rad =400 gradians =1 turn.
• Convert: radians = degrees ·(π)/(180); 1 rad ≈57.2958^\circ; degrees, minutes, seconds are sixtieths.
• Types: acute <90^\circ, right =90^\circ, obtuse between 90^\circ and 180^\circ, straight =180^\circ, reflex above.
• Pairs: complementary 90^\circ, supplementary 180^\circ, vertical angles equal, a linear pair supplementary.
• Parallel lines: corresponding, alternate and vertical pairs equal; co-interior and linear pairs add up to 180^\circ; equal corresponding angles prove the lines parallel.
• Polygons: interior sum (n-2)·180^\circ, exterior sum 360^\circ, regular interior ((n-2)·180^\circ)/(n); only the triangle, square and hexagon tile.
• Clock: |30h-5.5m|; 11 overlaps and 22 right angles in twelve hours.
• Directions: bearing =90^\circ-θ; sixteen compass points 22.5^\circ apart.
• Vectors and lines: \operatornameatan2(|cross|,dot), slopes with \tanθ=\left|(m_1-m_2)/(1+m_1m_2)\right|.
• Code: radians in trigonometric functions, normalize with \% (watch the sign), prefer atan2 to acos.

Glossary: key terms

Angle

The figure formed by two rays with a common endpoint, or the amount of turning from one ray to the other.

Vertex

The common endpoint of the two rays that form an angle.

Degree

A unit of angle equal to one 360th of a full turn, divided into 60 minutes of 60 seconds each.

Complementary angles

Two angles whose sizes add up to 90 degrees.

Supplementary angles

Two angles whose sizes add up to 180 degrees; a linear pair is supplementary.

Vertical angles

The opposite angles formed where two lines cross; they are equal.

Transversal

A line that crosses two or more other lines, making pairs of corresponding, alternate and co-interior angles.

Exterior angle of a polygon

The turn at a vertex when walking around the polygon, 180 degrees minus the interior angle; the exterior angles add up to 360 degrees.

Bearing

A direction measured as an angle clockwise from north, usually written with three digits.

Protractor

A semicircular or circular tool with a scale of degrees for measuring and drawing angles.

Frequently asked questions

References

1. Euclid, translated by Thomas L. Heath (1956). Euclid's Elements, Book I (definitions of angles, propositions 13, 15, 27, 29, 32). Dover.
2. Roger Cotes (1722). Harmonia mensurarum. Cambridge.
3. H. S. M. Coxeter, Samuel L. Greitzer (1967). Geometry Revisited, chapters on angles and circles. Mathematical Association of America.
4. Carl B. Boyer, Uta C. Merzbach (2011). A History of Mathematics (3rd ed.). Wiley.
5. IEEE (2019). IEEE Standard for Floating-Point Arithmetic (IEEE 754-2019). IEEE.

Related articles

• Triangles: Types, Angles, Area, Centers and Congruence: Triangles explained: types by sides and angles, the 180° angle sum, the triangle inequality, area and Heron's formula, centers, congruence and the law of sines.
• Quadrilaterals: Types, Properties, Perimeter and Area: Quadrilaterals explained: square, rectangle, rhombus, parallelogram, kite and trapezoid, their properties, perimeter, area and coordinates, with practice.
• Circles: Circumference, Area, Angles and Equations: Circles explained: pi, circumference and area, arcs and sectors, angle theorems, chords and tangents, the equation of a circle, and how lines and circles meet.
• Floating-Point Numbers: How floating-point numbers work: the IEEE 754 bits, spacing and rounding, NaN and infinity, accumulated errors, cancellation, comparisons, and which type to use.