Length of an arc between two angles
Length of an arc between two angles is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
- Task
Write a method (function, procedure etc.) in your language which calculates the length of the major arc of a circle of given radius between two angles.
In this diagram the major arc is colored green (note: this website leaves cookies).
Illustrate the use of your method by calculating the length of the major arc of a circle of radius 10 units, between angles of 10 and 120 degrees.
11l
F arc_length(r, angleA, angleB)
R (360.0 - abs(angleB - angleA)) * math:pi * r / 180.0
print(arc_length(10, 10, 120))- Output:
43.6332
Action!
INCLUDE "H6:REALMATH.ACT"
PROC ArcLength(REAL POINTER r,a1,a2,len)
REAL tmp1,tmp2,r180,r360,pi
IntToReal(360,r360)
IntToReal(180,r180)
ValR("3.14159265",pi)
RealAbsDiff(a1,a2,tmp1) ;tmp1=abs(a1-a2)
RealSub(r360,tmp1,tmp2) ;tmp2=360-abs(a1-a2)
RealMult(tmp2,pi,tmp1) ;tmp1=(360-abs(a1-a2))*pi
RealMult(tmp1,r,tmp2) ;tmp2=(360-abs(a1-a2))*pi*r
RealDiv(tmp2,r180,len) ;len=(360-abs(a1-a2))*pi*r/180
RETURN
PROC Main()
REAL r,a1,a2,len
Put(125) PutE() ;clear screen
Print("Length of arc: ")
IntToReal(10,r)
IntToReal(10,a1)
IntToReal(120,a2)
ArcLength(r,a1,a2,len)
PrintR(len)
RETURN- Output:
Screenshot from Atari 8-bit computer
Length of arc: 43.63323122
Ada
with Ada.Text_Io;
with Ada.Numerics;
procedure Calculate_Arc_Length is
use Ada.Text_Io;
type Angle_Type is new Float range 0.0 .. 360.0; -- In degrees
type Distance is new Float range 0.0 .. Float'Last; -- In units
function Major_Arc_Length (Angle_1, Angle_2 : Angle_Type;
Radius : Distance)
return Distance
is
Pi : constant := Ada.Numerics.Pi;
Circumference : constant Distance := 2.0 * Pi * Radius;
Major_Angle : constant Angle_Type := 360.0 - abs (Angle_2 - Angle_1);
Arc_Length : constant Distance :=
Distance (Major_Angle) / 360.0 * Circumference;
begin
return Arc_Length;
end Major_Arc_Length;
package Distance_Io is new Ada.Text_Io.Float_Io (Distance);
Arc_Length : constant Distance := Major_Arc_Length (Angle_1 => 10.0,
Angle_2 => 120.0,
Radius => 10.0);
begin
Put ("Arc length : ");
Distance_Io.Put (Arc_Length, Exp => 0, Aft => 4);
New_Line;
end Calculate_Arc_Length;
- Output:
Arc length : 43.6332
Ada can be simple and concise:
with ada.float_text_io; use ada.float_text_io; -- for put()
with ada.numerics; use ada.numerics; -- for pi
with ada.text_io; use ada.text_io; -- for new_line
procedure arc_length_simple is
function arc_length(radius, deg1, deg2: Float) return Float is
((360.0 - abs(deg1 - deg2)) * pi * radius / 180.0);
begin
put(arc_length(10.0, 120.0, 10.0), fore=>0, aft=>15, exp=>0);
new_line;
end arc_length_simple;
Ada can do automatic compile-time checking of types. In this example, degrees and radians cannot be accidentally mixed:
with ada.float_text_io; use ada.float_text_io; -- for put()
with ada.numerics; use ada.numerics; -- for pi
with ada.text_io; use ada.text_io; -- for new_line
procedure arc_length_both is
type Degree is new Float;
type Radian is new Float;
function arc_length(radius: Float; deg1, deg2: Degree) return Float is
((360.0 - abs(Float(deg1) - Float(deg2))) * radius * pi / 180.0);
function arc_length(radius: Float; rad1, rad2: Radian) return Float is
((2.0 * pi - abs(Float(rad1) - Float(rad2))) * radius);
d1 : Degree := 120.0;
d2 : Degree := 10.0;
r1 : Radian := Radian(d1) * pi / 180.0;
r2 : Radian := Radian(d2) * pi / 180.0;
begin
put(arc_length(10.0, d1, d2), fore=>0, aft=>15, exp=>0);
new_line;
put(arc_length(10.0, r1, r2), fore=>0, aft=>15, exp=>0);
new_line;
-- Next line will not compile as you cannot mix Degree and Radian
-- put(arc_length(10.0, d1, r2), fore=>0, aft=>15, exp=>0);
end arc_length_both;
ALGOL 68
BEGIN
# returns the length of the arc between the angles a and b on a circle of radius r #
# the angles should be specified in degrees #
PROC major arc length = ( REAL a, b, r )REAL:
BEGIN
REAL angle := ABS ( a - b );
WHILE angle > 360 DO angle -:= 360 OD;
IF angle < 180 THEN angle := 360 - angle FI;
( r * angle * pi ) / 180
END # majorArcLength # ;
# task test case #
print( ( fixed( major arc length( 10, 120, 10 ), -10, 4 ), newline ) )
END- Output:
43.6332
ALGOL W
Follows the Fortran interpretation of the task and finds the length of the major arc.
begin
% returns the length of the arc between the angles a and b on a circle of radius r %
% the angles should be specified in degrees %
real procedure majorArcLength( real value a, b, r ) ;
begin
real angle;
angle := abs( a - b );
while angle > 360 do angle := angle - 360;
if angle < 180 then angle := 360 - angle;
( r * angle * PI ) / 180
end majorArcLength ;
% task test case %
write( r_w := 10, r_d := 4, r_format := "A", majorArcLength( 10, 120, 10 ) )
end.- Output:
43.6332
APL
arc ← (○÷180)×⊣×360-(|(-/⊢))
- Output:
10 arc 10 120 43.6332313
Arturo
degToRad: function [deg]-> deg * pi // 180
doublePi: 2 * pi
arcLength: function [r, a, b][
d: (abs a-b) % doublePi
return r * (d >= pi)? -> d -> doublePi - d
]
print ["Arc length:" to :string .format:".5f" arcLength 10 degToRad 10.0 degToRad 120.0]
- Output:
Arc length: 43.63323
AutoHotkey
MsgBox % result := arcLength(10, 10, 120)
return
arcLength(radius, angle1, angle2){
return (360 - Abs(angle2-angle1)) * (π := 3.141592653589793) * radius / 180
}
- Output:
43.633231
AWK
# syntax: GAWK -f LENGTH_OF_AN_ARC_BETWEEN_TWO_ANGLES.AWK
# converted from PHIX
BEGIN {
printf("%.7f\n",arc_length(10,10,120))
exit(0)
}
function arc_length(radius,angle1,angle2) {
return (360 - abs(angle2-angle1)) * 3.14159265 / 180 * radius
}
function abs(x) { if (x >= 0) { return x } else { return -x } }
- Output:
43.6332313
BASIC
10 DATA 10, 10, 120
20 READ R, A1, A2
30 GOSUB 100
40 PRINT A
50 END
100 REM Calculate length of arc of radius R, angles A1 and A2
110 A = ATN(1)*R*(360-ABS(A1-A2))/45
120 RETURN
- Output:
43.6332
C
#define PI 3.14159265358979323846
#define ABS(x) (x<0?-x:x)
double arc_length(double radius, double angle1, double angle2) {
return (360 - ABS(angle2 - angle1)) * PI / 180 * radius;
}
void main()
{
printf("%.7f\n",arc_length(10, 10, 120));
}
- Output:
43.6332313
C++
#include <iostream>
#define _USE_MATH_DEFINES
#include <math.h>
double arcLength(double radius, double angle1, double angle2) {
return (360.0 - abs(angle2 - angle1)) * M_PI * radius / 180.0;
}
int main() {
auto al = arcLength(10.0, 10.0, 120.0);
std::cout << "arc length: " << al << '\n';
return 0;
}
- Output:
arc length: 43.6332
D
import std.math;
import std.stdio;
double arcLength(double radius, double angle1, double angle2) {
return (360.0 - abs(angle2 - angle1)) * PI * radius / 180.0;
}
void main() {
writeln("arc length: ", arcLength(10.0, 10.0, 120.0));
}
- Output:
arc length: 43.6332
Delphi
program Length_of_an_arc;
{$APPTYPE CONSOLE}
{$R *.res}
uses
System.SysUtils;
function arc_length(radius, angle1, angle2: Double): Double;
begin
Result := (360 - abs(angle2 - angle1)) * PI / 180 * radius;
end;
begin
Writeln(Format('%.7f', [arc_length(10, 10, 120)]));
Readln;
end.
- Output:
43.6332313
Factor
USING: kernel math math.constants math.trig prettyprint ;
: arc-length ( radius angle angle -- x )
- abs deg>rad 2pi swap - * ;
10 10 120 arc-length .
- Output:
43.63323129985824
FOCAL
01.10 S A1=10 ;C SET PARAMETERS
01.20 S A2=120
01.30 S R=10
01.40 D 2 ;C CALL SUBROUTINE 2
01.50 T %6.4,A,! ;C DISPLAY RESULT
01.60 Q
02.01 C CALCULATE LENGTH OF ARC OF RADIUS R, ANGLES A1 AND A2
02.10 S A=(360 - FABS(A2-A1)) * (3.14159 / 180) * R- Output:
= 43.6332
Fortran
The Fortran subroutine contains the MAX(DIF, 360. - DIF) operation. Other solutions presented here correspond to different interpretations of the problem. This subroutine computes the length of the major arc, which is not necessarily equal to distance traveling counter-clockwise.
*-----------------------------------------------------------------------
* given: polar coordinates of two points on a circle of known radius
* find: length of the major arc between these points
*
*___Name_____Type___I/O___Description___________________________________
* RAD Real In Radius of circle, any unit of measure
* ANG1 Real In Angle of first point, degrees
* ANG2 Real In Angle of second point, degrees
* MAJARC Real Out Length of major arc, same units as RAD
*-----------------------------------------------------------------------
FUNCTION MAJARC (RAD, ANG1, ANG2)
IMPLICIT NONE
REAL RAD, ANG1, ANG2, MAJARC
REAL FACT ! degrees to radians
PARAMETER (FACT = 3.1415926536 / 180.)
REAL DIF
* Begin
MAJARC = 0.
IF (RAD .LE. 0.) RETURN
DIF = MOD(ABS(ANG1 - ANG2), 360.) ! cyclic difference
DIF = MAX(DIF, 360. - DIF) ! choose the longer path
MAJARC = RAD * DIF * FACT ! L = r theta
RETURN
END ! of majarc
*-----------------------------------------------------------------------
PROGRAM TMA
IMPLICIT NONE
INTEGER J
REAL ANG1, ANG2, RAD, MAJARC, ALENG
REAL DATARR(3,3)
DATA DATARR / 120., 10., 10.,
$ 10., 120., 10.,
$ 180., 270., 10. /
DO J = 1, 3
ANG1 = DATARR(1,J)
ANG2 = DATARR(2,J)
RAD = DATARR(3,J)
ALENG = MAJARC (RAD, ANG1, ANG2)
PRINT *, 'first angle: ', ANG1, ' second angle: ', ANG2,
$ ' radius: ', RAD, ' Length of major arc: ', ALENG
END DO
END
- Output:
first angle: 120.000000 second angle: 10.0000000 radius: 10.0000000 Length of major arc: 43.6332321 first angle: 10.0000000 second angle: 120.000000 radius: 10.0000000 Length of major arc: 43.6332321 first angle: 180.000000 second angle: 270.000000 radius: 10.0000000 Length of major arc: 47.1238899
FreeBASIC
#define DEG 0.017453292519943295769236907684886127134
function arclength( r as double, a1 as double, a2 as double ) as double
return (360 - abs(a2 - a1)) * DEG * r
end function
print arclength(10, 10, 120)- Output:
43.63323129985824
Go
package main
import (
"fmt"
"math"
)
func arcLength(radius, angle1, angle2 float64) float64 {
return (360 - math.Abs(angle2-angle1)) * math.Pi * radius / 180
}
func main() {
fmt.Println(arcLength(10, 10, 120))
}
- Output:
43.63323129985823
Haskell
arcLength radius angle1 angle2 = (360.0 - (abs $ angle1 - angle2)) * pi * radius / 180.0
main = putStrLn $ "arcLength 10.0 10.0 120.0 = " ++ show (arcLength 10.0 10.0 120.0)
- Output:
arcLength 10.0 10.0 120.0 = 43.63323129985823
J
Interpreting the task requirements as asking for the length of a clockwise arc between two angles whose values are provided in degrees.
clockwise=: - + 360 * < NB. clockwise effective angle between two provided angles
length=: * 2r360p1 * ]
NB. use radius to find length of angle
Example use:
10 length 10 clockwise 120
43.6332
5 length 10 clockwise 120
21.8166
10 length 120 clockwise 90
5.23599
Java
public static double arcLength(double r, double a1, double a2){
return (360.0 - Math.abs(a2-a1))*Math.PI/180.0 * r;
}
JavaScript
function arc_length(radius, angle1, angle2) {
return (360 - Math.abs(angle2 - angle1)) * Math.PI / 180 * radius;
}
console.log(arc_length(10, 10, 120).toFixed(7));
- Output:
43.6332313
jq
Works with gojq, the Go implementation of jq
As noted in the entry for Julia, the function defined here does not correspond to the arc subtended by an angle.
In case you're wondering why `length` appears below where you might expect `abs`, rest assured that jq's `length` applied to a number yields its absolute value.
# Output is in the same units as radius; angles are in degrees.
def arclength(radius; angle1; angle2):
def pi: 1 | atan * 4;
(360 - ((angle2 - angle1)|length)) * (pi/180) * radius;
# The task:
arclength(10; 10; 120)- Output:
43.63323129985824
Julia
The task seems to be to find the distance along the circumference of the circle which is NOT swept out between the two angles.
arclength(r, angle1, angle2) = (360 - abs(angle2 - angle1)) * π/180 * r
@show arclength(10, 10, 120) # --> arclength(10, 10, 120) = 43.63323129985823
Kotlin
import kotlin.math.PI
import kotlin.math.abs
fun arcLength(radius: Double, angle1: Double, angle2: Double): Double {
return (360.0 - abs(angle2 - angle1)) * PI * radius / 180.0
}
fun main() {
val al = arcLength(10.0, 10.0, 120.0)
println("arc length: $al")
}
- Output:
arc length: 43.63323129985823
Lua
function arcLength(radius, angle1, angle2)
return (360.0 - math.abs(angle2 - angle1)) * math.pi * radius / 180.0
end
function main()
print("arc length: " .. arcLength(10.0, 10.0, 120.0))
end
main()
- Output:
arc length: 43.633231299858
Mathematica/Wolfram Language
ClearAll[MajorArcLength]
MajorArcLength[r_, {a1_, a2_}] := Module[{d},
d = Mod[Abs[a1 - a2], 360];
d = Max[d, 360 - d]; (* this will select the major arc *)
d Degree r
]
MajorArcLength[10, {10, 120}] // N
- Output:
43.6332
Nim
import math, strformat
const TwoPi = 2 * Pi
func arcLength(r, a, b: float): float =
## Return the length of the major arc in a circle of radius "r"
## between angles "a" and "b" expressed in radians.
let d = abs(a - b) mod TwoPi
result = r * (if d >= Pi: d else: TwoPi - d)
echo &"Arc length: {arcLength(10, degToRad(10.0), degToRad(120.0)):.5f}"
- Output:
Arc length: 43.63323
Perl
use strict;
use warnings;
use utf8;
binmode STDOUT, ":utf8";
use POSIX 'fmod';
use constant π => 2 * atan2(1, 0);
use constant τ => 2 * π;
sub d2r { $_[0] * τ / 360 }
sub arc {
my($a1, $a2, $r) = (d2r($_[0]), d2r($_[1]), $_[2]);
my @a = map { fmod( ($_ + τ), τ) } ($a1, $a2);
printf "Arc length: %8.5f Parameters: (%9.7f, %10.7f, %10.7f)\n",
(fmod(($a[0]-$a[1] + τ), τ) * $r), $a2, $a1, $r;
}
arc(@$_) for
[ 10, 120, 10],
[ 10, 120, 1],
[120, 10, 1],
[-90, 180, 10/π],
[-90, 0, 10/π],
[ 90, 0, 10/π];
- Output:
Arc length: 43.63323 Parameters: (2.0943951, 0.1745329, 10.0000000) Arc length: 43.63323 Parameters: (2.0943951, 0.1745329, 10.0000000) Arc length: 4.36332 Parameters: (2.0943951, 0.1745329, 1.0000000) Arc length: 1.91986 Parameters: (0.1745329, 2.0943951, 1.0000000) Arc length: 15.00000 Parameters: (0.0000000, -1.5707963, 3.1830989) Arc length: 5.00000 Parameters: (0.0000000, 1.5707963, 3.1830989)
Phix
with javascript_semantics function arclength(atom r, angle1, angle2) return (360 - abs(angle2 - angle1)) * PI/180 * r end function ?arclength(10, 10, 120) -- 43.6332313
Python
import math
def arc_length(r, angleA, angleB):
return (360.0 - abs(angleB - angleA)) * math.pi * r / 180.0
radius = 10 angleA = 10 angleB = 120 result = arc_length(radius, angleA, angleB) print(result) Output: 43.63323129985823
Raku
Taking a slightly different approach. Rather than the simplest thing that could possibly work, implements a reusable arc-length routine. Standard notation for angles has the zero to the right along an 'x' axis with a counter-clockwise rotation for increasing angles. This version follows convention and assumes the first given angle is "before" the second when rotating counter-clockwise. In order to return the major swept angle in the task example, you need to supply the "second" angle first. (The measurement will be from the first given angle counter-clockwise to the second.)
If you don't supply a radius, returns the radian arc angle which may then be multiplied by the radius to get actual circumferential length.
Works in radian angles by default but provides a postfix ° operator to convert degrees to radians and a postfix ᵍ to convert gradians to radians.
sub arc ( Real \a1, Real \a2, :r(:$radius) = 1 ) {
( ([-] (a2, a1).map((* + τ) % τ)) + τ ) % τ × $radius
}
sub postfix:<°> (\d) { d × τ / 360 }
sub postfix:<ᵍ> (\g) { g × τ / 400 }
say 'Task example: from 120° counter-clockwise to 10° with 10 unit radius';
say arc(:10radius, 120°, 10°), ' engineering units';
say "\nSome test examples:";
for \(120°, 10°), # radian magnitude (unit radius)
\(10°, 120°), # radian magnitude (unit radius)
\(:radius(10/π), 180°, -90°), # 20 unit circumference for ease of comparison
\(0°, -90°, :r(10/π),), # ↓ ↓ ↓ ↓ ↓ ↓ ↓
\(:radius(10/π), 0°, 90°),
\(π/4, 7*π/4, :r(10/π)),
\(175ᵍ, -45ᵍ, :r(10/π)) { # test gradian parameters
printf "Arc length: %8s Parameters: %s\n", arc(|$_).round(.000001), $_.raku
}
- Output:
Task example: from 120° counter-clockwise to 10° with 10 unit radius 43.63323129985824 engineering units Some test examples: Arc length: 4.363323 Parameters: \(2.0943951023931953e0, 0.17453292519943295e0) Arc length: 1.919862 Parameters: \(0.17453292519943295e0, 2.0943951023931953e0) Arc length: 5 Parameters: \(3.141592653589793e0, -1.5707963267948966e0, :radius(3.183098861837907e0)) Arc length: 15 Parameters: \(0e0, -1.5707963267948966e0, :r(3.183098861837907e0)) Arc length: 5 Parameters: \(0e0, 1.5707963267948966e0, :radius(3.183098861837907e0)) Arc length: 15 Parameters: \(0.7853981633974483e0, 5.497787143782138e0, :r(3.183098861837907e0)) Arc length: 9 Parameters: \(2.7488935718910685e0, -0.7068583470577035e0, :r(3.183098861837907e0))
REXX
This REXX version handles angles (in degrees) that may be > 360º.
/*REXX program calculates the length of an arc between two angles (stated in degrees).*/
parse arg radius angle1 angle2 . /*obtain optional arguments from the CL*/
if radius=='' | radius=="," then radius= 10 /*Not specified? Then use the default.*/
if angle1=='' | angle1=="," then angle1= 10 /* " " " " " " */
if angle2=='' | angle2=="," then angle2= 120 /* " " " " " " */
say ' circle radius = ' radius
say ' angle 1 = ' angle1"º" /*angles may be negative or > 360º.*/
say ' angle 2 = ' angle2"º" /* " " " " " " " */
say
say ' arc length = ' arcLength(radius, angle1, angle2)
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
arcLength: procedure; parse arg r,a1,a2; #=360; return (#-abs(a1//#-a2//#)) * pi()/180 * r
/*──────────────────────────────────────────────────────────────────────────────────────*/
pi: pi= 3.1415926535897932384626433832795; return pi /*use 32 digs (overkill).*/
- output when using the default inputs:
circle radius = 10
angle 1 = 10º
angle 2 = 120º
arc length = 43.6332313
Ring
decimals(7)
pi = 3.14159265
see "Length of an arc between two angles:" + nl
see arcLength(10,10,120) + nl
func arcLength(radius,angle1,angle2)
x = (360 - fabs(angle2-angle1)) * pi / 180 * radius
return x- Output:
Length of an arc between two angles: 43.6332313
RPL
The function also works when the difference between the two angles is greater than 180 degrees.
≪ DEG - ABS 360 OVER - MAX
* 180 / π * →NUM
≫ 'MAJARC' STO
10 10 120 MAJARC 10 10 350 MAJARC
- Output:
2: 43.6332312999 1: 59.3411945678
Ruby
def arc_length(radius, angle1, angle2)
return (360.0 - (angle2 - angle1).abs) * Math::PI / 180.0 * radius
end
print "%.7f\n" % [arc_length(10, 10, 120)]
- Output:
43.6332313
V (Vlang)
import math
fn arc_length(radius f64, angle1 f64, angle2 f64) f64 {
return (360 - math.abs(angle2-angle1)) * math.pi * radius/180
}
fn main() {
println(arc_length(10, 10, 120))
}- Output:
43.633231299858
Wren
var arcLength = Fn.new { |r, angle1, angle2| (360 - (angle2 - angle1).abs) * Num.pi / 180 * r }
System.print(arcLength.call(10, 10, 120))
- Output:
43.633231299858
XPL0
def Pi = 3.14159265358979323846;
func real ArcLen(Radius, Angle1, Angle2); \Length of major arc of circle
real Radius, Angle1, Angle2;
real Diff;
[Diff:= abs(Angle1 - Angle2);
Diff:= 360. - Diff;
return Pi * Radius / 180. * Diff;
];
RlOut(0, ArcLen(10., 10., 120.));- Output:
43.63323
zkl
fcn arcLength(radius, angle1, angle2){
(360.0 - (angle2 - angle1).abs()).toRad()*radius
}
println(arcLength(10,10,120));- Output:
43.6332