An Angle from an Arc

solved Units

Q. How big is an angle whose arc is $s$ long on a circle of radius $r$ ?

Inputs

arc length

s

1.00 m

radius

r

1.00 m

Outputs

angle in radians

θ

How it's solved

Solved with Units

Given

$s$ , $r$ yours to vary

Find $θ$ .

Working

arc over radius $θ = \frac{s}{r}$
$θ = \frac{1}{1}$
$θ = 1$

Conclusion

As big as the arc it cuts off, counted in radii: θ = s ⁄ r. An arc as long as the radius makes one radian, and a length over a length leaves a pure number, so an angle has no dimensions.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :angle_from_an_arc do4  describes "%{angle} whose arc is %{arc} long on a circle of radius %{radius}?"5  given :arc6  given :radius7  asks angle: "How big is an angle"8end

solution.rb

1require_relative "../questions/angle_from_an_arc"2require_relative "../science/units_and_measurement/units"34solve :angle_from_an_arc, using: Units do5  take arc: :s, radius: :r6  take angle: :theta7  find_value :angle89  conclusion "As big as the arc it cuts off, counted in radii: θ = s ⁄ r. An arc as long as the " \10             "radius makes one radian, and a length over a length leaves a pure number, so an " \11             "angle has no dimensions."12end

playground.rb

1require_relative "../solutions/angle_from_an_arc"23play_with :angle_from_an_arc do4  title "An Angle from an Arc"56  input :s, 0.0..20.0, in: "m", step: 0.1, default: 1.0, as: "arc length"7  input :r, 0.5..10.0, in: "m", step: 0.1, default: 1.0, as: "radius"89  output :theta, as: "angle in radians"10end

The laws it drew on

quantity	written	units	within	from
speed of light	$c$	m/s	—	Units
avogadro constant	$N_{A}$	mol⁻¹	—	Units
length	$l$	m	0 … ∞	Units
mass	$m$	kg	—	Units
time	$t$	s	0 … ∞	Units
electric current	$I$	A	—	Units
temperature	$T$	K	—	Units
amount of substance	$n$	mol	—	Units
luminous intensity	$I_{v}$	cd	—	Units
arc length	$s$	m	—	Units
radius	$r$	m	0 … ∞	Units
plane angle	$θ$	—	—	Units
area	$A$	m²	—	Units
solid angle	$Ω$	—	—	Units
light time	$t_{light}$	s	—	Units
light distance	$d_{light}$	m	—	Units
number of entities	$N$	—	—	Units
baseline	$B$	m	—	Units
distance away	$D$	m	—	Units
parallax angle	$θ_{p}$	—	—	Units

θ = \frac{s}{r}

arc over radius

Ω = \frac{A}{r^{2}}

area over radius squared

d_{light} = c \cdot t_{light}

distance at light speed

N = n \cdot N_{A}

entities in moles

θ_{p} = \frac{B}{D}

baseline over distance