randRange(5, 355) randRange(5, 10) 2 * Math.PI * R 2 * R + "\\pi" randRange(0, 359) ANGLE/360 * C fractionReduce(ANGLE * 2 * R, 360) + "\\pi"

A circle with circumference PRETTY_C has an arc with a ANGLE^\circ central angle.

What is the length of the arc?

init({ range: [[-R - 2, R + 2], [-R - 2, R + 2]], scale: [15, 15] }); circle([0, 0], R, { stroke: BLUE }); path([polar(R, ROTATE_ARC + ANGLE), [0, 0], polar(R, ROTATE_ARC)], { stroke: RED, "stroke-dasharray": "." }); arc([0, 0], R, ROTATE_ARC, ROTATE_ARC + ANGLE, { stroke: RED, "stroke-dasharray": "-" }); graph.cAngle = 180 + ((ROTATE_ARC + ANGLE) + ROTATE_ARC) / 2; graph.cL = label(polar(R, graph.cAngle), "\\color{"+BLUE+"}{"+PRETTY_C+"}", labelDirection(graph.cAngle)); graph.aAngle = (ROTATE_ARC * 2 + ANGLE) / 2; graph.angle = arc([0, 0], R * 0.12, ROTATE_ARC, ROTATE_ARC + ANGLE, { stroke: PINK }); graph.angleL = label([0, 0], "\\color{"+PINK+"}{"+ANGLE+"^\\circ}", labelDirection(graph.aAngle)); graph.arcL = label(polar(R, graph.aAngle), "\\color{"+RED+"}{"+PRETTY_A+"}", labelDirection(graph.aAngle)); $(graph.arcL).hide();

A

The ratio between the arc's central angle \theta and 360^\circ is equal to the ratio between the arc length s and the circle's circumference c.

\dfrac{\theta}{360^\circ} = \dfrac{s}{c}

\dfrac{ANGLE^\circ}{360^\circ} = \dfrac{s}{PRETTY_C}

PRETTY_A = s

$(graph.arcL).show();

A circle has a radius of R. An arc in this circle has a central angle of ANGLE^\circ.

$(graph.cL).hide(); graph.r = path([[0, 0], polar(R, graph.cAngle)], { stroke: BLUE }); graph.rL = label(polar(R/2, graph.cAngle), "\\color{"+BLUE+"}{"+R+"}", "above");

First, calculate the circumference of the circle.

c = 2\pi r = 2\pi (R) = PRETTY_C

A circle has a circumference of PRETTY_C. It has an arc of length PRETTY_A.

What is the central angle of the arc, in degrees?

$(graph.angle).hide(); $(graph.angleL).hide(); $(graph.arcL).show();
ANGLE

The ratio between the arc's central angle \theta and 360^\circ is equal to the ratio between the arc length s and the circle's circumference c.

\dfrac{\theta}{360^\circ} = \dfrac{s}{c}

\dfrac{\theta}{360^\circ} = \dfrac{PRETTY_A}{PRETTY_C}

\theta = ANGLE^\circ

$(graph.angle).show(); $(graph.angleL).show();