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();