randRange(2, 10) R * randFromArray([2, 3, 4, 5, 6, 8, 9, 10, 12, 15]) randRange(1, DENOMINATOR - 1) 360 * NUMERATOR / DENOMINATOR
fractionReduce(NUMERATOR * 2, DENOMINATOR) FRACTION !== '1' ? FRACTION + "\\pi" : "\\pi" 2 * NUMERATOR * Math.PI / DENOMINATOR rand(2) USE_RADIANS ? "2 \\pi" : "360 ^ \\circ" randRange(0, 359) 10 2 * Math.PI * R NUMERATOR / DENOMINATOR * C coefficient(fractionReduce(DEGREES * 2 * R, 360)) + "\\pi" "\\blue{" + (2 * R) + "\\pi}"

A circle with circumference `PRETTY_C` has an arc with a `RADIANS` radians `DEGREES^\circ` central angle.

What is the length of the arc?

init({ range: [[-DIAGRAM_R - 2, DIAGRAM_R + 2], [-DIAGRAM_R - 2, DIAGRAM_R + 2]], scale: [15, 15] }); circle([0, 0], DIAGRAM_R, { stroke: BLUE }); path([polar(DIAGRAM_R, ROTATE_ARC + DEGREES), [0, 0], polar(DIAGRAM_R, ROTATE_ARC)], { stroke: RED, "stroke-dasharray": "." }); arc([0, 0], DIAGRAM_R, ROTATE_ARC, ROTATE_ARC + DEGREES, { stroke: RED, "stroke-dasharray": "-" }); graph.cAngle = 180 + ((ROTATE_ARC + DEGREES) + ROTATE_ARC) / 2; graph.cL = label(polar(DIAGRAM_R, graph.cAngle), PRETTY_C, labelDirection(graph.cAngle)); graph.aAngle = (ROTATE_ARC * 2 + DEGREES) / 2; graph.angle = arc([0, 0], DIAGRAM_R * 0.12, ROTATE_ARC, ROTATE_ARC + DEGREES, { stroke: PINK }); if (USE_RADIANS) { graph.angleL = label(polar(0.5, graph.aAngle), "\\pink{" + RADIANS + "}", labelDirection(graph.aAngle)); } else { graph.angleL = label(polar(0.5, graph.aAngle), "\\pink{" + DEGREES + "^\\circ}", labelDirection(graph.aAngle)); } graph.arcL = label(polar(DIAGRAM_R, graph.aAngle), "\\red{" + PRETTY_A + "}", labelDirection(graph.aAngle)); $(graph.arcL).hide();

A

The ratio between the arc's central angle `\pink{\theta}` and `2 \pi` radians is equal to the the ratio between the arc length `\red{s}` and the circle's circumference `\blue{c}`.

`\dfrac{\pink{\theta}}{2 \pi} = \dfrac{\red{s}}{\blue{c}}`

`\pink{RADIANS} \div 2 \pi = \dfrac{\red{s}}{PRETTY_C}`

The ratio between the arc's central angle `\pink{\theta}` and `360^\circ` is equal to the ratio between the arc length `\red{s}` and the circle's circumference `\blue{c}`.

`\dfrac{\pink{\theta}}{360^\circ} = \dfrac{\red{s}}{\blue{c}}`

`\dfrac{\pink{DEGREES}^\circ}{360^\circ} = \dfrac{\red{s}}{\blue{PRETTY_C}}`

`fractionReduce(NUMERATOR, DENOMINATOR) = \dfrac{\red{s}}{PRETTY_C}`

`fractionReduce(NUMERATOR, DENOMINATOR) \times PRETTY_C = \red{s}`

`PRETTY_A = \red{s}`

$(graph.arcL).show();

A circle has a radius of `\blue{R}`. An arc in this circle has a central angle of `RADIANS` radians `DEGREES^\circ`.

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

First, calculate the circumference of the circle.

$(graph.cL).show();

`\blue{c} = 2\pi r = 2\pi (\blue{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 radians? What is the central angle of the arc, in degrees?

$(graph.angle).hide(); $(graph.angleL).hide(); $(graph.arcL).show();
RADIAN_VALUE DEGREES`^\circ`

The ratio between the arc's central angle `\theta` and `2 \pi` radians is equal to the the ratio between the arc length `s` and the circle's circumference `c`.

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{\pink{\theta}}{CIRCLE_ANGLES} = \dfrac{\red{s}}{\blue{c}}`

`\dfrac{\pink{\theta}}{CIRCLE_ANGLES} = \red{PRETTY_A} \div PRETTY_C`

`\dfrac{\pink{\theta}}{CIRCLE_ANGLES} = fractionReduce(NUMERATOR, DENOMINATOR)`

`\pink{\theta} = fractionReduce(NUMERATOR, DENOMINATOR) \times CIRCLE_ANGLES`

`\pink{\theta} = RADIANS` radians

`\pink{\theta} = DEGREES^\circ`

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