# Khan/khan-exercises

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 Circles and arcs
randRange(2, 10) R * randFromArray([2, 3, 4, 5, 6, 8, 9, 10, 12, 15]) randRange(1, DENOMINATOR - 1) 360 * NUMERATOR / DENOMINATOR
randRange(0, 359) 10 2 * Math.PI * R NUMERATOR / DENOMINATOR * C fractionReduce(DEGREES * 2 * R, 360) "\\blue{" + (2 * R) + "}"

A circle with circumference PRETTY_C has an arc with a 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 }); 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)); $(graph.arcL).hide(); NUMERATOR / DENOMINATOR * 2 * R The ratio between the arc's central angle \pink{\theta} and 360^\circ is equal to the ratio between the arc length \red{s} and the circle's circumference \blue{c}. \dfrac{\pink{\theta}}{360^\circ} = \dfrac{\red{s}}{\blue{c}} \dfrac{\pink{DEGREES}^\circ}{360^\circ} = \dfrac{\red{s}}{\blue{PRETTY_C}} fractionReduce(NUMERATOR, DENOMINATOR) = \dfrac{\red{s}}{PRETTY_C} fractionReduce(NUMERATOR, DENOMINATOR) \times PRETTY_C = \red{s} PRETTY_A = \red{s}$(graph.arcL).show();
coefficient(fractionReduce(DEGREES * 2 * R, 360)) + "\\pi" "\\blue{" + (2 * R) + "\\pi}"

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

$(graph.cL).hide(); graph.r = path([[0, 0], polar(DIAGRAM_R, graph.cAngle)], { stroke: BLUE }); graph.rL = label(polar(DIAGRAM_R / 2, graph.cAngle), "\\blue{" + R + "}", "above"); A First, calculate the circumference of the circle.$(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 degrees?

$(graph.angle).hide();$(graph.angleL).hide(); $(graph.arcL).show(); DEGREES^\circ The ratio between the arc's central angle \theta and 360^\circ is equal to the ratio between the arc length s and the circle's circumference c. \dfrac{\pink{\theta}}{360^\circ} = \dfrac{\red{s}}{\blue{c}} \dfrac{\pink{\theta}}{360^\circ} = \red{PRETTY_A} \div PRETTY_C \dfrac{\pink{\theta}}{360^\circ} = fractionReduce(NUMERATOR, DENOMINATOR) \pink{\theta} = fractionReduce(NUMERATOR, DENOMINATOR) \times 360^\circ \pink{\theta} = DEGREES^\circ$(graph.angle).show(); \$(graph.angleL).show();