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

A circle with area PRETTY_A_C has a sector with a ANGLE^\circ central angle.

What is the area of the sector?

init({ range: [[-R - 2, R + 2], [-R - 2, R + 2]], scale: [15, 15] }); circle([0, 0], R, { stroke: BLUE }); arc([0, 0], R, ROTATE_ARC, ROTATE_ARC + ANGLE, true, { stroke: ORANGE, fill: ORANGE, "fill-opacity": 0.1 }); graph.cAngle = 180 + ((ROTATE_ARC + ANGLE) + ROTATE_ARC) / 2; graph.aCL = label(polar(R/2, graph.cAngle), "\\color{"+BLUE+"}{"+PRETTY_A_C+"}", "below"); graph.aAngle = (ROTATE_ARC * 2 + ANGLE) / 2; graph.angle = arc([0, 0], R * 0.12, ROTATE_ARC, ROTATE_ARC + ANGLE, { stroke: PURPLE }); graph.angleL = label([0, 0], "\\color{"+PURPLE+"}{"+ANGLE+"^\\circ}", labelDirection(graph.aAngle)); graph.aSL = label(polar(R/2, graph.aAngle), "\\color{"+ORANGE+"}{"+PRETTY_A_S+"}"); $(graph.aSL).hide(); graph.arcL = label(polar(R, graph.aAngle), "\\color{"+ORANGE+"}{"+PRETTY_S+"}", labelDirection(graph.aAngle)); $(graph.arcL).hide();

A_S

The ratio between the sector's central angle \theta and 360^\circ equals the ratio between the sector's area, A_s, and the whole circle's area, A_c.

\dfrac{\theta}{360^\circ} = \dfrac{A_s}{A_c}

\dfrac{ANGLE^\circ}{360^\circ} = \dfrac{A_s}{PRETTY_A_C}

PRETTY_A_S = A_s

$(graph.aSL).show();

A circle with radius R has a sector with a ANGLE^\circ central angle.

$(graph.aCL).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 area of the whole circle.

Then the area of the sector is some fraction of the whole circle's area.

A_c = \pi r^2

A_c = \pi (R)^2

A_c = PRETTY_A_C

$(graph.aCL).show();

A circle has a sector with area PRETTY_A_S and central angle ANGLE^\circ.

What is the area of the circle?

$(graph.aCL).hide(); $(graph.aSL).show();

A_C

The ratio between the sector's central angle \theta and 360^\circ equals the ratio between the sector's area, A_s, and the whole circle's area, A_c.

\dfrac{\theta}{360^\circ} = \dfrac{A_s}{A_c}

\dfrac{ANGLE^\circ}{360^\circ} = \dfrac{PRETTY_A_S}{A_c}

PRETTY_A_C = A_c

$(graph.aCL).show();