# Khan/khan-exercises

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 Areas of circles and sectors
10
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" rand(2) PI * R * R DEGREES/360 * A_C R * R + "\\pi" fractionReduce(DEGREES * R * R, 360) + "\\pi" randRange(0, 359)

A circle with area \blue{PRETTY_A_C} has a sector with a central angle of \purple{RADIANS} radians \purple{DEGREES^\circ}.

What is the area of the sector?

init({ range: [[-DIAGRAM_R - 2, DIAGRAM_R + 2], [-DIAGRAM_R - 2, DIAGRAM_R + 2]], scale: [15, 15] }); circle([0, 0], DIAGRAM_R, { stroke: BLUE }); arc([0, 0], DIAGRAM_R, ROTATE_ARC, ROTATE_ARC + DEGREES, true, { stroke: GREEN, fill: GREEN, "fill-opacity": 0.1 }); graph.cAngle = 180 + ((ROTATE_ARC + DEGREES) + ROTATE_ARC) / 2; graph.aCL = label(polar(DIAGRAM_R/2, graph.cAngle), "\\blue{" + PRETTY_A_C + "}", "below"); graph.aAngle = (ROTATE_ARC * 2 + DEGREES) / 2; graph.angle = arc([0, 0], DIAGRAM_R * 0.12, ROTATE_ARC, ROTATE_ARC + DEGREES, { stroke: PURPLE }); if (USE_RADIANS) { label(polar(0.5, graph.aAngle), "\\purple{" + RADIANS + "}", labelDirection(graph.aAngle)); } else { label(polar(0.5, graph.aAngle), "\\purple{" + DEGREES + "^\\circ}", labelDirection(graph.aAngle)); } graph.aSL = label(polar(DIAGRAM_R * 0.65, graph.aAngle), "\\green{" + PRETTY_A_S + "}"); $(graph.aSL).hide(); A_S The ratio between the sector's central angle \purple{\theta} and 2 \pi radians is equal to the ratio between the sector's area, \green{A_s}, and the whole circle's area, \blue{A_c}. \dfrac{\purple{\theta}}{2 \pi} = \dfrac{\green{A_s}}{\blue{A_c}} \purple{RADIANS} \div 2 \pi = \dfrac{\green{A_s}}{\blue{PRETTY_A_C}} The ratio between the sector's central angle \purple{\theta} and 360^\circ is equal to the ratio between the sector's area, \green{A_s}, and the whole circle's area, \blue{A_c}. \dfrac{\purple{\theta}}{360^\circ} = \dfrac{\green{A_s}}{\blue{A_c}} \dfrac{\purple{DEGREES^\circ}}{360^\circ} = \dfrac{\green{A_s}}{\blue{PRETTY_A_C}} fractionReduce(NUMERATOR, DENOMINATOR) = \dfrac{\green{A_s}}{\blue{PRETTY_A_C}} fractionReduce(NUMERATOR, DENOMINATOR) \times \blue{PRETTY_A_C} = \green{A_s} PRETTY_A_S = \green{A_s}$(graph.aSL).show();

$(graph.aCL).hide(); graph.r = path([[0, 0], polar(DIAGRAM_R, graph.cAngle)], { stroke: PINK }); graph.rL = label(polar(DIAGRAM_R / 2, graph.cAngle), "\\pink{" + R + "}", "above"); First, calculate the area of the whole circle. Then the area of the sector is some fraction of the whole circle's area. \blue{A_c} = \pi \pink{r}^2 \blue{A_c} = \pi (\pink{R})^2 \blue{A_c} = PRETTY_A_C$(graph.aCL).show();

A circle has a sector with area PRETTY_A_S and central angle of \purple{RADIANS} radians \purple{DEGREES^\circ}.

What is the area of the circle?

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

A_C

The ratio between the sector's central angle \purple{\theta} and 2 \pi radians is equal to the ratio between the sector's area, \green{A_s}, and the whole circle's area, \blue{A_c}.

\dfrac{\purple{\theta}}{2 \pi} = \dfrac{\green{A_s}}{\blue{A_c}}

\purple{RADIANS} \div 2 \pi = \green{PRETTY_A_S} \div \blue{A_c}

The ratio between the sector's central angle \purple{\theta} and 360^\circ is equal to the ratio between the sector's area, \green{A_s}, and the whole circle's area, \blue{A_c}.

\dfrac{\purple{\theta}}{360^\circ} = \dfrac{\green{A_s}}{\blue{A_c}}

\dfrac{\purple{DEGREES^\circ}}{360^\circ} = \green{PRETTY_A_S} \div \blue{A_c}

fractionReduce(NUMERATOR, DENOMINATOR) = \green{PRETTY_A_S} \div \blue{A_c}

\blue{A_c} \times fractionReduce(NUMERATOR, DENOMINATOR) = \green{PRETTY_A_S}

\blue{A_c} = \green{PRETTY_A_S} \times fractionReduce(DENOMINATOR, NUMERATOR)

\blue{A_c} = PRETTY_A_C

\$(graph.aCL).show();