# public Khan /khan-exercises

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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120             Areas of circles and sectors

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();

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