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();

A circle with radius `\pink{R}` has a sector with a central angle of `\purple{RADIANS}` radians `\purple{DEGREES^\circ}`.

$(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();