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();