Sector area, arc length and chord explained
The difference between a sector's area, its arc and its chord, how the radian conversion works, and where each measurement is used in practice.
Three measurements of one pie slice
A circular sector has three distinct lengths and areas worth telling apart. The area is the flat region enclosed by the two radii and the arc, useful when you need paint, fabric or land coverage. The arc length is the curved outer boundary, the distance you would walk along the rim from one radius to the other. The chord is the straight shortcut between the two arc endpoints, always shorter than the arc it spans. Confusing the arc with the chord is a common error, since the arc bends outward while the chord cuts straight across.
Why the angle must be in radians
The tidy formulas 0.5 x r^2 x theta for area and r x theta for arc only hold when theta is measured in radians, because a radian is defined so that an angle of one radian subtends an arc equal to the radius. Degrees are a human convenience with no such link to the radius, so the calculator converts them first by multiplying by pi and dividing by 180. This is why a 90 degree quarter circle becomes pi over 2, roughly 1.570796 radians, before the area is worked out. Skipping the conversion is the most frequent mistake in sector problems.
Working the quarter circle by hand
Take the radius 5 and a 90 degree angle from the first example. Converting gives theta = pi / 2 = 1.570796 radians. The area is 0.5 times 25 times 1.570796, which is 19.634954 square units, exactly a quarter of the full disc area of 25 pi. The arc is 5 times 1.570796, or 7.853982, a quarter of the circumference. The chord is 2 times 5 times the sine of 45 degrees, which is 10 times 0.707107, giving 7.071068. Matching these against the tool confirms every formula at once.
Where sector geometry shows up
These measurements appear far beyond textbooks. Landscapers and architects use sector area to size curved flower beds, patios and fan-shaped rooms. Arc length matters for bending trim, laying curved track or cutting a strip that must follow a rounded edge. The chord governs where the straight span sits, which is what a bridge deck or a rafter across a curved opening actually follows. Reading all four outputs together lets you move between the material a curved region needs and the straight cuts required to build it.