Question 1

How do I calculate arc length?

Accepted Answer

If the angle is in radians: arc length = radius x angle. If in degrees: arc length = (angle/360) x 2 x pi x radius. For example, a circle with radius 10 and central angle 45 degrees: s = (45/360) x 2 x pi x 10 = (1/8) x 62.83 = 7.854 units.

Question 2

What is a sector of a circle?

Accepted Answer

A sector is the pie-shaped region enclosed by two radii and an arc. Its area is a fraction of the full circle's area, proportional to the central angle: A = (theta/360) x pi x r^2 in degrees, or A = (1/2) x r^2 x theta in radians. A semicircle is a sector with a 180-degree angle.

Question 3

What is the difference between arc length and chord length?

Accepted Answer

Arc length is the distance along the curved arc of the circle, while chord length is the straight-line distance between the two endpoints of the arc. The arc is always longer than the chord (unless the angle is 0). As the angle increases toward 180 degrees, the chord approaches the diameter while the arc approaches half the circumference.

Question 4

How do I convert between degrees and radians?

Accepted Answer

To convert degrees to radians: multiply by pi/180. To convert radians to degrees: multiply by 180/pi. Key values: 360 degrees = 2pi radians, 180 degrees = pi radians, 90 degrees = pi/2 radians, 60 degrees = pi/3 radians, 45 degrees = pi/4 radians.

Question 5

Why are radians used instead of degrees in calculus?

Accepted Answer

Radians simplify many formulas. The derivative of sin(x) is cos(x) only when x is in radians. The arc length formula s = r x theta is simplest in radians. The Taylor series for trigonometric functions require radian inputs. Radians are a dimensionless ratio (arc length divided by radius), making them the natural choice.

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