Central Angle Calculator
Solve central angle problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
theta = s / r (radians), s = r * theta, Sector Area = (1/2) * r^2 * theta
The central angle in radians equals arc length divided by radius. Arc length equals radius times the central angle. The sector area is half the radius squared times the angle in radians. These formulas connect the central angle to all other circle measurements.
Worked Examples
Example 1: Finding Central Angle from Arc Length
Problem:A circle has radius 10 cm and an arc length of 5 cm. Find the central angle.
Solution:theta = s / r = 5 / 10 = 0.5 radians\nConvert to degrees: 0.5 * (180/pi) = 28.6479 degrees\nSector area = (1/2)(10^2)(0.5) = 25 cm^2\nChord length = 2(10)sin(0.5/2) = 20 * sin(0.25) = 4.9479 cm
Result:Central angle = 28.65 degrees (0.5 radians). Sector area = 25 cm^2. Chord = 4.948 cm.
Example 2: Finding Arc Length from Central Angle
Problem:A circle has radius 8 m and a central angle of 60 degrees. Find the arc length and sector area.
Solution:Convert angle: 60 * (pi/180) = pi/3 = 1.0472 radians\nArc length = r * theta = 8 * 1.0472 = 8.378 m\nSector area = (1/2)(64)(1.0472) = 33.51 m^2\nChord = 2(8)sin(pi/6) = 16 * 0.5 = 8 m
Result:Arc length = 8.378 m. Sector area = 33.51 m^2. Chord length = 8 m (equal to radius for 60-degree angle).
Frequently Asked Questions
What is a central angle and how is it measured?
A central angle is an angle whose vertex is at the center of a circle and whose sides (rays) extend to two points on the circle. The central angle is measured by the arc it intercepts, and there is a direct proportional relationship between the central angle and the arc length. In degrees, a full rotation around the center is 360 degrees, so a central angle of 90 degrees intercepts one-quarter of the circle. In radians, a full rotation is 2*pi radians, and the central angle in radians equals the arc length divided by the radius. Central angles are fundamental in circle geometry and appear in applications ranging from clock design to satellite orbit calculations.
What is the formula for finding a central angle from arc length?
The central angle in radians equals the arc length divided by the radius: theta = s/r, where s is the arc length and r is the radius. To convert to degrees, multiply by 180/pi. This formula comes directly from the definition of a radian: one radian is the angle subtended when the arc length equals the radius. So if the arc length is twice the radius, the angle is 2 radians. This formula is beautifully simple and is the reason radians are considered the natural unit for measuring angles. The formula also rearranges easily: s = r*theta for finding arc length, and r = s/theta for finding the radius.
How do you calculate the sector area from a central angle?
The area of a sector (the pizza-slice region bounded by two radii and an arc) equals (1/2) * r^2 * theta, where theta is the central angle in radians. Alternatively, if the angle is in degrees, the formula is (theta/360) * pi * r^2. This formula makes intuitive sense: the sector area is the same fraction of the full circle area as the central angle is of a full rotation. A 90-degree central angle creates a sector that is 1/4 of the full circle, so its area is (1/4) * pi * r^2. For practical applications like calculating the area of a lawn sprinkler coverage or a radar sweep, this formula is indispensable.
How is the chord length related to the central angle?
The chord connecting two points on a circle is related to the central angle by the formula: chord = 2r * sin(theta/2), where r is the radius and theta is the central angle in radians. This formula comes from trigonometry applied to the isosceles triangle formed by the two radii and the chord. As the central angle increases from 0 to 180 degrees, the chord length increases from 0 to 2r (the diameter). For angles beyond 180 degrees, the chord begins to decrease, and at 360 degrees it returns to 0. The chord is always shorter than the corresponding arc length, and the ratio of chord to arc approaches 1 as the central angle approaches 0.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy