This all-in-one online Arc Length Calculator allows you to calculate the missing parameters of a circular sector if any two parameters of the sector are known, except for the following parameter pairs: chord, arc and chord, area.
What is a Circular Sector?
A circular sector is a portion of a circle enclosed by two radii and the arc between them. Imagine slicing a pizza; each slice, from the center of the pizza to the crust, forms a circular sector. The central angle, formed by the radii, determines the size of the sector. The larger the angle, the bigger the sector.
In the diagram \(\theta\) is the central angle, \(r\) is the radius of the circle, \(s\) is the arc length of the sector, and \(c\) is the chord connecting the endpoints of the arc.
Understanding Arc Length
The arc length of a circular sector is the distance along its curved boundary. To put it simply, if you were to walk along the edge of the sliced pizza, the arc length would be the total distance covered. This measurement plays a crucial role in various fields, from physics to engineering, providing a precise way to quantify the extent of a curved path.
The sector arc length can be expressed using the following straightforward formula:
$$s = r \cdot \theta,$$
where \(\theta\) is the angle in radians. This formula offers a precise and efficient means of determining the arc length for any circular sector.
Length of the Respective Chord
In addition to the arc length, another essential parameter of a circular sector is the length of the chord, the straight line connecting the two ends of the arc. We can find this length using the following formula:
$$c = 2r \cdot sin\frac {\theta}{2}.$$
The sine function is employed to account for the trigonometric relationship between the angle and the sides of the sector.
Area of a Circle Sector
The total area of a circle is known to be \(\pi r^2\). The area of the sector can be obtained by multiplying the circle’s area by the ratio of the angle \(\theta\) (expressed in radians) and \(2\pi\) (because the area of the sector is directly proportional to its angle, and \(2\pi\) is the angle for the whole circle in radians). So we come to the following circular sector area formula:
$$A = \pi r^2 \cdot \frac {\theta} {2 \pi} = { \frac {r^2 \theta}{2}}.$$
This formula elegantly captures the essence of the sector, emphasizing the influence of both the radius and the central angle on its area. It’s a fundamental equation in geometry, providing a foundation for understanding the spatial extent of circular sectors.
The circle sector area in terms of the arc length \(s\) can be obtained by multiplying the total area of the circle \(\pi r^2\) by the ratio of \(s\) to the total circle perimeter \(2\pi r\):
$$A = \pi r^2 \cdot \frac {s} {2 \pi r} = { \frac {r s}{2}}.$$
These formulas are used in our Arc Length Calculator. With this calculator you can easily find all the parameters of a circle sector (\(r\), \(\theta\), \(c\), \(s\), \(A\)) if any two of these parameters are known, except for the following parameter pairs: chord, arc and chord, area. The reason for the latter is that knowledge of these pairs of parameters does not allow, in the general case, to unambiguously find the remaining parameters.
The angle \(\theta\) can be specified both in degrees and in radians. Please note that angles greater than 360 degrees (\(2\pi\)) are taken modulo 360 degrees (\(2\pi\)).
Note that we don’t specify units of measure in our calculator. We assume that area is measured in square units of length and all the lengths are measured in the same length unit.
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