Arc length
- What Do You Mean by Arc Length?
- Formula for Arc Length
- Major and Minor Arc
- Real-Life Applications of Arc Length
- Solved Examples
- Practice Problems
- Frequently Asked Questions
What Do You Mean by Arc Length?
An arc can be any portion of the circumference of a circle, akin to a slice of the circle. The length of an arc is termed arc length. The length of an arc is the distance between two points in the section of a curve. The arc length formula makes it easier to determine the length of a circle's arc. The process of determining the length of an irregular arc is known as the rectification of a curve.
The arc length formula is used to compute the distance along the curved line that forms the arc which is a segment of a circle. The arc length is simply the distance that passes through the curved line of the circle forming the arc. It should be observed that the arc length exceeds the distance of a straight line between its endpoints.
The arc length can be defined as the distance between two points on a curve segment. A circle's arc is any section of its circumference. An arc's angle at any point is the angle created by the two line segments from the center to the arc's endpoints.
Formula for Arc Length
How to find arc length? The arc length depends on the radius or the diameter of the circle and the central angle that the endpoints of the arc subtended at the center of the circle. It also depends on the unit of the central angle of the arc.
Measurement of the central angle is often given in radians or degrees.
There is an arc length formula that helps us find the measure of the distance along the curved line making up the arc.do the same.
The length of an arc `(s)` can be calculated using the formula:
`s = rθ`, where `θ` is the central angle in radians and `r` is the radius
`s = 2πr * (θ/360^\circ)`, where `θ` is the central angle in degrees and `r` is the radius
In a reverse manner, we can use the arc length formula to find the measure of the arc angle.
`θ = s/r`, where `θ` is the central angle in radians and `r` is the radius
`θ = s/2πr * 360^\circ`, where `θ` is the central angle in degrees and `r` is the radius
Major and Minor Arc
The central angle divides a circle into two arcs. The smaller of the two arcs is called the minor arc. The larger of the two arcs is called the major arc.
Major arc: A major arc is greater than half the circumference.
Minor arc: A minor arc is less than half the circumference.
Real-Life Applications of Arc Length
Arc length finds numerous applications in real-world scenarios.
- In architecture, arc length is essential for designing and constructing bridges, arches, and other curved structures.
- Engineers use arc length measurements in designing conveyor belts, roller coasters, and other machinery that involve circular or curved components.
- In astronomy, arc length plays a crucial role in understanding the trajectory of celestial bodies and the curvature of space-time. Calculating the arc length of planetary orbits and the curvature of gravitational fields aids scientists in predicting and analyzing various celestial phenomena.
- In physics, some physical quantities depend on the distance traveled, energy or work. The distance traveled can be calculated using the arc length formula.
- In mechanical engineering, arc length calculations are used in various mechanical systems to control the motion of mechanisms such as engines, pumps, and valves. This helps them to produce desired movements and performance.
- Artists and designers often incorporate curved lines and arcs into their paintings, sculptures, or digital graphics. Understanding arc length helps artists accurately depict curves and arcs in their compositions, adding depth, perspective, and visual interest to their artwork.
- Many maps use curves to represent a road or a pathway. To understand how far one has to drive to reach his or her destination, it is important to know the length of the curve.
- In navigation and GPS (Global Positioning System) technology, arc length calculations are used to calculate the shortest path between two points on the Earth's surface, taking into account the curvature of the Earth. GPS devices rely on arc length calculations to provide accurate distance measurements and route guidance for navigation purposes.
Solved Examples
Example `1`: Calculate the arc length of a circle (in inches) having a central angle of `15` radians with a radius of `8` inches.
Solution:
Center angle `θ = 15` radians
Radius `r = 8` inches.
Arc length `s = θ × r`
`s = 15 × 8`
`s = 120` inches
Example `2`: Calculate the arc length of a circle if the radius is `8` units and the central angle is `18^\circ`. Consider `π = 3.14`.
Solution:
Circumference of circle `= 2πr`
`C = 2π × 8`
`C = 16 π`
Arc length `s = (θ/360) × C`
`s = \frac{18^\circ}{360^\circ} × 16π`
`s = 2.512` Units
Example `3`: Calculate the radius of a circle, if the central angle is `74°` and the arc length is `3.66` units. Consider `π = 3.14`.
Solution:
\( s = \frac{\theta}{360} \times C \)
\( s = \frac{\theta}{360} \times 2\pi r \)
And the given numerical example:
\( 3.66 = \left(\frac{74^\circ}{360^\circ}\right) \times 2\pi \times r \)
To solve for \( r \):
\( r = 3.66 \times \frac{360^\circ}{74^\circ \times 2\pi} \)
\( r = 2.835 \)
The radius of the circle is `2.835` units.
Example `4`: Determine the central angle of a circle with a radius of `9` units and the arc length is `15` units. Consider `π = 3.14`.
Solution:
\( \text{Arc length} = \frac{\theta}{360} \times C \)
\( \text{Arc length} = \frac{\theta}{360} \times 2\pi r \)
And the given numerical example:
\( 15 = \left(\frac{\theta}{360}\right) \times (2 \times \pi \times 9) \)
\( 15 = \left(\frac{\theta}{360}\right) \times (18\pi) \)
To solve for \( \theta \):
\( \theta = \frac{15 \times 360}{18\pi} \)
\( \theta = 95.54^\circ \)
The central angle of the circle is `95.54^\circ`.
Example `5`: Calculate the arc length of a semicircle with a radius of `14` cm. Consider `π = 22/7`.
Solution:
For a semi-circle the central angle `θ = 180°`
`\text{Arc length}= (θ/360) × C`
\(\begin{align*}
\text{Arc length} & = \left(\frac{\theta}{360}\right) \times 2\pi r \\
& = \left(\frac{180}{360}\right) \times \left(2 \times \frac{22}{7} \times 14\right) \\
& = \frac{1}{2} \times (88) \\
& = 44
\end{align*}\)
The arc length of the semicircle is `44` cm.
Example 6: The central angle of `60°` subtends an arc of `10` meters on a circle. Find the diameter of the circle. Consider `π = 3.14`.
Solution:
\(\begin{align*}
\text{Arc length} & = \left(\frac{\theta}{360}\right) \times C \\
& = \left(\frac{\theta}{360}\right) \times 2\pi r \\
10 & = \left(\frac{60^\circ}{360^\circ}\right) \times 2\pi r \\
r & = \frac{10 \times 360^\circ}{60^\circ \times 2\pi} \\
r & = \frac{1800}{120\pi} \\
r & = \frac{15}{\pi} \\
r & \approx 4.77 \text{ meters}
\end{align*}\)
The radius of the circle is approximately `4.77` meters.
Practice Problems
Q`1`. How long would an arc formed by `27^\circ` of a circle with a diameter of `3` cm be? Consider `π = 3.14`.
- `0.725`
- `0.7065`
- `0.82`
- `0.92`
Answer: b
Q`2`. Calculate the radius of a circle, if the central angle is `105^\circ` and the arc length is `3.50` units. Consider `π = 3.14`.
- `1.91`
- `3.56`
- `2.54`
- `1.75`
Answer: a
Q`3`. Determine the central angle of a circle with a diameter of `10` units and the arc length is `12` units. Consider `π = 3.14`.
- `150.45^\circ`
- `92.65^\circ`
- `137.58^\circ`
- `106.54^\circ`
Answer: c
Frequently Asked Questions
Q`1`. What does an arc of a circle mean?
Answer: The length of a portion of a circle's circumference that exists between any two points on it is defined as its arc. i.e., A circle's arc is any portion of its circumference. The angle formed by an arc at any point is the angle formed by the two line segments connecting that point to the arc's endpoints.
Q`2`. Is it necessary to have arc length in radians?
Answer: No, arc length cannot be expressed in radians. Because it is a distance measurement, it cannot be expressed in radians. The central angle at the center can be expressed in radians, degrees, or arcsecs.
Q`3`. How long is a major arc using the arc length formula?
Answer: A major arc is broader than a semicircle in a circle. The central angle is greater than 180°. We may calculate the circumference of a circle arc using the formula `l = rθ`, where `θ` is in radians.