What Is the Length of the Arc Shown in Red?
The length of the arc shown in red depends on the circle’s radius and the central angle subtended by that arc. If the red arc is part of a circle, its length can be calculated using the formula (s=r\theta), where (s) is arc length, (r) is the radius, and (\theta) is the angle measured in radians. Which means when the angle is given in degrees, the equivalent formula is (s=\frac{\theta}{360^\circ}\times 2\pi r). A numerical answer requires the radius and the angle representing the red portion of the circle Most people skip this — try not to..
Understanding Arc Length
An arc is a curved portion of a circle. Also, imagine tracing a complete circle with a piece of string: the total length of the string is the circle’s circumference. If only part of the string covers the red section, that covered portion is the arc length.
The length of an arc is therefore related to:
- The radius of the circle
- The central angle subtended by the arc
- The proportion of the full circle represented by the red arc
If the red arc covers one-quarter of a circle, for example, its length is one-quarter of the circumference. But if it covers one-third, its length is one-third of the circumference. This makes it possible to calculate an arc even when the angle is not stated directly, as long as the fraction of the circle is known Not complicated — just consistent. Practical, not theoretical..
The Basic Arc Length Formula
The most direct formula is:
[ s=r\theta ]
In this equation:
- (s) represents the arc length
- (r) represents the radius
- (\theta) represents the central angle in radians
Radians are often unfamiliar to students, so the degree-based version is more common in geometry problems:
[ s=\frac{\theta}{360^\circ}\times 2\pi r ]
This formula works because the circumference of a complete circle is:
[ C=2\pi r ]
The fraction (\frac{\theta}{360^\circ}) represents the portion of the circle occupied by the red arc. Multiplying that fraction by the full circumference gives the arc length.
An equivalent degree-based formula is:
[ s=\frac{\pi r\theta}{180^\circ} ]
Example Using Degrees
Suppose the red arc has a radius of (12) units and a central angle of (60^\circ). Then:
[ s=\frac{60}{360}\times 2\pi(12) ]
[ s=\frac{1}{6}\times24\pi=4\pi ]
Thus, the length of the red arc is:
[ \boxed{4\pi\text{ units}} ]
Using (\pi\approx3.1416), the decimal approximation is:
[ 4\pi\approx12.57\text{ units} ]
If the problem requests an exact answer, (4\pi) is preferable. If it requests a decimal answer, (12.57) units is appropriate Turns out it matters..
How to Find the Length of the Red Arc
To determine the length of an arc shown in a diagram, follow these steps.
1. Identify the center of the circle
The central angle must have its vertex at the center of the circle. If the diagram includes a point at the center, check whether the two radii extending to the endpoints of the red arc form the required angle.
If the marked angle is located somewhere else on the circumference, it may be an inscribed angle rather than a central angle. An inscribed angle is half the central angle that intercepts the same arc The details matter here..
Here's one way to look at it: if an inscribed angle measures (40^\circ), the red arc’s central angle is:
[ 2(40^\circ)=80^\circ ]
2. Determine the radius
Look for a line segment from the center to the circumference. This segment is the radius Easy to understand, harder to ignore..
If the diagram gives the diameter instead, divide the diameter by two:
[ r=\frac{d}{2} ]
Here's a good example: if the diameter is (18) units, the radius is (9) units.
3. Identify the central angle
Find the degree measure of the angle whose sides are the two radii bordering the red arc. The angle may be printed in the diagram or calculated from other geometric information.
The red arc may represent either a **
