Arc Length Radius X Angle

Arc Length and Central Angle

Problem

Pictured is a circumvolve with central angle x, radius 100, and an arc whose length is somewhere between 70 and 90. What are the possible values, in degrees, of x?

Solution

An arc is a fraction of a circumference. What fraction? An arc's length is the same proportion of a circumvolve'southward circumference equally its central angle is of a whole circle.And a circle's circumference equals 2π times the circle'south radius.

And then $\frac{L}{2\pi r}=\frac{x}{360}$ , whereL is arc length andx is a central angle, then $x=\frac{180L}{\pi r}$ .

Consider the extremes of the range you're given: $L=70$ and $L=90.$ What isx for those values of arc length?

Soten measures between 40.1 and 51.half dozen degrees.

This problem had to practise with an angle measured in degrees. What if y'all're asked nearly an angle measured in radians?

Mini-lesson: What's a radian?

Radian is a unit of angle measure out. Think of a circle. Imagine you cut a cord equal to the circle'south radius and then lay the string out on the circle. The arc covered past the string is one radian. The central angle that corresponds to that arc is too 1 radian.

One radian is the measure out of a fundamental angle whose sides intersect an arc that is equally long as the circle'south radius.

In other words, one radian is ane/(2π) of the circle's circumference. A whole circumvolve's arc is 2π radians — just a whole circle is likewise 360°. So 2π radians = 360°.

For an angle $\theta$ measured in radians, the length of a circle'south arc, L, equals the cardinal angle times the radius. $L=r\theta$ .