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# Finding Arc Lengths

We may know how to find the perimeter and area of various shapes.. But do we know how to find arc lengths? Although this concept might seem challenging at first, we can find arc lengths easily by following a few easy rules:

## How to find arc lengths

An arc is a section of a circle. Remember that circles have just one side -- but we can examine specific sections of a circle's perimeter by dividing the shape into pieces. If we slice our circle into pieces like a pie, the arc length would be the outer, curved edge of each pie slice. In other words, it's the "crust."

But how do we measure these arc lengths? One method is to measure the arc length in degrees. If we know the central angle, this method is easy. Why? Because the arc's degree is exactly the same as the degree of the central angle. Consider the following diagram:

Here we can see that the central angle is 100 degrees and the radius is 3.

But what if we needed to find the distance or length of the arch rather than the degree measure? We know that the arc length must be a fraction of a larger whole. If we find the circumference of the circle, we can determine its value based on this fraction. First, let's find the circumference of the circle:

We are given that the radius is 3, so let's use our formula for finding the circumference:

$2\pi r$

$2\pi \left(3\right)$

$=6\pi$

Let's stop there for a moment.

To find the arc length, we need to multiply this value by a handy fraction:

The degree measure of $\mathrm{AB}÷360$

Why does this work? Remember that an entire circle equals 360 degrees. If we find the proportional relationship between the interior angle and 360 degrees, this will be the same proportional relationship between the arc length and the entire circumference.

In other words, we can find our answer by multiplying our fraction by the circumference:

$\frac{6\pi ×100}{360}$

Let's simplify that fraction:

$\frac{6\pi ×5}{18}$

This leaves us with:

$\frac{5\pi }{3}$

This is equivalent to approximately 5.236 units.

What if we have an interior angle that is expressed in radians rather than degrees? No problem -- we can use this formula instead:

Arc length = interior angle multiplied by the radius. Yes, it really is that simple! This is one of the many reasons mathematicians like work with radians rather than degrees -- especially as they get into more advanced mathematics.

Radians: Length of an $\mathrm{Arc}=\theta ×r$

Degrees: Length of an $\mathrm{Arc}=\theta ×\frac{\pi }{180}×r$

## Arc lengths of general curves

For smooth curves defined by a function y = f(x), where x ranges from a to b, we can find the arc length using an integral-based formula. The arc length, L, of this curve can be found using the following formula:

$L=\underset{a}{\overset{b}{\int }}\sqrt{1+{\left({f}^{\text{'}}\left(x\right)\right)}^{2}}\mathrm{dx}$

In this formula, ${f}^{\text{'}}\left(x\right)$ represents the derivative of $f\left(x\right)$ with respect to x. By integrating the square root of the sum of 1 and the square of the derivative from a to b, we can find the arc length of the given smooth curve.

Let's find the arc length of the curve $y={x}^{2}$ from $x=0$ to $x=1$ .

First, we need to find the derivative of the function with respect to x:

$f\left(x\right)={x}^{2}$

$f ' ( x ) = 2 x$

Now, we can plug the derivative into the arc length formula:

$L=\underset{a}{\overset{b}{\int }}\sqrt{1+{\left(2x\right)}^{2}}\mathrm{dx}$

$L=\underset{a}{\overset{b}{\int }}\sqrt{1+{4x}^{2}}\mathrm{dx}$

This integral can be quite difficult to solve by hand, so we often use a calculator or computer software to evaluate it numerically. Using a calculator, we get:

$L\approx 1.47894$

So, the arc length of the curve $y={x}^{2}$ from $x=0$ to $x=1$ is approximately 1.47894.

Arcs