Find the arc length of the curve

To calculate the distance, S, along a curve C between points A and B.

This distance is called arc length of C between A and B.

Let Ds be the distance along the curve between M and N and Dx, Dy their difference in coordinates.

When M and N are very close to each other,

Find the arc length of the curve

and by the Pythagorean theorem we get

.

Find the arc length of the curve

We approximate S by a succession of chords, so that the sum of their lengths tends to S as their number increases.

The length S becomes the integral of ds from A to B.

Find the arc length of the curve

Length S

Find the arc length of the curve

If the curve C is expressed by parametric equations x(t), y(t):

Find the arc length of the curve

If the curve C is expressed by y = f(x):

Find the arc length of the curve

Examples:

Circle

The parametric equations of a circle of radius b are

Find the arc length of the curve

Calculate the arc length S of the circle.

Find the arc length of the curve

Astroid

The parametric equations of an astroid are

x = cos3t

y = sin3t

Calculate the arc length of 1 / 4 of the astroid (0

Find the arc length of the curve
t
Find the arc length of the curve
Find the arc length of the curve
/ 2).

Find the arc length of the curve

Cycloid

A cycloid is the curve traced out by a point on the circumference of a circle when the circle rolls along a straight line in its own plane. The equations of a cycloid created by a circle of radius 1 are

x(t) = t - sin t

y(t) = 1 - cos t

Calculate the arc length S of the cycloid for 0

Find the arc length of the curve
t
Find the arc length of the curve
Find the arc length of the curve
/ 2.

Find the arc length of the curve

Ball trajectory

The equations of the trajectory of a thrown ball are

x = at

y = bt - 16t2

Find the arc length of the curve

Example 5

Calculate the arc length of y = x2 between x = 0 and x = 2.

Find the arc length of the curve

Example 6

Calculate the arc length of the curve y = x3/2 between x = 0 and x = 1.

Find the arc length of the curve

Comment

Created by Bogna Szyk

Reviewed by Steven Wooding and Jack Bowater

Last updated: Mar 11, 2022

This arc length calculator is a tool that can calculate the length of an arc and the area of a circle sector. This article explains the arc length formula in detail and provides you with step-by-step instructions on how to find the arc length. You will also learn the equation for sector area.

In case you're new to circles, calculating the length and area of sectors could be a little advanced, and you need to start with simpler tools, such as circle length and circumference and area of a circle calculators.

Arc length formula

The length of an arc depends on the radius of a circle and the central angle θ. We know that for the angle equal to 360 degrees (2π), the arc length is equal to circumference. Hence, as the proportion between angle and arc length is constant, we can say that:

L / θ = C / 2π

As circumference C = 2πr,

L / θ = 2πr / 2πL / θ = r

We find out the arc length formula when multiplying this equation by θ:

L = r * θ

Hence, the arc length is equal to radius multiplied by the central angle (in radians).

Area of a sector of a circle

We can find the area of a sector of a circle in a similar manner. We know that the area of the whole circle is equal to πr². From the proportions,

A / θ = πr² / 2πA / θ = r² / 2

The formula for the area of a sector is:

A = r² * θ / 2

How to find the length of an arc and sector area: an example

  1. Decide on the radius of your circle. For example, it can be equal to 15 cm. (You can also input the diameter into the arc length calculator instead.)
  2. What will be the angle between the ends of the arc? Let's say it is equal to 45 degrees, or π/4.
  3. Calculate the arc length according to the formula above: L = r * θ = 15 * π/4 = 11.78 cm.
  4. Calculate the area of a sector: A = r² * θ / 2 = 15² * π/4 / 2 = 88.36 cm².
  5. You can also use the arc length calculator to find the central angle or the circle's radius. Simply input any two values into the appropriate boxes and watch it conducting all calculations for you.

Make sure to check out the equation of a circle calculator, too!

FAQ

How do you find arc length without the radius?

To calculate arc length without radius, you need the central angle and the sector area:

  1. Multiply the area by 2 and divide the result by the central angle in radians.
  2. Find the square root of this division.
  3. Multiply this root by the central angle again to get the arc length.
  4. The units will be the square root of the sector area units.

Or the central angle and the chord length:

  1. Divide the central angle in radians by 2 and perform the sine function on it.
  2. Divide the chord length by double the result of step 1. This calculation gives you the radius.
  3. Multiply the radius by the central angle to get the arc length.

How do you find arc length using radians?

  1. Multiply the central angle in radians by the circle’s radius.
  2. That’s it! The result is simply this multiplication.

How do you calculate arc length without the angle?

To calculate arc length without the angle, you need the radius and the sector area:

  1. Multiply the area by 2.
  2. Then divide the result by the radius squared (make sure that the units are the same) to get the central angle in radians.

Or you can use the radius and chord length:

  1. Divide the chord length by double the radius.
  2. Find the inverse sine of the result (in radians).
  3. Double the result of the inverse sine to get the central angle in radians.
  4. Once you have the central angle in radians, multiply it by the radius to get the arc length.

Does arc length have to be in radians?

Arc length is a measurement of distance, so it cannot be in radians. The central angle, however, does not have to be in radians. It can be in any unit for angles you like, from degrees to arcsecs. Using radians, however, is much easier for calculations regarding arc length, as finding it is as easy as multiplying the angle by the radius.

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