To calculate the distance, S, along a curve C between points A and B. Show 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,
and by the Pythagorean theorem we get . 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.
Length S
If the curve C is expressed by parametric equations x(t), y(t):
If the curve C is expressed by y = f(x):
Examples: Circle The parametric equations of a circle of radius b are
Calculate the arc length S of the circle.
Astroid The parametric equations of an astroid are x = cos3t y = sin3t Calculate the arc length of 1 / 4 of the astroid (0
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
Ball trajectory The equations of the trajectory of a thrown ball are x = at y = bt - 16t2
Example 5 Calculate the arc length of y = x2 between x = 0 and x = 2.
Example 6 Calculate the arc length of the curve y = x3/2 between x = 0 and x = 1.
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 formulaThe length of an arc depends on the radius of a circle and the central angle L / θ = C / 2π As circumference L / θ = 2πr / 2π L / 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 circleWe 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 / 2π A / 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
Make sure to check out the equation of a circle calculator, too! FAQHow do you find arc length without the radius?To calculate arc length without radius, you need the central angle and the sector area:
Or the central angle and the chord length:
How do you find arc length using radians?
How do you calculate arc length without the angle?To calculate arc length without the angle, you need the radius and the sector area:
Or you can use the radius and chord 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. Area of a circleCircle calc: find c, d, a, rCircumference… 5 more |