How to find arc length of a circle

Arc length formula is used to calculate the measure of the distance along the curved line making up the arc (a segment of a circle). In simple words, the distance that runs through the curved line of the circle making up the arc is known as the arc length. It should be noted that the arc length is longer than the straight line distance between its endpoints.

Table of Contents Show

Formulas for Arc Length
Denotations in the Arc Length Formula
Example Questions Using the Formula for Arc Length
Practice Questions Based on Arc Length Formula
Also Check:
Arc length formula
Area of a sector of a circle
How to find the length of an arc and sector area: an example
How do you find arc length without the radius?
How do you find arc length using radians?
How do you calculate arc length without the angle?
Does arc length have to be in radians?

Formula to Calculate Arc Length

Formulas for Arc Length

The formula to measure the length of the arc is –

Arc Length Formula (if θ is in degrees)s = 2 π r (θ/360°)Arc Length Formula (if θ is in radians)s = ϴ × rArc Length Formula in Integral Forms=

\(\begin{array}{l}\int^{b}_a\sqrt{1+(\frac{dy}{dx})^2}dx\end{array} \)

Denotations in the Arc Length Formula

s is the arc length
r is the radius of the circle
θ is the central angle of the arc

Example Questions Using the Formula for Arc Length

Question 1: Calculate the length of an arc if the radius of an arc is 8 cm and the central angle is 40°.

Solution:

Radius, r = 8 cm

Central angle, θ = 40°

Arc length = 2 π r × (θ/360°)

So, s = 2 × π × 8 × (40°/360°)

= 5.582 cm

Question 2: What is the arc length for a function f(x) = 6 between x = 4 and x = 6?

Solution:

Since the function is a constant, the differential of it will be 0. So, the arc length will now be-

\(\begin{array}{l}s=\int^{6}_4\sqrt{1 + (0)^2}dx\end{array} \)

So, arc length (s) = (6 – 4) = 2.

Practice Questions Based on Arc Length Formula

What would be the length of the arc formed by 75° of a circle having the diameter of 18 cm?
The length of an arc formed by 60° of a circle of radius “r” is 8.37 cm. Find the radius (r) of that circle.
Calculate the perimeter of a semicircle of radius 1. cm using the arc length formula.

Also Check:

Arc of a Circle
Arc Length Calculator
Circles
List of Maths Formulas

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An arc’s length means the same commonsense thing length always means — you know, like the length of a piece of string (with an arc, of course, it’d be a curved piece of string). Make sure you don’t mix up arc length with the measure of an arc which is the degree size of its central angle.

A circle is 360° all the way around; therefore, if you divide an arc’s degree measure by 360°, you find the fraction of the circle’s circumference that the arc makes up. Then, if you multiply the length all the way around the circle (the circle’s circumference) by that fraction, you get the length along the arc. So finally, here’s the formula you’ve been waiting for.

To find arc length, start by dividing the arc's central angle in degrees by 360. Then, multiply that number by the radius of the circle. Finally, multiply that number by 2 × pi to find the arc length. If you want to learn how to calculate the arc length in radians, keep reading the article!

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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

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.)
What will be the angle between the ends of the arc? Let's say it is equal to 45 degrees, or π/4.
Calculate the arc length according to the formula above: L = r * θ = 15 * π/4 = 11.78 cm.
Calculate the area of a sector: L / θ = C / 2π0.
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:

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

Or the central angle and the chord length:

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

How do you find arc length using radians?

Multiply the central angle in radians by the circle’s radius.
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:

Multiply the area by 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:

Divide the chord length by double the radius.
Find the inverse sine of the result (in radians).
Double the result of the inverse sine to get the central angle in radians.
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.