Arc Length Calculator
Arc Length Calculator
m
deg
Result
Note! This is not a diagnosis. The calculations that are provided are estimates based on averages.
Angle in radians
| Sector Area | |
| Arc Length | |
| Chord d | |
| Diameter |
Angle in degrees
| Sector Area | |
| Arc Length | |
| Chord d | |
| Diameter |
This arc length calculator helps you determine the length of an arc as well as the area of a circle sector. In this guide, we explain the arc length formula clearly and give you step-by-step instructions for calculating arc length. You will also find the formula for calculating the area of a sector.
If you are new to working with circles, finding the length and area of sectors can seem a bit challenging at first. It is better to start with basic tools like a calculator for the circumference and area of a circle before moving on to arc and area.
Arc Length Formula
The arc length depends on the radius of the circle and the central angle θ. For an angle of 360 degrees (2π), the arc length is equal to the circumference of the circle. Since the ratio of angle to arc length is constant, we can write:
L / θ = C / 2π
As the circumference C = 2πr,
L / θ = 2πr / 2π
L / θ = r
Multiplying both sides by θ gives the arc length formula:
L = r × θ
Thus, arc length is equal to multiplying the radius by the central angle (in radians).
How to calculate arc length and area: Example
Choose the radius of your circle. For example, let’s say it’s 15 cm. (You can also enter the diameter into the arc length calculator.)
Determine the angle between the endpoints of the arc. Let’s say it’s 45 degrees or π/4 radians.
Calculate the arc length using this formula:
L = r × θ = 15 × π/4 = 11.78 cm
Find the area:
A = r² × θ / 2 = 15² × π/4 / 2 = 88.36 cm²
You can also use the arc length calculator to find the radius or central angle. Just enter any two known values, and it will calculate the rest instantly.
How to Find Arc Length
The arc length represents a portion of the circumference of a complete circle. To find it, calculate the fraction of the circle formed by the central angle, then use the radius to determine the total circumference. Finally, multiply them together. Let’s work through an example where the central angle measures 72° and the radius is 3 meters.
First, find what fraction of the circumference of the circle the arc covers. A complete circle measures 360°. Our angle is 72°. Write that as a fraction: 72/360. This simplifies to 1/5. That means the arc length equals one-fifth of the total circumference. Next, let’s calculate that circumference.
We use the formula for circumference: C = πd when we know the diameter, or C = 2πr when we know the radius. Here, the radius is 3, so C = 6π meters, which is about 18.8495559 meters.
Now multiply that result by 1/5 (or 0.2). The arc length is 3.769911 meters. Note that the unit here is always a measure of length.
How to Find the Area of a Circle
Just as the arc length represents part of the circumference, the area represents part of the total area of a circle. To find it, determine the fraction of the circle based on the central angle, then calculate the area of the circle using the radius. Finally, multiply the areas. Let’s try the same example with a 72° central angle and a radius of 3 meters.
First, calculate the fraction of the area of the circle. A full circle is equal to 360°. Our fraction is 72°. The fraction is 72/360, which is 1/5. This means that the area is one-fifth of the total area. Now we need that area.
The formula for area is A = πr². With a radius of 3, A = 9π square meters, which is approximately 28.27433388 square meters.
Multiply it by 1/5 (or 0.2) to get the sector area: 5.654867 square meters. Keep in mind that sector areas always have square units because they represent space.
FAQs: Frequently Asked Questions
Q1. What is the use of arc length?
Arc length is used in many real-world situations. For example, if a rocket travels on a parabolic path, we can determine how far it has traveled. Similarly, if the path on a map is curved, we often need to know how far we must drive to reach our destination.
Q2. When to Use the Arc Length Formula?
We use the arc length formula to measure distance along a circle or any curved segment, called an arc. Arc length represents part of the circumference of a circle. To calculate arc length, the circumference of the circle must be known.
Q3. What Affects Arc Length?
In GMAW, an effective way to control arc length is by adjusting the arc voltage. With a constant-current power supply and variable wire-feed speed (WFS), the arc voltage controls the wire feed motor, which in turn corrects the arc length.
Q4. Is arc length always positive?
Arc length is always a positive value because it measures distance along a curve, which cannot be negative. Distances never go below zero, so arc length is always positive or zero.
