Step-by-step explanation:
Arc length formula
arc length
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 / 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 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 / 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
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: A = r² * θ / 2 = 15² * π/4 / 2 = 88.36 cm².
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.
