What will be the length of the arc of the circle of radius 10 cm subtending a central angle measuring 45?

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To use the arc length calculator, simply enter the central angle and the radius into the top two boxes. If we are only given the diameter and not the radius we can enter that instead, though the radius is always half the diameter so it’s not too difficult to calculate. 

The calculator will then determine the length of the arc. It will also calculate the area of the sector with that same central angle.

How to Calculate the Area of a Sector and the Length of an Arc

Our calculators are very handy, but we can find the arc length and the sector area manually. It’s good practice to make sure you know how to calculate these measurements on your own.

How to Find the Arc Length

An arc length is just a fraction of the circumference of the entire circle. So we need to find the fraction of the circle made by the central angle we know, then find the circumference of the total circle made by the radius we know. Then we just multiply them together. Let’s try an example where our central angle is 72° and our radius is 3 meters. 

First, let’s find the fraction of the circle’s circumference our arc length is. The whole circle is 360°. Let’s say our part is 72°. We make a fraction by placing the part over the whole and we get \(\frac{72}{360}\), which reduces to \(\frac{1}{5}\). So, our arc length will be one fifth of the total circumference. Now we just need to find that circumference.

The circumference can be found by the formula C = πd when we know the diameter and C = 2πr when we know the radius, as we do here. Plugging our radius of 3 into the formula, we get C = 6π meters or approximately 18.8495559 m.

Now we multiply that by \(\frac{1}{5}\) (or its decimal equivalent 0.2) to find our arc length, which is 3.769911 meters. Note that our units will always be a length.

How to Find the Sector Area

Just as every arc length is a fraction of the circumference of the whole circle, the sector area is simply a fraction of the area of the circle. So to find the sector area, we need to find the fraction of the circle made by the central angle we know, then find the area of the total circle made by the radius we know. Then we just multiply them together. Let’s try an example where our central angle is 72° and our radius is 3 meters.

First, let’s find the fraction of the circle’s area our sector takes up. The whole circle is 360°. Our part is 72°. We make a fraction by placing the part over the whole and we get \(\frac{72}{360}\), which reduces to \(\frac{1}{5}\). So, our sector area will be one fifth of the total area of the circle. Now we just need to find that area.

The area can be found by the formula A = πr2. Plugging our radius of 3 into the formula we get A = 9π meters squared or approximately 28.27433388 m2. 

Now we multiply that by \(\frac{1}{5}\) (or its decimal equivalent 0.2) to find our sector area, which is 5.654867 meters squared.  Note that our answer will always be an area so the units will always be squared.

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.

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:

As circumference C = 2πr,

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

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

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,

The formula for the area of a sector is:

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!

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.

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

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.

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.