When working with circles, you may come across a situation where you do not need to know the entire circumference of it, only the arc length between two points around it. Generally speaking, an arc of a circle is a smooth curved bounded by two distinct points. Alternatively, any arc is a portion of the circumference of a circle, bounded by two points, rather than the entire circle. Arc length is the distance of the arc bounded by two points. A sector of a circle is the area that is between the two radii of the circle that uses the endpoints of the arc.
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In the above image, the endpoints of arc BC are connecting from center point A, forming radii AC and AB. The sector is the area between those two radii.
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To find the length of the arc, we need two things: the length of the radius and the measure of the central angle. We will label arc BC as {eq}s {/eq}. Additionally, we need the length of one of the radii ({eq}r {/eq}) and the measure of the central angle (the angle whose vertex is at the center of the circle), which we will label as {eq}\theta {/eq}. The formula looks like this: $$s=r\theta $$
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Keep in mind that the measure of the central angle must be expressed in radians rather than in degrees. To convert from degrees to radians use the following formula:
$$\theta= d^o( \frac{\pi}{180^o}) $$ where {eq}d^o {/eq} is the measure of the central angle, in degrees. If we were to substitute this piece by {eq}\theta {/eq} we would have this expanded form of the arc length formula:
$$s=rd( \frac{\pi}{180}) $$.
Once you know your radius length and the degree (or radian) measure of your central angle, follow these steps:
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First, we identify our angle measure and convert it into radians. In this example, the angle measure is 82 degrees, so we must convert that into radians as shown below:
$$\theta = 82(\frac{\pi}{180}) $$ We then simplify our fraction {eq}\frac{82}{180} {/eq}: {eq}\pi {/eq}, leading to this measure in radians:
$$\theta=\frac{41}{90}\pi $$ Now we plug in the angle measure and our radius measure in the arc length formula and calculate the length:
$$s=r\theta $$ $$s=3.5(\frac{41}{90}\pi) $$$$s\approx3.5(0.4556)(3.14) $$$$s\approx5.0 $$
Therefore, the length of arc BC is about 5.0 centimeters.
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We first convert the angle measure into a radian measure.
$$\theta=\frac{45}{180}\pi=\frac{1}{4}\pi $$Then we plug in our values to find the measure of arc XY:
$$s=r\theta $$ $$s=7.3(\frac{1}{4}\pi) $$ $$s\approx7.3(0.25)(3.14) $$ $$s\approx5.7305 $$ $$s\approx5.7 $$ Thus, the length of arc XY is approximately 5.7 units.
The formula for the area of a sector, or the area of the arc. We need the same things as with the formula to find the length of a sector (arc length): the length of the radius and the measure of the central angle. The area of the sector is the region bounded between the sides of the central angle. The formula is shown below:
$$A=\frac{\theta}{360^o}\times{\pi}{r^2}\;or\;A=\frac{{\pi}{r^2}\theta}{360^o} $$
In this formula, you do not need to convert the angle measure into a radian measure.
To find the area of the arc bounded by the central angle of a circle, follow these steps:
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$$A={\pi}r^2\approx3.14(15.5^2)\approx754.39 $$
So, the area of the sector bounded by angle XZY is about 125.76 square units.
To calculate the length of the arc or the length of a sector, follow these steps:
To calculate the area of the sector bounded by a central angle, follow these steps:
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The formula to find the arc length of a sector is as follows: {eq}s=r\theta {/eq}, where you would need the length of the radius between the endpoint of the arc and the center of the circle, and the angle's measure in radians.
The formula to calculate the area of the sector is $$A=\frac{\theta}{360^o}\times{\pi}r^2 $$, where you would need the measure of the central angle in degrees and the radius length.
To find arc length given an angle measure and a radius, you would need to first convert the angle measure into radians, then multiply that number by the length of the radius.
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