Study Guide Review Arc Length and Sector Area Module 16
Sectors, Areas, and Arcs
Equally you may recall from geometry, the area A of a circle having a radius of length r is given:
The circumference C (that is, the length around the exterior) of that aforementioned circumvolve is given past:
These are the formulas give us the area and arc-length (that is, the length of the "arc", or curved line) for the entire circumvolve. But sometimes we need to work with just a portion of a circle's revolution, or with many revolutions of the circle. What formulas practise nosotros utilise so?
If we start with a circle with a marked radius line, and turn the circle a bit, the area marked off looks something similar a wedge of pie or a slice of pizza; this is called a "sector" of the circle, and the sector looks like the green portion of this picture:
The angle marked off by the original and terminal locations of the radius line (that is, the angle at the center of the pie / pizza) is the "subtended" bending of the sector. This angle can too be referred to as the "cardinal" bending of the sector. In the film above, the central angle is labelled as "θ" (which is pronounced as "THAY-tuh").
What is the surface area A of the sector subtended by the marked central bending θ? What is the length s of the arc, being the portion of the circumference subtended by this angle?
To determine these values, let'south first take a closer await at the area and circumference formulas. The area and circumference are for the entire circle, one full revolution of the radius line. The subtended bending for "one total revolution" is 2π. So the formulas for the expanse and circumference of the whole circle can be restated equally:
What is the point of splitting the angle value of "one time around" the circumvolve? I did this in gild to highlight how the angle for the whole circle (being 2π) fits into the formulas for the whole circle. This and then allows us to run into exactly how and where the subtended bending θ of a sector will fit into the sector formulas. At present we tin replace the "one time around" angle (that is, the 2π) for an unabridged circle with the measure of a sector's subtended angle θ, and this will give us the formulas for the area and arc length of that sector:
Note: If you are working with angles measured in degrees, instead of in radians, then you lot'll need to include an extra conversion factor:
Confession: A large office of the reason that I've explained the human relationship between the circle formulas and the sector formulas is that I could never keep rails of the sector-area and arc-length formulas; I was always forgetting them or messing them up. But I could e'er remember the formulas for the area and circumference of an entire circle. So I learned (the difficult way) that, by keeping the higher up relationship in mind, noting where the angles go in the whole-circle formulas, information technology is possible always to go along things directly.
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Given a circle with radius r = 8 units and a sector with subtended angle measuring 45°, observe the surface area of the sector and the length of the arc.
They've given me the radius and the primal bending, and so I can simply plug straight into the formulas, and simplify to get my answers. For convenience, I'll starting time convert "45°" to the corresponding radian value of . And so I'll do my plug-n-chug:
And then my answer is:
area A = 8π foursquare units, arc-length southward = 2π units
Notice how I put "units" on my answers. If they'd stated a specific unit for the radius, like "centimeters" or "miles" or whatever, and so I could have been more specific in my respond. As information technology was, I had to be generic.
Many times, if the question doesn't state a unit, or just says "units", and then yous can probably get abroad without putting "units" on your answer. However, this often leads to the bad habit of ignoring units entirely, and then — surprise! — the instructor counts off on the test because you didn't include any units. It's probably better to err on the side of caution, and always put some unit, fifty-fifty if it'due south only "units", on your answers.
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Given a sector with radius r = 3 cm and a corresponding arc length of 5π radians, discover the expanse of the sector.
For this exercise, they've given me the radius and the arc length. To find the area of the sector, I need the measure of the central angle, which they did non give me. However, the formula for the arc length includes the cardinal bending. Then I can plug the radius and the arc length into the arc-length formula, and solve for the measure out of the subtended bending. Once I've got that, I can plug-n-chug to find the sector area.
So the fundamental angle for this sector measures . Then the area of the sector is:
And this value is the numerical portion of my answer. Since this value stands for "area", which is a square dimension, I'll desire to remember to put "squared" on the units they gave me for the radius.
Sometimes, an exercise will give you information, but, similar the above, information technology might non seem like it's the information that you lot really need. Don't be afraid to fiddle with the values and the formulas; attempt to see if y'all can figure out a dorsum door in to a solution, or some other manipulation that'll give you lot want you need. It's okay non to know, correct at the offset, how you're going to reach the end. (And, if they requite you lot, or inquire for, the diameter, remember that the radius is half of the bore, and the diameter is twice the radius.)
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A circumvolve'due south sector has an expanse of 108 cm2 , and the sector intercepts an arc with length 12 cm. Detect the diameter of the circumvolve.
They've asked me for the diameter. The formulas I've learned use the radius. But I tin can find the radius, and then double it to get the diameter, so that'south not a problem. All the same, they've asked me for a length, given the arc length and the expanse, each of which uses the radius and the subtended bending. And I have neither of those values. So what do I do?
When I tin't recollect of anything else to practice, I plug whatsoever they've given me into any formulas might relate, and I hope something drops out of it that I can use. So:
I tin can substitute from the second line in a higher place into the first line above (after some rearrangement), and see if the upshot helps me at all:
Ha! I found the value for the radius! I don't take the value for the cardinal bending, simply they didn't ask for that, and it turns out that I didn't need information technology anyhow. They asked me for the bore, which is twice the radius, then my answer (including the units!) is:
d = 36 cm
Source: https://www.purplemath.com/modules/sectors.htm
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