Sectors, Areas, and Arcs (page 1 of 2)


The area A of a circle with radius r is given by A = πr2. The circumference C of that same circle is given by C = 2πr. But these are the formulas for the entire circle. Sometimes you will need to work with just a portion of a circle's revolution, or with many revolutions of the circle.

If you start with a circle with a marked radius line, and turn the circle a bit, the angle marked off by the original and final locations of the radius line is the "subtended" angle of the "sector", the sector being the pie-wedge-shaped section of the circle. This angle can also be referred to as the "central" angle.

What is the area A of the sector subtended by the marked central angle θ? What is the length s of the arc, being the portion of the circumference subtended by this angle?

 

circle with central angle 'theta', radius 'r' marked, delineating sector

To determine these values, take a closer look at the area and circumference formulas. The area and circumference are for the entire circle, one full revolution of the radius line. The subtended angle for "one full revolution" is . So the formulas for the whole circle can be restated as:

    A = ((2pi)/2) r^2, C = (2pi) r

Now that the use of the angle is clearly delineated, we can replace "once around" () with "the subtended angle" (θ) and get the formulas we need for the sector:

    A = (theta / 2) r^2, s = (theta) r

Note: If you are working with angles measured in degrees, instead of in radians, then you'll need to include a conversion:

    A = ((theta) / 2) * (pi /1 80) r^2, s = (pi / 180) (theta) r

I could never keep track of the sector-area and arc-length formulas. But I always remembered the formulas for the area and circumference of a circle. If you keep the above relationship in mind, noting where the angles go in the whole-circle formulas, you should be able to keep things straight.

  • Given a circle with radius r = 8 and a sector with subtended angle measuring 45°, find the area of the sector and the arc length.
  • They've given me the radius and the central angle, so I can just plug into the formulas. For convenience, I'll first convert "45°" to the corresponding radian value of π/4:

      A = ((pi/4)/2)(8^2) = (pi/8)(8^2) = 8pi, s = (pi/4)(8) = 2pi

    area A = 8π, arc-length s = 2π

  • Given a sector with radius r = 3 and a corresponding arc length of , find the area of the sector.
  • For this exercise, they've given me the radius and arc length. From this, I can work backwards to find the subtended angle. Then I can plug-n-chug to find the sector area.

      5pi = (theta)(3), (theta) = (5/3)pi

    So the central angle is (5/3)π. Then the area of the sector is:

      A = ((5/3)pi / 2)*(3^2) = ((5/6)pi)*(9) = (15/2)pi

    A = (15 pi) / 2


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