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The area of a circular sector is five-eighths of the area of the circle.
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The radius is 27 centimeters.
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Find the central angle in radians giving the answer to two decimal places and the perimeter of the sector giving the answer to the nearest centimeter.
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Let’s sketch this out to get an idea what’s going on.
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We start with the circle.
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We know that the sector area is five-eighths the circle area.
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Five-eighths is a little bit more than half.
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If the sector was four-eighths, if it was one-half of the circle, it would be this much.
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But we have five-eighths, so we’ll make it a little larger.
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We can identify the sector and label the radius.
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We need to know what this angle is.
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For now, we can call it 𝜃.
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And we’re interested in the angle 𝜃 given in radians.
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In radians, a full turn equals two 𝜋.
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Two 𝜋 is the radian measure of 360 degrees.
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To find our central angle, we know that we’re going five-eighths of the way around the circle.
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We’re interested in radians.
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And that means we want five-eighths times two 𝜋.
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Multiplying that together, we can reduce a little bit.
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Two goes into eight four times.
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Our central angle 𝜃, in radians, is five 𝜋 over four.
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However, we need to give this answer to two decimal places.
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So, we’ll plug five 𝜋 divided by four into the calculator.
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And we’ll get 3.9269 continuing.
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To round this to the second decimal place, we’ll look to the deciding digit on the right.
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This six is larger than five.
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We need to round up.
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So, we get the central angle 𝜃 is 3.93 radians.
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We’re also interested in the perimeter of this sector.
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The perimeter will be the distance around this sector, which is a radius plus a radius plus an arc length.
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We usually use the variable 𝑠 to represent the arc length.
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To find that arc length, we start with our central angle in radians and we multiply it by the radius.
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I’m gonna plug in five 𝜋 over four for our central angle.
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You could also use this rounded value.
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But since we’re already going to round at the end of this step, it’s best to use a value that hasn’t already been rounded.
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And the radius is 27.
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So, we need to multiply five 𝜋 over four times 27.
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And we’ll get the arc length equals 106.028 continuing.
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If we know that the perimeter equals the radius plus the radius plus the arc length, we plug in 27 for the radius and 106.028 continuing for the arc length.
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The best way to do this is to keep this value, this answer, in your calculator and just add 27 plus 27 to it.
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This will allow us to round in the final step.
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Now we have 160.028.
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Rounding to the nearest centimeter is the nearest whole number.
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The digit on the right, the deciding digit, is a zero.
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So, we round down.
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The perimeter equals 160.
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And our units were being measured in centimeters.
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That means the perimeter, the distance around this sector, is 160 centimeters.