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The radius of a circle is 10 centimeters and the perimeter of a sector is 25 centimeters.
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Find the area of the sector.
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Letβs begin with a diagram to visualize the situation.
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We have a circle with a radius of 10 centimeters.
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Weβre then told that there is a sector of this circle with a perimeter of 25 centimeters.
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Now a sector of a circle is an area enclosed between two radii.
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So, each of these lines, which enclose the sector, are 10 centimeters.
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The perimeter of this sector is the distance all the way around its edge.
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So, thatβs the sum of two radii and then the arc length.
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We can use the information weβve been given to form an equation.
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The perimeter is 25 centimeters and the radius is 10 centimeters.
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So, we have the equation 25 is equal to two multiplied by 10 plus the arc length.
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Two multiplied by 10 is 20.
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And then subtracting this value from each side of the equation, we see that the arc length is five centimeters.
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So, we know the arc length, but how does this help us with working out the area of this sector?
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Well, we know that the area of any circle can be found using the formula ππ squared, where again π represents the radius.
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To find the area of a sector, we multiply this value by the fraction of the full circle that the sector represents.
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So, if the angle at the center of the sector is π, then the fraction of the full circle will be π over 360, as there are 360 degrees in a full turn.
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We know the radius of the circle; itβs 10 centimeters.
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So, if we can work out the angle π at the center of this sector, then weβll be able to use this formula to find its area.
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We have a similar formula for working out an arc length.
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Two ππ gives us the circumference of a full circle.
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And then we multiply this by the fraction of the circle that we have, which again is π over 360.
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So, as we know the radius of our circle, and weβve worked out the arc length of this sector, we can use this formula for arc length to work out the angle π at the center of the sector, which weβll then be able to substitute into our formula for its area.
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Substituting 10 for the radius and five for the arc length into this last formula gives π over 360 multiplied by two multiplied by π multiplied by 10 is equal to five.
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We can solve this equation to find the value of π.
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First, we multiply by 360 to bring this out of the denominator on the left-hand side, which will give five multiplied by 360 in the numerator on the right.
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We also need to divide by two and π and 10, which means weβre dividing by 20π.
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So, we have that π is equal to five multiplied by 360 over 20π.
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Simplifying the numbers in this value gives 90 over π because 90 is five multiplied by 360 over 20.
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And if we want to work this out as a decimal, itβs equal to 28.647 continuing.
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So, now that we know the value of π, the central angle for this sector, as well as the circleβs radius, we can apply our formula.
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Our formula, remember, is π over 360 multiplied by ππ squared.
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Now, we could substitute our decimal value of π.
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But if we use the exact value of π as 90 over π, then weβll get an exact value as our answer.
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We can recall that π over 360 is equal to π multiplied by one over 360.
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So, substituting 90 over π for π and 10 for π, we have that the area of the sector is equal to 90 over π multiplied by one over 360 multiplied by π multiplied by 10 squared.
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The factor of π in the denominator of our value for π will cancel with the π in the numerator of the formula, which is why we used this exact value.
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90 and 360 can also be canceled because they share a common factor of 90.
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90 divided by 90 is one, and 360 divided by 90 is four.
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So, now weβre left with one multiplied by one multiplied by 10 squared in the numerator and just four in the denominator.
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10 squared is 100.
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So, we have 100 over four.
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And 100 divided by four is 25.
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The units for this area will be centimeters squared.
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So, we found that the exact area of this sector is 25 centimeters squared.