ShmoopTube

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Area of a Sector, a la Shmoop.

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The King of Bavaria has a circular plot of land with a radius of 50 meters

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that he’d like to turn into a garden, where he can grow his own Bavarian crème donuts.

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Shh, don’t tell him. It would break his heart.

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He left his gardener the instruction that the garden should be divided into fourteen

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“slices” of equal area, and all of them planted with a different variation of crème donut.

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Before the gardener could ask any questions, the King of Bavaria had skipped away to look

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after his swans. Fortunately, this particular gardener remembered

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his geometry, specifically the formula for the area of a sector, or “slice” of a circle.

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Area equals one-half the radius of the circle times the arc length of the desired sector.

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To find the arc length of the desired sector, we need the radius and central angle in radians:

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An entire circle encompasses 2 pi radians, and the king wants it divided 14 ways,

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so the central angle of each sector would be 2 pi divided by 14.

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2 divided by 14 simplifies to 1 over 7, so we can reduce 2 pi over 14 to pi over 7 radians.

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Multiply this by the radius of the garden, 50 meters, and we have 50 pi over 7 meters

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for the arc length.

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Now we can find the area of the sector.

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1/2 times 50 meters times 50 pi over 7 meters is the same as 25 meters times 50pi over 7 metes.

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This equals 1250 pi over 7 meters squared…

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…roughly 561 meters squared per sector of the garden.

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And now to figure out a way to cultivate donuts.

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The gardener had better come up with something.

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The King gets cranky when he hasn’t gotten his sugar fix.

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