ShmoopTube

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Perimeter of Irregular Shapes, a la Shmoop. Arnold the Alien buys a square patch of land

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with a perimeter of 9 yards. He’s a small alien.

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His supervisor Bill tells him that the land is not nearly interesting enough for a crop

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circle. Arnold immediately goes out and buys four

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semicircular plots of land around the square…

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… where the diameter of each circle is equal to one of the square's sides.

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Before he can make the crop circle, Arnold needs to know the total area of his new plot

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of land. First we’ll find the area of the inner square

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of land. We’ll call the length of each side of the square “a”.

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Now, we know that the Perimeter equals nine yards.

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First we want to find out the length of one side, so we write out our equation as “a”

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plus “a” plus “a” plus “a”...or put another way… 4a equals 9.

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By dividing four from both sides of the equation...we get the length of “a” which equals nine-fourths…

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Now we need to square “a” to get the area of our square. We square 9 fourths to get…

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… eighty-one sixteenths, the area of the original plot of land.

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Now onto the 4 semi-circles.

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The diameter of the circle is equal to the length of the square, which if you’ll remember

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was nine-fourths…

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… so the radius of the circle, is half of that, or nine-fourths times one-half...which

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equals nine-eighths.

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The formula for the area of a semicircle is one-half pi times the radius squared.

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By plugging in nine-eighths as the radius of the circle, we get one-half pi times nine-eighths

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squared…

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When we square nine eighths we get 81 over 64….times one half pi….

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Next we multiply 81 64ths by 1 half pi to get 81 pi over 128…the total area of 1 semicircle…

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Now we add them all to get the total area…

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Since we have 4 semicircles and one semicircle equals 81pi over 128, we multiply 81pi over

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128 by 4 to get….324 pi over 128… the total area of all four semicircles.

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We need to add the total area of the square… which is 81 16ths… to our four semicircles.

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By estimating that pi equals 3.14, we can type into our nifty calculators... eighty-one

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sixteenths plus 324 pi over 128 equals... thirteen point seven.

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Bingo! The area of the new land parcel is 13.7 yards.

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Good crop circle, Arnold. Way to stick it to your supervisor.

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