WEBVTT
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𝐴𝐵𝐶𝐷 is a square of side 140.
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Calculate the perimeter of the shaded region, taking 22 over seven as an approximation for 𝜋.
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We’re asked to find the perimeter of the shaded region.
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We can remember that the perimeter is the distance around the outside.
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Therefore, we’ll need to calculate the distance of these two curved sections.
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So, let’s have a think about one of these curved sections.
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This will in fact be made of a circle that has the same radius as the length of the side in the square.
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Therefore, we know that the radius of this circle would be 140 units.
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As 𝐴𝐵𝐶𝐷 is a square, we know that the angle at 𝐷𝐶𝐵 will be 90 degrees, which means that the shape is a quarter circle.
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The arc created at 𝐷𝐵 is also part of a quarter circle.
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In order to answer the question, to find the perimeter of the shaded region, we’ll then need to calculate the perimeter of two of these quarter circles.
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In order to find the perimeter or circumference of a whole circle, we use the formula that this is equal to 𝜋 times the diameter.
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To find the perimeter of a quarter circle, we take the circumference and divide it by four.
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So, we’ll be calculating 𝜋 times the diameter over four.
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In the question, we’re given that the radius is 140.
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So, to find the diameter, we multiply 140 by two.
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We can then plug this into our calculation to give us the perimeter is equal to 𝜋 times 280 over four.
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We can simplify this calculation so that we’re calculating 70 times 𝜋.
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We’re told to take 22 over seven as an approximation for 𝜋.
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So, we’ll have the calculation of 70 times 22 over seven.
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This will simplify to give us 220 units.
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And now we’ve found the perimeter of a quarter circle, we can find the perimeter of two quarter circles.
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We take our value of 220, and we multiply it by two, which gives us a value of 440 units.
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And so, our answer for the perimeter of the shaded region would be 440 units.
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There is, of course, an alternative method that we could’ve used.
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And that is by understanding that two quarter circles added together would give us half of a circle.
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In this method, we take the circumference 𝜋 times the diameter and divide it by two.
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We would’ve calculated 22 over seven multiplied by 280 all over two.
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The numerator of the fraction would cancel to give us 22 times 40 over two.
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This can be further simplified to 11 times 40, which would also give us the answer of 440 units.
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So, we could use either method of finding two quarter circles or finding one half circle to give us the perimeter of this shaded region.