WEBVTT
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The perimeter of a regular pentagon is 85 centimeters.
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Find the area, giving the answer to the nearest square centimeter.
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The key piece of information given in this question is that this pentagon is regular.
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We have a formula that we can use for calculating the area of any regular polygon.
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For a regular polygon with π sides each of length π₯, its area is equal to one-quarter multiplied by π multiplied by π₯ squared multiplied by cot of π over π.
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Remember, cot is the reciprocal of tan.
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Cot of π over π is equal to one over tan of π over π.
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This question is about the pentagon.
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And therefore, the number of sides π is equal to five.
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We havenβt been given the side length directly in the question.
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But instead, weβve been told the perimeter of the pentagon.
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However, we can form a simple equation to calculate this side length.
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The regular pentagon has five sides, all of length π₯.
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And therefore, its perimeter will be equal to five π₯.
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So we have the equation five π₯ is equal to 85.
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Dividing both sides of this equation by five tells us that the side length of this pentagon is 17 centimeters.
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Now that we know the values of both π and π₯, we can substitute them into our formula for the area.
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This gives the area of the pentagon is equal to one-quarter multiplied by five multiplied by 17 squared multiplied by cot of π over five.
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Now we need to use a calculator to evaluate this.
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And the important thing is that you make sure the calculator is in the correct mode.
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As the angle has been specified using π, this means that weβre using radians to measure angles, not degrees.
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So the first part of this simplifies to 1445 over four and then multiplied by cot π over five.
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The simplest way to evaluate this on your calculator is probably to recall that cot π over five is equal to one over tan of π over five.
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So this whole thing is equal to 1445 divided by four tan π over five.
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Evaluating this gives 497.21796.
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Now if you havenβt got that and youβre certain that youβve typed it into your calculator correctly, then again check that your calculator is in the correct mode.
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It needs to be in radians.
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Now the question has asked for the answer to the nearest square centimeter.
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So the final step is to round this value.
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And we have that the area of this regular pentagon to the nearest square centimeter is 497 square centimeters.