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Find the area of a regular hexagon with a side length of 35 centimetres giving the answer to two decimal places.
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So, let’s take a look at a hexagon.
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So, here I’ve drawn a regular hexagon.
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And each of the sides are 35 centimetres.
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And we know that because a regular hexagon is a hexagon that has all of the sides the same length.
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But it also has each of the interior angles the same.
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So, now what I’ve done is I’ve divided our hexagon into six equal triangles.
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And each of the angles made by these triangles at the centre of the circle are gonna be the same because, as we know, each of these triangles are gonna be identical cause we’ve got a regular hexagon.
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Since working out what 𝑥 is or each one of these angles, we’re gonna divide 360 by six.
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And that’s cause it makes a complete circle in the centre.
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And this makes 60 degrees.
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So, we know that each of those angles is gonna be 60 degrees.
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But what we also know about each of our triangles is that the two lines that are the interior parts of our triangles, so the lines from the edge to the centre, are gonna be the same length.
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And because they’re all the same length, then therefore the two base angles of our triangles must also be equal.
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And to work these out, what we’re gonna do is do 180, because that’s the angles in a triangle, minus 60, because that’s the top angle, then divided by two, because there’re two base angles.
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What it’s gonna give us is 120 over two.
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And 120 over two is equal to 60 degrees.
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So therefore, we can now say that each of our triangles that make up our regular hexagon are going to be equilateral triangles because we’ve got 60, 60, 60 as their interior angles.
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So therefore, if we want to find out what the area is going to be of our regular hexagon, then all we need to do is find the area of one of our triangles and then multiply it by six because they’re all identical.
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Well, if we take a look at the triangle we’ve got, well if we want to find the area of a triangle, what we need to do is multiply the base by the perpendicular height.
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So, now if we take a look at the triangle we’ve got, well if we want to find the area of a triangle, well this is gonna be equal to is a half multiplied by the base times the height.
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And the height is the perpendicular height and I’ll mark this with an orange dash line here.
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And we can see this perpendicular height cause we’ve got a right angle at the bottom.
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Okay, so how do we work out what ℎ is?
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Well, the way that we could work it out is because when we take a look at the triangle that was split into two, what we’ve got is two right triangles.
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We’ll take a look at the right triangle.
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What it is is a triangle with a height, or a perpendicular height, ℎ.
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We’ve got hypotenuse, which is 35 centimetres.
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We’ve got the base, which is 17.5 centimetres cause that’s half of 35 centimetres.
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Now, because it’s a right triangle, we can use the Pythagorean theorem, which tells us that 𝑎 squared plus 𝑏 squared equals 𝑐 squared.
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So, the sum of the two smaller sides squared is equal to the larger or hypotenuse squared.
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Well, if we rearrange this to find one of the short sides, then we can say the short side squared is equal to the hypotenuse squared minus the other short side squared.
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So, we get ℎ squared is equal to 35 squared minus 17.5 squared.
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So therefore, ℎ squared is equal to 918.75.
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So therefore, if we take the square root of both sides of our equation, what we’re gonna get is ℎ is equal to 30.31, et cetera.
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And that’s our ℎ, that’s our perpendicular height.
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At this point, we wouldn’t round because we wanna keep accuracy.
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So, now what we can do is work out the area of our triangle.
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And we can work that out cause it’s a half multiplied by the base multiplied by the height.
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So, it’s gonna be half multiplied by 35 multiplied by 30.31 continued, which gonna give us 530.440 continued.
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Again, not rounding at this stage cause we wanna keep the accuracy.
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As we’re saying that the height would’ve been in centimetres and this area would’ve been in centimetre squared, but I’m gonna put these on at the end when we work out the final area.
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So now, to work out what the area of the hexagon is gonna be, what we do is we multiply our area of our triangle, which is 530.440 continued, by six.
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And that’s because we’ve got six identical triangles making up our hexagon.
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So then, when we calculate that, what we get is 3182.64 centimetres squared, and that’s to two decimal places.
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So, that’s the area of our regular hexagon.
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Well, what we can do is we can check our answer.
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We can do that because there is, in fact, a formula that you might come across for the area of a regular hexagon.
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That is, the area is equal to three root three over two multiplied by 𝑎 squared, where 𝑎 is the length of one of the sides.
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So therefore, we can say that the area of the hexagon is gonna be equal to three root three over two multiplied by 35 squared with 𝑎 being 35, which again will give us 3182.64.
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So, we can definitely say yes, that’s the correct area for our hexagon.