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Circle area formula1/6/2024 ![]() We can make the pattern of the calculation clear by writing things algebraically. Try it out on your caculator (but don't use the trigonometric function keys) and hence find and. ![]() We can not quite do that, but we do know the formulae (from the standard trigonometric identities) Ideally, we would like to know how to calculate from. (in other words we double the number of sides each time), we begin to see a way forward. ![]() However, if instead of trying to calculate for all, we concentrate on,. How can we calculate ? The answer is that we cannot with the tools presented here. The second is that we cheated when we used our calculator to evaluate since the calculator uses hidden mathematics substantially more complicated than occurs in our discussion. The first is that we need to take large to get a good approximation to. There are two problems with our formula for. If you calculated the length of the perimeter for an inscribed square or triangle, does our general formula for an n-sided polygon agree for n = 3 and 4? If you try to use your calculator to calculate, , you'll observe that the results aren't a very good approximation for. Trying out our formula on a hand calculator, we get When is large, we expect that will be close to. Since we are interested in which is half the length of the perimeter of the circle of radius, we consider If is the mid point of then is a right angled triangle with hypotenuse of length one and angle If we look at one of these triangles, say, we see that (as we have drawn things) Observe that the polygon is made up of triangles of exactly the same shape. We use trigonometry to find a general formula for the length of the perimeter of an inscribed -sided regular polygon. We can approximate the circle with n-sided polygon,
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