![]() #Area in math circles in rectangle how toGo back to Calculators page How to Calculate the Area of a Rectangle Jump to Real-life Applications of these Calculations Jump to Calculating the Diagonal of a Rectangle Jump to Calculating the Perimeter of a Rectangle Although has now been calculated to a trillion (that is to say 1,000,000,000,000) places, the hunt for better ways to do the computation will continue.Jump to Calculating the Area of a Rectangle Since then even faster methods have been discovered. Using this algorithm, it is easy to compute the first few approximations, say P 1, P 2, P 3 However, the method is simple enough to be given here. To show that it works requires hard work and first year university calculus. In the 1970's the mathematical world was stunned by the discovery by Brent and Salamin of method which roughly doubled the number of correct decimal places at each step. Places after the nth step was proportional to n. Other methods were developed using calculus, but it remained true for these methods that number of correct decimal Roughly speaking the number of correct decimal places after the nth step is proportional to n. The accuracy of this approximation increases fairly steadily, as you will have seen if you used your calculator to compute the successive values of s n. Using the rule above to write the successive values for (starting with ), we obtain Vieta's formula:įrom an elementary point of view this formula is nonsense, but it is beautiful nonsense and the theory of the calculus shows that, from a more advanced standpoint, we can make complete sense of such formulae. Where, in some sense, we multiply together an infinite number of terms. We have shown thatĪnd the larger is, the better the approximation. Vieta used his method to calculate to 10 decimal places. Nowadays we leave computation of square roots to electronic calculators, but, already in Vieta's time, there were methods for calculating square roots by hand which were little more complicated than long division. If you have a computer or programmable calculator, see if you can compute. The definitions of and lead to the rulesĪnd we can use these to easily compute successive values of. Let us take and writeĪnd, continuing as far as we like, we see thatĪlthough Vieta lived long before computers, this approach is admirably suited to writing a short computer program. 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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