 From the Rhine Papyrus, we have a number of examples of ancient Egyptian geometric computation. The examples are fairly clear, but they do provide us with some difficulties in the interpretation of the words involved. For example, one of the problems is that Achmos computes the area of a triangle with a base of four khet and a mirrored of ten khet. So there are many unfamiliar words here, but we do know that a khet is a unit of length about 150 feet. And Achmos computes the area, takes half a four to get two, and then multiplies by ten to get twenty the area. And if we think about our area formula for a triangle, this suggests that the term mirrt must mean the altitude of the triangle. And that makes perfect sense, until you look at the picture that accompanies the problem. And we can read the hieratic number four, which indicates our base here, and the hieratic number ten, which indicates the mirrt. And given the picture, there is no reasonable interpretation of mirrt other than the side of a triangle. And so here's the problem. While reading mirrt as altitude makes Achmos' computation correct, the figure shows the mirrt should be read as side. And so Achmos' computation of the area should be regarded as incorrect, except he might not be computing area. In other words, the problem may be the mistranslation of the word area. We tend to think about area as a geometric quantity, but a common use for a computation like this is valuation of land for purposes of taxation. This is called a cadastral account. And while these quantities appear to be and are often called by the same names as geometric quantities like area, perimeter, and so on, they're a purely legal concept, suitable for use by government officials. We'll see more examples of this type of cadastral computation as we go along. So if we want to think about parcels of land, another type of parcel of land might be trapezoidal. And so Achmos also talks about the area of a truncated triangle with base six ket, mirrt 20 ket, and four ket, the truncating line. And Achmos' computation, we're going to add four and six to make ten, divide by two to get five, and then multiply by 20 to get 100 the area. And again, this computation is correct if we regard the mirrt as the altitude of a trapezoid. But again, Achmos' figure is consistent with the idea that mirrt is the length of one side. How about circles? Oddly enough, Achmos computes the volume of a cylinder before finding the area of a circle. And Achmos does the following computation to find the area of a circle with a diameter of nine ket. And so Achmos takes away one ninth of the diameter, leaving eight ket, then multiplies eight by itself to obtain 64 the area. Now there's a lot of speculation over how the Egyptians derive this result, all of which is, well, speculation. More importantly, this leads to a rather contentious question, which is who discovered pi? If we compare the Egyptian formula for the area of a circle to our own, we find the following. Our formula, pi r squared, and Achmos' formula, will be the square of eight ninths of the diameter. And since the diameter is twice the radius, we can substitute, simplify, and here's the odd part, solve for pi, and we get. And you'll see in many sources that the Egyptians approximated pi with 3.16. But did they? And here's the thing to remember. In the context of geometry, pi is the ratio of the circumference of a circle to its diameter. This ratio appears nowhere in Egyptian mathematics. At no point do the Egyptians ever compare the circumference of a circle to its diameter. And so we have to say that the Egyptians did not have a value for pi, they had a way of computing the area of a circle, and the area they found might have been a cadastral one, not a geometric one. And finally, there's one last bit of geometry. There's a claim that the ancient Egyptians made right angles by making a 3, 4, 5 triangle with a knotted rope. Now the evidence for the claim is below. And, well, actually there's no evidence that the ancient Egyptians did this. And in fact there's no evidence that the Egyptians were aware the 3, 4, 5 triangle was right angled.