Geometry

Polygon Coordinates and Areas

We’ve been collecting techniques for finding areas of polygons, mostly using their side lengths. We started with triangles (Heron’s formula), then quadrilaterals (Bretschneider’s formula and Brahmagupta’s formula), and the fact that the largest possible area is attained when the vertices lie on a circle. We’ll look at one more way to find area, using coordinates …

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Area of a Triangle: Heron’s Formula II

Last time we looked at a very useful formula for finding the area of any triangle, given only the lengths of its sides. Today I want to look at several problems in which the formula has been used, some of them surprising.

Tangents Without Calculus

I always like solving advanced problems with basic methods. For example, many problems that we usually think of as “algebra problems” can be solved by creative thinking without algebra; and some “calculus problems” can be solved using only algebra or geometry. Using simple tools for a big job requires more thought than using “the right …

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Distances to an Arc: Exact and Approximate Formulas

(A new question of the week) It can be an interesting challenge to be presented with a formula and asked how it was derived. This becomes a bigger challenge when the formula is only approximate, so we have to figure out how to arrive at this particular approximation. But it is impressive when several different …

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More on Faces, Edges, and Vertices: The Euler Polyhedral Formula

Last time we looked at how to count the parts of a polyhedron, and a mention was made of Euler’s Formula (also called the Descartes-Euler Polyhedral Formula), which says that for any polyhedron, with V vertices, E edges, and F faces, V – E + F = 2. We should take a close look at that simple, yet amazing, …

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