Measuring An Angle

Today we will focus on a topic from elementary geometry namely the concept of measurement of angles. The idea of an angle is a simple one in the sense that it is made by two rays emanating from the same point. But the measurement of angles is not that simple as it appears. Many theorems in elementary geometry deal with ideas which involve the concept of measurement of angles but they assume the understanding of this measurement in an implicit fashion.

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Cavalieri's Principle and its Applications

Introduction

In this post we will discuss something is which is very elementary and fascinating, yet not available in a high school curriculum. More precisely we will study a part of solid geometry related to calculation of volume of solids. In so doing we will need the famous Cavalieri's Principle which relates volumes of two solids under certain conditions.

Cavalieri's Principle

The Cavalieri's Principle states that:
If two solids lie between two parallel planes and any plane parallel to these planes intersects both the solids into cross sections of equal areas then the two solids have the same volume.

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Conics and the Cone: Part 3

After having dealt with the case of circular sections of an oblique cone in previous post, let's now focus on the conics. We will first treat the case of a parabola as this is the simplest case after the case of a circular section.

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Conics and the Cone: Part 2

In the previous post we established that sections of a right circular cone are the familiar curves (ellipse, parabola, hyperbola) having the focus directrix property. Now we will have a look at the more general case when the cone is not right but oblique. Our approach will be identical to the one followed by the great Greek geometer Apollonius. However we will not be developing a systematic theory of conics as described by Apollonius, but rather focus on the interesting results which will help us to connect them with the modern definition of conics. In doing so we will observe that the main tool used by Apollonius is the similarity of triangles.

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Conics and the Cone: Part 1

Introduction

While studying co-ordinate geometry (aka analytic geometry) in intermediate classes we normally arrive at the study of conic sections or in short "conics". Three new curves namely "ellipse", "parabola", and "hyperbola" come into picture and their theory is quite unlike those of the elementary geometrical objects (line, triangle, circle etc) studied in secondary classes. In case of the elementary geometrical objects like points, lines, triangles, circles we have two approaches: 1) using the axioms of Euclid and then deducing the properties of these objects logically from Euclid's axioms and, 2) using the language of coordinate geometry which transforms the subject of geometry into algebra. Unfortunately the beautiful approach of using Euclid's axioms is discarded in higher secondary classes in favor of the approach using coordinate geometry which makes the subject dull with huge amount of laborious algebraical manipulations. In fact most students think that the only way to study new curves like ellipses, parabola and hyperbola is through coordinate geometry.

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