The Circle and Regular Polygons

Let’s turn now to the simplest of all curved lines, the circle. We shall study its properties and its relation to straight lines and to figures made up of straight lines, especially polygons.

In a plane all the points at a given distance from a given fixed point are said to form a circle. A circle is a set of points in a plane, all of which are the same distance from a given point.

The fixed point Ο is called the CENTER of the circle, from which all other points are equidistant. The distance r is called the RADIUS. A radius is a line segment from the center of a circle to a point on the circle.

Every point at a distance r from Οis said to be on the circle. Every point at a distance less than r from Οis said to be inside the circle, and every point at a distance greater than r from Οis said to be outside the circle. If the center of the circle is taken asthe origin of a rectangular network, it follows from the Pythagorean Theorem that the coordinates (x, y) of every point P of the circle will satisfy the equation x ² +y² = r². This equation is the equation of the circle.

On any half-line with end-point Ο there is a point at the Fig. 1. distance r from Ο. We may select one such half-line – for example, OX in Fig.1 as a reference line from which to measure the angles to all other such half-lines. If we measure these angles in degrees, then on every half-line which makes an angle of between 0 and 360 degrees with OX there is a point of the circle.

All the points of the circle which lie on half-lines from p to q (Fig.2) are said to form an arc PQ of the circle.The word " arc " comes from a Latin word meaning " bow. "

In Fig.2 arc PQ corresponds to angle POQ. Angle POQ is called a central angle because its vert ex is at the center of the circle.

A chord of a circle is a line segment whose two endpoints lie on the circle. The diameter, passing through the circle centre, is the longest chord in a circle. A tangent to a circle is a straight line that touches the circle at a single point, thus guaranteening that all tangents are perpendicular to the radius and diameter. A secant is an Fig. 2. extended chord: a straight line

cutting the circle at two points.

In Fig.2notice that the half-lines OP and OQ form two angles whose sum is 360°. Ordinarily when we speak of angle POQ we refer to the lesser of these two angles; only rarely do we mean the greater angle. Similarly, when we speak of the arc PQ, we ordinarily mean the arc that corresponds to the lesser central angle POQ; but occasionally we mean the arc that corresponds to the greater central angle. Except for the end-points P and Q, all the points of the first arc PQ aresometimes called the minor arc which are distinct from the points of the second arc PQ which are sometimes called the major arc. If the two central angles POQ are equal, Fig. 3. each of the two corresponding areaPQ

is called a semicircle.

Instead of speaking of the perimeter of a circle, we usually use the term circumference to mean the distance around a circle. We cannot find the circumference of a circle by adding the measure of the segments, because a circle does not contain any segments. No matter how short an arc is, it is curved at least slightly. Mathematicians have discovered, that the ratio of the circumference (c) to a diameter(d) is the same for all circles and is expressed . The number or (since d=2r – the length of a diameter is equal to twice the length of a radius), which is the same for all circles, is designated by π.

= π or = π

By using the multiplication property of equation, we obtain the following:

c= πd or c= 2πr.