Thomas Industrial Library

THE CIRCLE

Circles have long intrigued humans. Often called the perfect geometric form,

the circle occurs frequently in engineering design, art, architecture, and applied

mathematics. Almost 2200 years ago, Archimedes sought to find a square with

the same area as a given circle. This problem is called squaring the circle.

The circle forms the base for systems of navigation on the sea and in the air. The

geometry of circles involves not only the relationships of the principal parts of

the circle but also the relation of the circle to lines, angles, and polygons.

Circle

Circles are symbolic of infinity, eternity, and completeness. They appear in the literature and architecture of many cultures.

A circle is the set of all points in a plane equidistant from a given point. This distance is the length of the radius and the given point is the circle’s center . A circle is named by its center point. For example, a circle with center at point O is referred to as “circle O .”

Figure 19.1 is a drawing of a circle with nearby points. Note that a point may lie either on the circle; or it may lie inside the circle, making it an interior point of the circle; or it may lie outside the circle, making it an exterior point of the circle. A common mistake is to view interior points, including the center, of a circle as part of the circle. The circle consists of only those points a given distance from the center.

The plural of “radius” is “radii.”

The radius is any line segment joining the center of a circle with a point on the circle’s circumference.

A chord is a line segment joining any two points on the circle’s circumference.

The diameter is any line segment through the center of the circle joining two points on the circumference. The diameter’s length is twice the length of the radius. Thus, the diameter is the longest chord that can be drawn in a circle.

The words “radius,” “diameter,” and “circumference” can mean either these physical aspects of the circle or the specific measurement assigned to each.

An angle formed by two radii is a central angle . We say that a chord or an angle subtends an arc, or an arc is intercepted by an angle or a chord.

An arc is a portion of the circle lying between two points on the circle. A chord’s endpoints split the circle into two arcs, the minor arc and the major arc , as shown in Figure 19.2. Since the diameter is the longest chord in a circle, it subtends two arcs equal in measure. These arcs are semicircles .

F IGURE 19.2 Arcs of a chord

A semicircle is an arc measuring one-half the circumference of a circle, a half circle.

A wedge-of-pie shaped area bounded by an arc and its two related radii is a sector .

The area bounded by an arc and its chord is a segment .

A secant is a line that intersects the circle at two points. The part of a secant that is not outside the circle is a chord.

A line that intersects a circle at one and only one point is called a tangent . Note that a tangent does not cut through the circle. Referring to Figure 19.3, a tangent at point A is perpendicular to the radius OA . In fact, it is the only line perpendicular to radius OA through point A .

F IGURE 19.3 Tangent line and radi

An inscribed angle in a circle has its vertex on the circle and its sides in the interior of the circle. It subtends an arc, as shown in Figure 19.4 where angle ACB is an inscribed angle and angle AOB is the corresponding central angle . In fact, angle AOB is said to be the central angle subtended by inscribed angle ACB and arc AB is the intercepted arc . Figure 19.4 illustrates an angle inscribed in a circle.

F IGURE 19.4 An inscribed angle

The important geometric features of a circle are summarized in Figure 19.5.

F IGURE 19.5 Important geometric features of a circle

Concentric circles are circles with a common center but different radii, as shown in Figure 19.6.

F IGURE 19.6 Concentric circles

Externally tangent and internally tangent circles are shown in Figures 19.7 A and 19.7 B .

F IGURE 19.7 A Externally tangent circles

F IGURE 19.7 B Internally tangent circles

Polygon/Circle Relationships

A polygon is said to be inscribed in a circle if all its sides are chords of a circle. This is the same as requiring that each vertex of the polygon be a point on the circle, as shown in Figure 19.8A. We could also say that the circle circumscribes the polygon.

F IGURE 19.8 A A regular hexagon inscribed in a circle

F IGURE 19.8 B A regular hexagon circumscribed on a circle

A polygon is circumscribed on a circle if all of its sides are tangent to the circle. A circumscribed polygon is shown in Figure 19.8B. The circle is inscribed in the polygon. Although regular hexagons are shown in Figures 19.8A and 19.8B, an inscribed or circumscribed polygon does not have to be regular. Figure 19.9 shows a nonregular quadrilateral inscribed in a circle.

F IGURE 19.9 A nonregular quadrilateral inscribed in a circle

Ellipse

An ellipse is shown in Figure 19.10. Like a circle, an ellipse is a closed curve. Unlike a circle, which has a single focus , its center, an ellipse has two focal points, called foci (pronounced ) , making it appear flattened, or egg shaped. An ellipse can be drawn by positioning pins at two points, F 1 and F 2, and loosely tying a length of string between these points, the foci. Using a pencil point to hold the string taut, follow the string to form the shape shown in Figure 19.10. Notice that the distance remains constant.

The ellipse also has a center. It is the point of intersection of the major and minor axes. These are special line segments that intersect at right angles in the el lipse. The major axis runs through the foci and center; the minor axis is the perpendicular bisector of the major axis.

A circle is a special kind of ellipse in which the two focal points are coincident.

F IGURE 19.10 Important features of an ellipse

Some examples of ellipses in technical fields are:

• The shape of the end section on a pipe that has been cut on a slant.

• The projection of a circle in an isometric view of a part to be machined.

• The orbits of the planets around the sun.

Circles and ellipses are examples of conic sections . The other conic sections are the parabola and the hyperbola. Conic sections are the shapes obtained by passing a plane through a right circular cone or cones at various angles.

EXERCISES

19.1 Define the terms:

19.2 What is an inscribed polygon?

19.3 What is a circumscribed polygon?

19.4 How many foci does an ellipse have?

19.5 Draw an ellipse whose major and minor axes are equal. What figure has been drawn?