Chapter 9 — An Introduction to Circles
Topics in this chapter:
9.1 Circle Definitions
Circle. A circle is the set of all points in a plane that are a fixed distance from a specific point.
Key terminology:
| Term | Definition |
|---|---|
| Center | The fixed point from which every point on the circle is equidistant (point $O$). |
| Radius | The distance from the center to any point on the circle ($OA$). |
| Chord | A segment whose endpoints are both on the circle ($\overline{AC}$). |
| Diameter | A chord that passes through the center of the circle ($\overline{AB}$). The diameter $d = 2r$. |
| Tangent | A line that touches the circle at exactly one point. |
| Secant | A line that intersects the circle at two points. |
| Concentric circles | Circles that share the same center. |
Arc. A part of the curve of a circle is called an arc, denoted $\overset{\frown}{AC}$. If $A$ and $C$ can refer to two different arcs (the long way around or the short way), three points are used: $\overset{\frown}{ABC}$.
- The shorter arc is the minor arc.
- The longer arc is the major arc.
Sector. The region inside a circle cut off by two radii is called a sector.
Circular Segment. The area between a chord and the arc it intercepts is called a circular segment.
9.2 Circumference and Area
For any circle, the ratio of the circumference $C$ to the diameter $d$ is the same constant, denoted by the Greek letter $\pi \approx 3.14159\ldots$
Example 9-1. The area of a circle is $16\pi$. What is the circumference of the circle?
Solution: Let the radius be $r$, so $\pi r^2 = 16\pi$. Then $r^2 = 16$, giving $r = 4$. Thus:
\[C = 2\pi r = 2\pi(4) = 8\pi\]Example 9-2. What is the maximum area that can be enclosed by 12 feet of fencing? (MAΘ 1992)
Solution: The largest area is enclosed when the fence is circular. The circle has circumference 12, hence diameter $12/\pi$. The radius is $6/\pi$ and the area is:
\[A = \pi\left(\frac{6}{\pi}\right)^2 = \frac{36}{\pi}\]Key Insight. Among all closed curves of fixed perimeter, the circle encloses the maximum area. This is the isoperimetric inequality.
9.3 Arcs and Sectors
Arcs can be measured by their length or by the central angle that subtends them.
Arc Length. If $\theta$ is the central angle in radians and $r$ is the radius:
\[\text{Arc length} = r\theta\]This comes from the proportion:
\[\frac{\text{arc length}}{2\pi r} = \frac{\theta}{2\pi}\]Sector Area. The area of a sector with central angle $\theta$ (in radians):
\[A_{\text{sector}} = \frac{\theta}{2\pi} \cdot \pi r^2 = \frac{1}{2}r^2\theta\]Circular Segment Area. The area of a circular segment equals the area of the sector minus the area of the triangle formed by the two radii and the chord:
\[A_{\text{segment}} = A_{\text{sector}} - A_{\triangle}\]9.4 Tangent Lines
A tangent line touches the circle at exactly one point, called the point of tangency. A key property is:
A tangent to a circle is perpendicular to the radius drawn to the point of tangency.
Tangent segments from an external point. If two tangent segments are drawn from the same external point to a circle, they are equal in length.
9.5 Angular and Linear Velocity
Example. A fly sits on the edge of a spinning record with radius 10 cm. If the record makes 150 revolutions per minute, what is the speed of the fly?
Solution: Each revolution, the fly travels the circumference $2\pi(10) = 20\pi$ cm. In 150 revolutions:
\[\text{Distance} = 150 \times 20\pi = 3000\pi \text{ cm}\]The fly’s speed is $3000\pi$ cm/min.
Converting angular velocity to linear velocity. If an object moves along a circle of radius $r$ with angular velocity $\omega$ (in radians per unit time), then its linear speed is:
\[v = r\omega\]Exercises & Problems
Exercise 9-1. What is the circumference of a circle whose area is $8\pi$?
Exercise 9-2. In the figure, circle $B$ is tangent to circle $A$ at $X$, circle $C$ is tangent to circle $A$ at $Y$, and circles $B$ and $C$ are tangent to each other. If $AB = 6$, $AC = 5$, and $BC = 9$, what is $AX$? (MAΘ 1987)
Exercise 9-3. A piece of wire 72 cm long is cut into two equal pieces and each is formed into a circle. What is the sum, in square centimeters, of the areas of the circles? (MATHCOUNTS 1991)
Exercise 9-4. Circles $A$, $B$, and $C$ are externally tangent. Express the radius of circle $A$ in terms of $BC$, $AC$, and $AB$. (MAΘ 1992)