10.1 Parts of circles and their relationships

Circle Properties and Relationships

A circle is the set of all points in a plane that are equidistant from a single point called the center. Every measurement and relationship in this unit builds from that definition, so keep it front and center.

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Parts of a Circle

• Center: The fixed point from which every point on the circle is equidistant.
• Radius: A segment from the center to any point on the circle. All radii of a given circle are congruent.
• Diameter: A chord that passes through the center. Its endpoints lie on the circle, and it's the longest possible chord.
• Chord: A segment whose endpoints both lie on the circle. A diameter is a special case of a chord.
• Secant: A line (not a segment) that intersects a circle at two points. You can think of it as a chord that extends infinitely in both directions.
• Tangent: A line that touches the circle at exactly one point, called the point of tangency. A tangent is always perpendicular to the radius drawn to that point.

Radius–Diameter Relationship

The diameter is simply twice the radius:

$d = 2r \qquad \text{or equivalently} \qquad r = \frac{d}{2}$

This looks simple, but watch for problems that give you the diameter when a formula needs the radius (especially the area formula). Always check which one you've been given before plugging in.

Circumference and Area

Circumference is the distance around the circle:

$C = 2\pi r = \pi d$

Area is the space enclosed by the circle:

$A = \pi r^2$

In both formulas, $\pi \approx 3.14159$. Unless a problem says otherwise, leave answers in terms of $\pi$ for exact work.

Arc length is a fraction of the circumference. If a central angle measures $\theta$ degrees:

$\text{arc length} = \frac{\theta}{360} \times 2\pi r$

Sector area works the same way, but as a fraction of the total area:

$\text{sector area} = \frac{\theta}{360} \times \pi r^2$

The logic behind both formulas is identical: you're taking the ratio of the central angle to the full $360°$ and applying it to the whole circumference or whole area.

Inscribed and Circumscribed Figures

These terms describe how a circle and a polygon relate to each other.

• An inscribed polygon has all of its vertices on the circle. The circle is called the circumscribed circle (or circumcircle) of that polygon. Its center is the intersection of the perpendicular bisectors of the polygon's sides, and its radius connects the center to any vertex.
• A circumscribed polygon has each of its sides tangent to the circle. The circle is called the inscribed circle (or incircle) of that polygon. Its center is the intersection of the polygon's angle bisectors, and its radius is the perpendicular distance from the center to any side.

Inscribed Angles

An inscribed angle is formed by two chords that share an endpoint on the circle. That shared endpoint is the vertex of the angle.

The key theorem: the measure of an inscribed angle is half the measure of its intercepted arc (which equals half the central angle that subtends the same arc).

$m(\text{inscribed angle}) = \frac{1}{2} \times m(\text{intercepted arc})$

For example, if the intercepted arc measures $80°$, the inscribed angle measures $40°$.

Two inscribed angles that intercept the same arc are always congruent, regardless of where their vertices sit on the circle. This is a powerful tool in proofs: if you can show two inscribed angles share an intercepted arc, you've proven them equal.