Circles: Radius, Diameter, and Central Angles

5 Questions

What is the relationship between a central angle and the arc length?

They are directly proportional

A full circle is equivalent to how many radians?

$2\pi$

If an arc represents half of the circle's circumference, what is the measure of the central angle in degrees?

$180^\circ$

What happens to the central angle if the radius of a circle is doubled?

It remains the same

How are angles $ABC$ and $CDA$ related if arcs $AB$ and $CD$ are congruent?

They are congruent

Study Notes

Circles: Radius, Diameter, and Central Angles

Understanding Circles

Circles are two-dimensional shapes with every point on their edge, called the circumference, equidistant from the center. Circles are described by their unique properties like radius, diameter, and central angles, which help us measure and analyze them.

Radius

The radius of a circle is a straight line drawn from the center of the circle to the circumference. A circle has two radii, and they are always equal in length. The symbol for radius is (r).

Example: In the image below, (AB) and (AC) are the radii, each with length (r).

Diameter

The diameter of a circle is the length of a line segment passing through the center of the circle and perpendicular to the circumference. The diameter is always twice the length of the radius and is denoted by (d).

Example: In the image below, (AD) is the diameter, with length (d = 2r).

Central Angles

Central angles are the angles subtended by a circular arc at the center of a circle. Central angles help us measure the amount of the circle's circumference that a specific arc covers.

Example: In the image below, angle (ABC) is a central angle, and it measures the portion of the circle's circumference that arc (BC) represents.

Measuring Angles

Central angles are measured in degrees or radians. The degree is a unit of angular measurement, and a full circle (360 degrees) is equivalent to (2\pi) (approximately 3.14159) radians. The unit radian is a more convenient measure for mathematical calculations, as it is directly proportional to the arc length.

Example: An arc that is (1/4) of the circle's circumference forms an angle of (90^\circ) or (\pi/2) radians at the center.

Relating Radii, Diameters, and Central Angles

The relationship between radius, diameter, and central angles is that the angle subtended by an arc is proportional to the arc length, and the arc length is proportional to the radius.

Example: If arc (AB) and arc (CD) in the image below are congruent, then angles (ABC) and (CDA) are also congruent.

Understanding and using these properties of circles aid in solving a wide variety of problems related to circle geometry, and they form a solid foundation for advanced mathematical concepts.