Circles and Circular Arcs

Circle

A circle is the set of all points in a plane that are a fixed distance from one point. That point is the center, and the fixed distance is the radius.

If the center is $O(a,b)$ and the radius is $r$, the distance formula gives the standard equation:

Parts of a Circle

The diagram turns that definition into the parts we use in calculations:

The four parts to recognize first:

• Center ($O$): the fixed point from which every radius is measured
• Radius ($r$): a segment from the center to a point on the circle
• Diameter ($d$): a chord through the center, so $d = 2r$
• Chord: a segment joining two points on the circle

Circular Arcs

A circular arc is the part of a circle between two points on its circumference. We write an arc with a curved mark over its endpoints, for example $\overset{\frown}{AB}$.

Types of Arc

Classifying arcs by length:

• Minor arc: shorter than half the circumference
• Major arc: longer than half the circumference
• Semicircle: exactly half the circumference

Central Angle and Inscribed Angle

A central angle opens at the centre of the circle, and an inscribed angle opens on the circle itself. Both subtend the same arc, and a fixed ratio links the two sizes.

Central Angle

A central angle has its vertex at the center of the circle, with two radii as its sides.

The key central-angle fact:

• A central angle and its intercepted arc have the same angular measure.
• Therefore, if the central angle measures $\alpha$, its intercepted arc also measures $\alpha$.

Inscribed Angle

An inscribed angle has its vertex on the circle, with two chords as its sides.

Relationship Between Central Angle and Inscribed Angle

When a central angle and an inscribed angle intercept the same arc, the inscribed angle is half the central angle.

Example application:

Arc Length and Sector Area

An arc and its sector both take the same fraction of the whole circle. That fraction is the central angle divided by $360^\circ$, so one ratio fixes both the arc length and the sector area.

Arc Length

An arc occupies the same fraction of the circumference as its central angle occupies of a full turn. For a central angle of $\alpha$ degrees:

Where:

• $l$ = arc length
• $\alpha$ = measure of central angle (in degrees)
• $r$ = radius of circle

Sector Area

A sector is the region bounded by two radii and the arc between them. It occupies the same fraction of the circle's area as its central angle occupies of a full turn.

The diagram shows the two radii, the intercepted arc, and the sector they enclose.

Calculating Arc Length and Sector Area

A circle has a radius of $14 \text{ cm}$. If the central angle that subtends an arc is $90^\circ$, determine:

1. Arc length
2. Sector area

Solution:

Given: $r = 14 \text{ cm}$, $\alpha = 90^\circ$

1. Arc length:

   $$l = \frac{\alpha}{360^\circ} \times 2\pi r$$

   $$l = \frac{90^\circ}{360^\circ} \times 2\pi \times 14$$

   $$l = \frac{1}{4} \times 28\pi$$

   $$l = 7\pi \text{ cm} \approx 21.99 \text{ cm}$$

2. Sector area:

   $$L = \frac{\alpha}{360^\circ} \times \pi r^2$$

   $$L = \frac{90^\circ}{360^\circ} \times \pi \times 14^2$$

   $$L = \frac{1}{4} \times 196\pi$$

   $$L = 49\pi \text{ cm}^2 \approx 153.94 \text{ cm}^2$$

Practice Problems

The problems combine central angles, inscribed angles, and arc measurements. Decide which relation connects them before substituting numbers.

1. A circle has a diameter of $20 \text{ cm}$. If an inscribed angle that subtends an arc is $30^\circ$, determine the measure of central angle that subtends the same arc!

2. In a circle with center $O$ and radius $21 \text{ cm}$, there is an arc $AB$ with central angle $120^\circ$. Calculate:

   • Arc length $AB$
   • Sector area $AOB$

3. Two inscribed angles subtend the same arc. If one inscribed angle measures $45^\circ$, determine the measure of the other inscribed angle!

Worked Solutions

1. The diameter is not needed here. What matters is that both angles intercept the same arc. The central angle is twice the inscribed angle:

   $$\begin{aligned} \text{central angle} &=2(30^\circ) \\ &=60^\circ \end{aligned}$$

   So the central angle measures $60^\circ$.

2. The angle $120^\circ$ is one third of a full turn because $\frac{120^\circ}{360^\circ}=\frac{1}{3}$. Apply that fraction first to the circumference, then to the area of the circle.

   $$\begin{aligned} l &=\frac{120^\circ}{360^\circ}(2\pi)(21) \\ &=14\pi\text{ cm} \\ &\approx43.98\text{ cm} \end{aligned}$$

   $$\begin{aligned} L &=\frac{120^\circ}{360^\circ}\pi(21)^2 \\ &=147\pi\text{ cm}^2 \\ &\approx461.81\text{ cm}^2 \end{aligned}$$

   Therefore, the arc length is $14\pi\text{ cm}$ and the sector area is $147\pi\text{ cm}^2$.

3. Inscribed angles that intercept the same arc are equal. Since the first angle is $45^\circ$, the second angle is also $45^\circ$.