Circle and Arc Circle

Definition of Circle

A circle is the set of all points on a plane that have the same distance to a fixed point. The fixed point is called the center of the circle, while the same distance is called the radius.

Mathematically, a circle with center $O(a,b)$ and radius $r$ can be expressed with the equation:

Elements of Circle

How is the visualization of the circle equation?

Important elements of circle:

• Center of circle (O): Fixed point that becomes the reference of the circle
• Radius (r): Distance from center to any point on the circle
• Diameter (d): Chord that passes through the center of the circle, $d = 2r$
• Chord: Line segment that connects two points on the circle

Arc of Circle

Arc of circle is a part of the circumference of the circle that is bounded by two points on the circle. Arc is denoted with a curved symbol above the letters, for example $\overset{\frown}{AB}$.

Types of Arc

Types of arc based on their length:

• Minor arc: Arc whose length is less than half the circumference of the circle
• Major arc: Arc whose length is more than half the circumference of the circle
• Semicircle: Arc whose length is exactly half the circumference of the circle

Central Angle and Inscribed Angle

Central Angle

Central angle is an angle whose vertex is at the center of the circle and whose sides are radii of the circle.

Properties of central angle:

• The measure of central angle equals the measure of the arc it subtends
• If central angle = $\alpha$, then arc = $\alpha$

Inscribed Angle

Inscribed angle is an angle whose vertex is on the circle and whose sides are chords.

Relationship Between Central Angle and Inscribed Angle

If central angle and inscribed angle subtend the same arc, then the measure of inscribed angle is half the measure of central angle.

Example application:

Arc Length and Sector Area

Arc Length

Arc length is directly proportional to the measure of central angle that subtends it.

Where:

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

Sector Area

Sector is the region bounded by two radii and an arc of circle.

We can visualize the sector area using the equation above.

Calculating Arc Length and Sector Area

A circle has a radius of 14 cm. If the central angle that subtends an arc is 90°, determine:

1. Arc length
2. Sector area

Solution:

Given: $r = 14$ cm, $\alpha = 90°$

1. Arc length:

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

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

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

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

2. Sector area:

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

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

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

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

Practice Problems

1. A circle has a diameter of 20 cm. If an inscribed angle that subtends an arc is 30°, determine the measure of central angle that subtends the same arc!

2. In a circle with center O and radius 21 cm, there is an arc AB with central angle 120°. Calculate:

   • Arc length AB
   • Sector area AOB

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

Answer Key

1. Central angle = 2 × inscribed angle = 2 × 30° = 60°

2. Given: r = 21 cm, α = 120°

   • Arc length AB = $\frac{120°}{360°} \times 2\pi \times 21 = \frac{1}{3} \times 42\pi = 14\pi$ cm ≈ 43.98 cm
   • Sector area AOB = $\frac{120°}{360°} \times \pi \times 21^2 = \frac{1}{3} \times 441\pi = 147\pi$ cm² ≈ 461.81 cm²

3. Inscribed angles that subtend the same arc have the same measure, so the other inscribed angle = 45°