Circles - Advanced Topics

Circle Parts Review

Circle: All points equidistant from center

Radius (r): Distance from center to edge

Diameter (d): Distance across through center (d = 2r)

Circumference (C): Distance around (C = 2πr)

Area (A): Space inside (A = πr²)

Central Angles

Central angle: Angle with vertex at center of circle

Measured in degrees

Full circle: 360°

Key property: Central angle = Arc measure

Example 1: Central Angle

Angle from center = 60°

The arc it creates also measures 60°

Example 2: Multiple Angles

Three angles at center: 100°, 120°, and x

Find x:

100 + 120 + x = 360
220 + x = 360
x = 140°

Answer: 140°

Arcs

Arc: Part of the circle's circumference

Types:

• Minor arc: Less than 180° (smaller part)
• Major arc: Greater than 180° (larger part)
• Semicircle: Exactly 180° (half circle)

Arc measure = Central angle measure

Example: Arc Measures

Circle with central angle 80°

Minor arc: 80° Major arc: 360° - 80° = 280°

Arc Length

Arc length: Actual distance along the arc

Formula: Arc length = (θ/360°) × 2πr

Where:

• θ = central angle in degrees
• r = radius

Or: Arc length = (θ/360°) × C

Example 1: Find Arc Length

Radius = 12 cm, Central angle = 60°

Calculate:

Arc length = (60/360) × 2π(12)
Arc length = (1/6) × 24π
Arc length = 4π cm
Arc length ≈ 12.57 cm

Answer: 4π cm or ≈ 12.57 cm

Example 2: Semicircle

Radius = 8 m, Angle = 180°

Arc length = (180/360) × 2π(8)

= (1/2) × 16π
= 8π m

Answer: 8π m (half the circumference)

Example 3: Find Radius

Arc length = 10π cm, Angle = 90°

Set up equation:

10π = (90/360) × 2πr
10π = (1/4) × 2πr
10π = (π/2)r
r = 20 cm

Answer: Radius = 20 cm

Sectors

Sector: Pie-slice shaped region

Bounded by: Two radii and an arc

Like a slice of pizza!

Area of sector = (θ/360°) × πr²

Where θ = central angle

Example 1: Sector Area

Radius = 10 cm, Angle = 72°

Calculate:

Area = (72/360) × π(10)²
Area = (1/5) × 100π
Area = 20π cm²
Area ≈ 62.8 cm²

Answer: 20π cm²

Example 2: Semicircular Sector

Radius = 6 ft, Angle = 180°

Area = (180/360) × π(6)²

= (1/2) × 36π
= 18π ft²

Answer: 18π ft² (half the circle's area)

Example 3: Find Angle

Sector area = 15π m², Radius = 6 m

Set up:

15π = (θ/360) × π(6)²
15π = (θ/360) × 36π
15 = (θ/360) × 36
15 = θ/10
θ = 150°

Answer: 150° central angle

Segment of a Circle

Segment: Region between chord and arc

Area = Sector area - Triangle area

Example: Segment Area

Radius = 10, Angle = 90°

Sector area:

(90/360) × π(10)² = 25π

Triangle area: (Isosceles right triangle)

(1/2) × 10 × 10 = 50

Segment area:

25π - 50 ≈ 78.5 - 50 = 28.5

Answer: ≈ 28.5 square units

Inscribed Angles

Inscribed angle: Vertex on the circle

Inscribed Angle Theorem:

• Inscribed angle = (1/2) × Central angle
• Both intercept same arc

Example: Inscribed vs. Central

Central angle = 80° Inscribed angle intercepting same arc = 40°

Chord Properties

Chord: Line segment connecting two points on circle

Properties:

• Perpendicular from center bisects chord
• Equal chords are equidistant from center

Example: Chord Length

Radius = 5, Distance from center to chord = 3

Half the chord forms right triangle:

5² = 3² + (half-chord)²
25 = 9 + (half-chord)²
half-chord = 4

Full chord = 2 × 4 = 8

Tangent Lines

Tangent: Line touching circle at exactly one point

Property: Tangent perpendicular to radius at point of contact

Example: Tangent Length

From external point to circle:

• Two tangents from same point are equal length
• Form right angle with radius

Converting Between Degrees and Radians

Radian measure (advanced):

Conversion:

• 180° = π radians
• 360° = 2π radians

To convert:

• Degrees to radians: multiply by π/180
• Radians to degrees: multiply by 180/π

Example: Convert 90°

90° × (π/180) = π/2 radians

Real-World Applications

Pizza: Sector area = slice size

• 1/8 of 14-inch pizza

Clocks: Arc between hour marks

• 30° per hour (360°/12)

Architecture: Arched doorways

• Calculate arc length for materials

Sports: Track lanes

• Arc length determines running distance

Engineering: Gears and pulleys

• Rotation angles and distances

Practice