Circles and Arc Measures (SAT)

Circle Basics

Key Components

• Center: Point equidistant from all points on circle
• Radius ($r$): Distance from center to any point on circle
• Diameter ($d$): Distance across circle through center ($d = 2r$)
• Chord: Line segment connecting two points on circle
• Tangent: Line touching circle at exactly one point

Circumference (Perimeter)

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

Example: If $r = 5$, then $C = 2\pi(5) = 10\pi$

Area

$A = \pi r^2$

Example: If $r = 3$, then $A = \pi(3)^2 = 9\pi$

Circle Equations

Standard Form

$(x - h)^2 + (y - k)^2 = r^2$

• Center: $(h, k)$
• Radius: $r$

Example: $(x - 2)^2 + (y + 3)^2 = 25$

• Center: $(2, -3)$
• Radius: $\sqrt{25} = 5$

Finding Center and Radius

Given equation, identify $h$, $k$, and $r$

Watch out for signs!

• $(x - 2)$ means $h = +2$
• $(y + 3)$ means $k = -3$

Arcs and Sectors

Arc Length

Fraction of circumference

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

Where $\theta$ is the central angle in degrees

Example: $r = 6$, $\theta = 60°$ $\text{Arc length} = \frac{60}{360} \times 2\pi(6) = \frac{1}{6} \times 12\pi = 2\pi$

Sector Area

Fraction of circle area

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

Example: $r = 6$, $\theta = 60°$ $\text{Sector area} = \frac{60}{360} \times \pi(6)^2 = \frac{1}{6} \times 36\pi = 6\pi$

Angles in Circles

Central Angle

Vertex at center of circle

• Measure = arc it intercepts

Inscribed Angle

Vertex on circle

• Measure = half the arc it intercepts

Inscribed Angle Theorem: $\text{Inscribed angle} = \frac{1}{2} \times \text{central angle}$

Angle in Semicircle

Any angle inscribed in a semicircle is a right angle ($90°$)

Tangent Lines

Properties:

1. Tangent ⊥ radius at point of tangency
2. Two tangents from external point have equal length

Power of a Point

For tangent from external point:

If tangent has length $t$ and point is distance $d$ from center with radius $r$: $t^2 = d^2 - r^2$

(This is Pythagorean theorem!)

Circle Problems on SAT

Type 1: Find Area or Circumference

Given radius or diameter, apply formulas

Type 2: Arc Length and Sector Area

Use fraction of circle based on angle

Type 3: Equation of Circle

Identify center and radius from standard form

Type 4: Inscribed Angles

Remember: inscribed angle = ½ central angle

Type 5: Tangent Lines

Use perpendicularity and Pythagorean theorem

SAT Strategies

Leave in Terms of π

Unless told to approximate, leave π in answer

Example: Area = $25\pi$ (not 78.5)

Check Units

Radius vs diameter - easy to confuse!

Use Fractions for Arcs

Arc = fraction × whole circle

$\frac{60°}{360°} = \frac{1}{6}$ of circle

Draw Radii

Creates right triangles with tangents!

Memorize Formulas

• Circumference: $C = 2\pi r$
• Area: $A = \pi r^2$
• Standard form: $(x-h)^2 + (y-k)^2 = r^2$

Common SAT Traps

Trap 1: Radius vs Diameter

Area given diameter 10:

• Wrong: $A = \pi(10)^2 = 100\pi$ ❌
• Right: $r = 5$, so $A = \pi(5)^2 = 25\pi$ ✓

Trap 2: Sign Errors in Equation

$(x + 3)^2$ means center is at $x = -3$ (not $+3$)

Trap 3: Central vs Inscribed Angle

Inscribed = ½ central

Trap 4: Forgetting to Square Radius

Area = $\pi r^2$ not $\pi r$

Trap 5: Arc vs Sector

Arc length = distance along edge Sector area = area of "pizza slice"

SAT Tips

• Circumference: $2\pi r$ or $\pi d$
• Area: $\pi r^2$ (square the radius!)
• Arc length: Fraction of circumference
• Sector area: Fraction of total area
• Inscribed angle = ½ central angle
• Tangent ⊥ radius at point of contact
• Leave answers in terms of $\pi$ unless told otherwise
• Standard form: $(x-h)^2 + (y-k)^2 = r^2$ with center $(h,k)$