SAT Math – Geometry and Trigonometry
Circle Theorems
Mastering circle equations, arc length, sectors, and angle relationships
Circle theorems encompass equations defining circles, relationships between central and inscribed angles, properties of chords and tangents, and calculations involving arcs and sectors. On the SAT, you'll identify circle equations, find centers and radii, calculate arc lengths and sector areas, apply inscribed angle theorems, and use tangent-radius perpendicularity.
Success requires memorizing the standard circle equation form, understanding that inscribed angles are half their intercepted arcs, recognizing tangent-radius relationships, and applying area and circumference formulas. These theorems aren't just geometric curiosities—they model planetary orbits, wheel mechanics, radar ranges, wireless signal coverage, and any circular or rotational phenomenon.
Understanding Circle Properties
Circle Equation
The standard form defines all points equidistant from center.
Center: Point (h, k)
Radius: r (always positive)
All points: Distance r from center
Expanded form: \(x^2 + y^2 + Dx + Ey + F = 0\) (complete the square to convert)
Central and Inscribed Angles
Angles related to the same arc have a predictable relationship.
Inscribed angle: Vertex on circle, equals half the arc
Key theorem: Inscribed angle = ½ × central angle (same arc)
Semicircle: Inscribed angle in semicircle = 90°
Tangent Properties
Tangent lines touch circles at exactly one point.
From external point: Two tangent segments equal length
Right angle: Always forms 90° with radius
Use: Create right triangles for calculations
Arc Length and Sector Area
Portions of circles proportional to central angle.
Sector area: \(A = \frac{\theta}{360°} \times \pi r^2\)
Proportion: Arc/Circumference = Sector/Total Area = Angle/360°
Remember: All three ratios equal
Essential Circle Formulas
Basic Circle Measurements
Circumference: \(C = 2\pi r\) or \(C = \pi d\)
Area: \(A = \pi r^2\)
Diameter: \(d = 2r\)
Radius: \(r = \frac{d}{2}\)
Circle Equation Forms
Standard form: \((x - h)^2 + (y - k)^2 = r^2\)
Center at origin: \(x^2 + y^2 = r^2\)
To find center and radius: Identify h, k, r from standard form
To convert: Complete the square for both x and y terms
Angle Relationships
Inscribed angle theorem: Inscribed angle = ½ × intercepted arc
Central angle: Central angle = intercepted arc (measured in degrees)
Semicircle: Any angle inscribed in semicircle = 90°
Full circle: All angles around center sum to 360°
Arc and Sector Formulas
Arc length: \(s = \frac{\theta}{360°} \times 2\pi r\)
Sector area: \(A_{sector} = \frac{\theta}{360°} \times \pi r^2\)
Relationship: \(\frac{\text{Arc}}{\text{Circumference}} = \frac{\text{Sector}}{\text{Total Area}} = \frac{\theta}{360°}\)
Common Pitfalls & Expert Tips
❌ Confusing radius and diameter
Circle formulas use RADIUS. If given diameter = 8, radius = 4. Don't plug diameter into \(\pi r^2\)!
❌ Forgetting to halve inscribed angles
Inscribed angle = ½ arc, not equal to arc! If arc is 80°, inscribed angle is 40°.
❌ Sign errors in circle equations
\((x - 3)^2\) means center at x = 3, not -3. Form is \((x - h)^2\), so \((x - (+3))^2\).
❌ Using degrees in radian formulas
SAT typically uses degrees. Arc length formula shown uses degrees (÷360°). Don't mix with radian formulas!
✓ Expert Tip: Recognize (x-h)² pattern
In \((x-h)^2 + (y-k)^2 = r^2\), center is (h, k). Watch signs! \((x+2)^2\) means h = -2.
✓ Expert Tip: Tangent creates right angle
Tangent perpendicular to radius at contact point creates 90° angle—use Pythagorean theorem!
✓ Expert Tip: Use proportions for arcs
If angle is 60° out of 360°, arc is 1/6 of circumference. Use fractions to simplify!
Fully Worked SAT-Style Examples
What are the center and radius of the circle \((x - 4)^2 + (y + 2)^2 = 25\)?
Solution:
Compare to standard form: \((x - h)^2 + (y - k)^2 = r^2\)
Rewrite: \((x - 4)^2 + (y - (-2))^2 = 5^2\)
Identify values:
h = 4, k = -2
\(r^2 = 25\) → \(r = 5\)
Answer: Center (4, -2); Radius 5
A circle has radius 12. What is the length of an arc that subtends a central angle of 60°?
Solution:
Apply arc length formula:
\(s = \frac{\theta}{360°} \times 2\pi r\)
\(s = \frac{60°}{360°} \times 2\pi(12)\)
Simplify:
\(s = \frac{1}{6} \times 24\pi = 4\pi\)
Answer: \(4\pi\) (approximately 12.57)
An inscribed angle intercepts an arc of 100°. What is the measure of the inscribed angle?
Solution:
Apply inscribed angle theorem:
Inscribed angle = ½ × intercepted arc
Inscribed angle = ½ × 100° = 50°
Answer: 50°
A sector of a circle with radius 10 has a central angle of 90°. What is the area of the sector?
Solution:
Apply sector area formula:
\(A = \frac{\theta}{360°} \times \pi r^2\)
\(A = \frac{90°}{360°} \times \pi(10)^2\)
Simplify:
\(A = \frac{1}{4} \times 100\pi = 25\pi\)
Answer: \(25\pi\) (approximately 78.54)
An angle is inscribed in a semicircle (inscribed angle's endpoints are diameter endpoints). What is the measure of this angle?
Solution:
Apply semicircle theorem:
Semicircle = 180° arc
Inscribed angle = ½ × 180° = 90°
Key theorem:
Any angle inscribed in semicircle is a right angle
Diameter always subtends 90° angle from circle
Answer: 90°
Write the equation of a circle with center (3, -5) and radius 7.
Solution:
Use standard form: \((x - h)^2 + (y - k)^2 = r^2\)
h = 3, k = -5, r = 7
Substitute:
\((x - 3)^2 + (y - (-5))^2 = 7^2\)
\((x - 3)^2 + (y + 5)^2 = 49\)
Answer: \((x - 3)^2 + (y + 5)^2 = 49\)
A circle has center O and radius 5. A tangent line touches the circle at point T. If OT = 5 and the distance from O to an external point P is 13, what is the length of the tangent segment PT?
Solution:
Recognize right triangle:
Tangent ⊥ radius at T → creates right triangle OTP
OT = 5 (radius), OP = 13 (hypotenuse)
Use Pythagorean theorem:
\(OT^2 + PT^2 = OP^2\)
\(5^2 + PT^2 = 13^2\)
\(25 + PT^2 = 169\)
\(PT^2 = 144\)
\(PT = 12\)
Recognition:
This is a 5-12-13 Pythagorean triple!
Answer: 12
Quick Reference Guide
Essential Formulas
Circumference: \(2\pi r\)
Area: \(\pi r^2\)
Equation: \((x-h)^2+(y-k)^2=r^2\)
Arc: \(\frac{\theta}{360°} \times 2\pi r\)
Key Theorems
Inscribed = ½ arc
Semicircle → 90°
Tangent ⊥ radius
Circle Theorems: The Geometry of Perfect Curves
Circle theorems represent mathematical relationships governing perfect curves—shapes where all points maintain constant distance from a center, creating symmetry and proportional relationships fundamental to geometry, physics, and engineering. The SAT tests these concepts because circles model countless phenomena: planetary orbits follow elliptical paths approximating circles, wheels and gears depend on circular motion, wireless coverage areas form circles around transmitters, and radar ranges create circular detection zones. The standard circle equation \((x-h)^2 + (y-k)^2 = r^2\) algebraically defines all points satisfying the distance condition—understanding that h and k shift the center while r controls size connects algebraic manipulation to geometric visualization. The inscribed angle theorem—that angles inscribed in circles equal half their intercepted arcs—provides a powerful tool for angle calculations, while the special case that semicircle-inscribed angles always equal 90° creates right triangles useful for applying Pythagorean relationships. Tangent properties—that tangent lines perpendicular to radii at contact points create right angles—enable indirect distance measurements and optimization problems. Arc length and sector area formulas reveal that these measurements scale proportionally with central angles: a 60° angle captures 60/360 = 1/6 of the full circle, whether measuring circumference or area. Understanding that central angles equal their intercepted arcs while inscribed angles equal half illustrates how position affects angle measurement. These theorems transcend geometric exercises, empowering you to calculate satellite orbital parameters, design circular tracks and racecourses, understand how pizza slices relate angle to area, analyze circular motion in physics, calculate gear ratios in machinery, and recognize circular patterns in art and architecture from rose windows to amphitheaters. The symmetry of circles—where rotation around center preserves shape—makes them fundamental to rotational motion, periodic phenomena, and any system exhibiting radial symmetry from sunflowers to galaxies.
