SAT Math – Geometry and Trigonometry
Circle Equations
Mastering standard form, completing the square, and finding centers and radii
Circle equations algebraically define all points equidistant from a center point, using the distance formula to create quadratic expressions in x and y. On the SAT, you'll recognize standard form, convert expanded equations by completing the square, identify centers and radii, write equations from geometric information, and determine whether points lie on circles.
Success requires fluency with completing the square, understanding the relationship between algebraic form and geometric properties, manipulating equations strategically, and interpreting constants as geometric measurements. These skills aren't just algebraic manipulation—they enable modeling circular boundaries, wireless coverage areas, blast radii, planetary orbits, and any scenario requiring distance-based definitions.
Understanding Circle Equation Forms
Standard Form
The most useful form, revealing center and radius directly.
Center: Point (h, k)
Radius: r (take square root of right side)
Key: Watch signs! \((x - 3)^2\) means h = 3, not -3
Special case: \(x^2 + y^2 = r^2\) when center at origin
General/Expanded Form
Expanded form that requires completing the square to interpret.
No squared terms mixed: Coefficient of x² and y² must be equal
To find center/radius: Complete the square for x and y
Strategy: Group x terms, group y terms, move constant
Completing the Square
Technique to convert general form to standard form.
Creates: \(\left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2\)
Same for y: Group y terms and complete the square
Remember: Balance both sides of equation
Writing Circle Equations
Given geometric information, construct the equation.
From diameter endpoints: Midpoint = center, distance/2 = radius
From three points: Set up system (advanced, rare on SAT)
From graph: Read center coordinates, count radius units
Essential Formulas and Techniques
Standard Form Template
\((x - h)^2 + (y - k)^2 = r^2\)
Center: (h, k)
Radius: \(r = \sqrt{r^2}\)
Sign rule: \((x - 3)\) means h = +3; \((x + 3)\) means h = -3
Distance Formula (for Radius)
\(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
Distance between center and any point on circle equals radius
Use to verify if point lies on circle
Completing the Square Steps
Step 1: Group x terms and y terms separately
Step 2: For x² + bx, add \(\left(\frac{b}{2}\right)^2\) inside and outside
Step 3: Repeat for y terms
Step 4: Factor perfect squares, simplify right side
Point on Circle Test
Method: Substitute point coordinates into equation
If true: Point lies on circle
If false: Point not on circle
Alternative: Calculate distance from center and compare to radius
Common Pitfalls & Expert Tips
❌ Sign errors in standard form
\((x + 4)^2\) means center at x = -4, not +4! Form is \((x - h)^2\), so \((x - (-4))^2 = (x + 4)^2\).
❌ Forgetting to take square root for radius
If equation ends with = 36, radius is 6 (not 36). Right side is r², not r!
❌ Completing square incorrectly
For x² + 6x, add (6/2)² = 9, not 6. Take half the coefficient, then square it!
❌ Not balancing when completing square
If you add 9 to left side, must add 9 to right side too! Keep equation balanced.
✓ Expert Tip: Check your signs twice
Write out \((x - h)^2\) explicitly. If you see \((x - 3)^2\), h = +3. If \((x + 5)^2\), h = -5.
✓ Expert Tip: Group terms before completing square
Organize: (x² + bx) + (y² + dy) = constant. Work with each group separately, much cleaner!
✓ Expert Tip: Verify with a point
After writing equation, test with a known point. If center is (2,3) and radius 5, point (2,8) should satisfy equation!
Fully Worked SAT-Style Examples
Find the center and radius of the circle \((x + 2)^2 + (y - 5)^2 = 16\).
Solution:
Compare to standard form: \((x - h)^2 + (y - k)^2 = r^2\)
Rewrite: \((x - (-2))^2 + (y - 5)^2 = 4^2\)
Identify values:
h = -2, k = 5
Center: (-2, 5)
\(r^2 = 16\) → \(r = 4\)
Answer: Center (-2, 5); Radius 4
Write the equation of a circle with center (3, -4) and radius 7.
Solution:
Use standard form: \((x - h)^2 + (y - k)^2 = r^2\)
h = 3, k = -4, r = 7
Substitute:
\((x - 3)^2 + (y - (-4))^2 = 7^2\)
\((x - 3)^2 + (y + 4)^2 = 49\)
Answer: \((x - 3)^2 + (y + 4)^2 = 49\)
Find the center and radius of \(x^2 + y^2 - 6x + 8y - 11 = 0\).
Solution:
Group x and y terms:
\((x^2 - 6x) + (y^2 + 8y) = 11\)
Complete the square for x:
Coefficient is -6, half is -3, square is 9
\((x^2 - 6x + 9) = (x - 3)^2\)
Complete the square for y:
Coefficient is 8, half is 4, square is 16
\((y^2 + 8y + 16) = (y + 4)^2\)
Add to both sides:
\((x - 3)^2 + (y + 4)^2 = 11 + 9 + 16 = 36\)
Identify center and radius:
Center: (3, -4)
Radius: \(\sqrt{36} = 6\)
Answer: Center (3, -4); Radius 6
Does the point (5, 2) lie on the circle \((x - 1)^2 + (y + 3)^2 = 25\)?
Solution:
Substitute point into equation:
\((5 - 1)^2 + (2 + 3)^2 = 25\)
\(4^2 + 5^2 = 25\)
\(16 + 25 = 25\)
\(41 \neq 25\)
Answer: No, the point does not lie on the circle
Write the equation of a circle centered at the origin with radius 10.
Solution:
Origin means center at (0, 0):
h = 0, k = 0, r = 10
Substitute into standard form:
\((x - 0)^2 + (y - 0)^2 = 10^2\)
\(x^2 + y^2 = 100\)
Answer: \(x^2 + y^2 = 100\)
A circle has center (2, -1) and passes through point (6, 2). What is the radius?
Solution:
Use distance formula:
\(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
\(r = \sqrt{(6 - 2)^2 + (2 - (-1))^2}\)
\(r = \sqrt{4^2 + 3^2}\)
\(r = \sqrt{16 + 9} = \sqrt{25} = 5\)
Answer: Radius = 5
Find the center and radius of \(x^2 + y^2 + 4x - 10y + 13 = 0\).
Solution:
Group and move constant:
\((x^2 + 4x) + (y^2 - 10y) = -13\)
Complete squares:
For x: (4/2)² = 4
For y: (-10/2)² = 25
\((x^2 + 4x + 4) + (y^2 - 10y + 25) = -13 + 4 + 25\)
Factor and simplify:
\((x + 2)^2 + (y - 5)^2 = 16\)
Answer: Center (-2, 5); Radius 4
Quick Reference Guide
Standard Form
\((x-h)^2 + (y-k)^2 = r^2\)
Center: (h, k)
Radius: r
Complete Square
For x² + bx:
Add (b/2)²
Circle Equations: Algebraic Definitions of Perfect Curves
Circle equations translate geometric definitions—all points equidistant from a center—into algebraic language using the distance formula, creating quadratic expressions connecting x and y coordinates. The SAT tests these skills because circles represent fundamental mathematical objects modeling countless real phenomena from wireless coverage zones to planetary orbits, while the algebraic techniques—completing the square, manipulating equations, extracting geometric meaning from symbolic form—develop problem-solving flexibility essential across mathematics. The standard form \((x-h)^2 + (y-k)^2 = r^2\) directly encodes center position (h, k) and radius r, emerging from the distance formula \(\sqrt{(x-h)^2 + (y-k)^2} = r\) by squaring both sides. Understanding why \((x+3)^2\) indicates h = -3 (rewriting as \((x-(-3))^2\)) prevents sign errors plaguing students who mechanically copy values without understanding the template structure. Completing the square—the systematic technique converting general form \(x^2 + y^2 + Dx + Ey + F = 0\) into standard form—represents algebraic power: transforming opaque expressions into transparent geometric information by creating perfect square trinomials. Recognizing that adding (b/2)² completes x² + bx into (x + b/2)² connects memorized procedures to conceptual understanding. These manipulations transcend circle problems, appearing throughout algebra when converting between equivalent forms revealing different information—quadratic vertex form, completing squares in calculus integration, eigenvalue problems in linear algebra. Testing whether points satisfy circle equations—substituting coordinates and checking equality—develops verification habits ensuring solution validity. Writing equations from geometric data—given centers and radii, calculating radii from center-point distances—connects spatial visualization to algebraic representation. Every completed square, every extracted center, every calculated radius represents facility with the fundamental mathematical skill of moving fluidly between geometric concepts and algebraic expressions—translations essential for advanced mathematics where abstract symbolic manipulation must maintain connection to concrete geometric or physical meaning, from conic sections to differential equations to complex analysis where circles in the complex plane encode rotation and modulus.
