SAT Geometry: Circles and Coordinate Geometry

The standard-form circle equation, completing the square to find a center, arc length and sector area, the distance and midpoint formulas, and the degrees-to-radians conversions you must memorize.

9 분 TEKS GEO SAT Math

Circles and coordinate geometry appear 2–3 times per Digital SAT — circle equations, arc length, and the distance/midpoint formulas. The algebra is small once you know the templates.

Circle basics

Circumference: C = 2πr (or πd) Area: A = πr² Diameter: d = 2r Both on the reference sheet. Watch carefully whether the problem gives r or d.

Equation of a circle

Standard-form circle equation. The signs inside the parentheses are opposite the center's coordinates.

(x − h)² + (y − k)² = r² Center: (h, k) — opposite signs from the equation Radius: r — square root of the right side Example: (x − 3)² + (y + 2)² = 25 → center (3, −2), radius 5.

Completing the square for circles

When the SAT gives a circle in expanded form, complete the square on both x and y to find the center.

x² + y² − 6x + 4y − 12 = 0 (x² − 6x) + (y² + 4y) = 12 (x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4 (x − 3)² + (y + 2)² = 25 Center (3, −2), radius 5 Add (b/2)² for each variable. Add the same amount to the right side.

Arc length and sector area

Arc length = (θ/360) · 2πr (θ in degrees) Sector area = (θ/360) · πr² The arc/sector is a fraction θ/360 of the full circle.

A circle has radius 6 and a 60° sector. Find sector area. Sector area = (60/360) · π · 6² = (1/6) · 36π = 6π Always reduce the angle fraction first — it's easier to multiply.

Distance and midpoint

Distance: d = √((x₂ − x₁)² + (y₂ − y₁)²) Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2) Distance is the Pythagorean theorem with x- and y-differences as the legs.

Distance from (1, 2) to (4, 6): d = √((4 − 1)² + (6 − 2)²) = √(9 + 16) = √25 = 5 Midpoint of (1, 2) and (4, 6) = (2.5, 4) Notice the 3-4-5 triple hiding in the distance calculation.

A taste of radians

The Digital SAT uses degrees mostly, but occasionally asks for radians. Conversion:

180° = π radians degrees → radians: multiply by π/180 radians → degrees: multiply by 180/π Common: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2.

Desmos for coordinate geo

Plot the points and circle directly. Desmos shows intersections, distances, and the visual layout — often answering "which of the following points lies on the circle?" instantly.

Test-day tip

If a question gives a circle equation and asks whether a point is on/inside/outside, plug the point's coordinates into the left side and compare to r². Less than r² = inside, equal = on, greater = outside.

Tangent lines to a circle

A tangent line touches a circle at exactly one point. The key fact: the radius drawn to the point of tangency is perpendicular to the tangent line.

Circle centered at (0, 0), radius 5. A tangent line touches at (3, 4). Slope of radius = (4 − 0) / (3 − 0) = 4/3 Slope of tangent line = −3/4 (perpendicular) Equation: y − 4 = −3/4 (x − 3) → y = −(3/4)x + 25/4 Tangent ⊥ radius — the cornerstone of every SAT tangent problem.

Slope-intercept form on coordinate geo

Coordinate-geometry questions often mix circles with linear concepts.

Find the slope of the line through (−2, 5) and (4, −1). m = (−1 − 5) / (4 − (−2)) = −6 / 6 = −1 Equation: y − 5 = −1(x − (−2)) → y = −x + 3 Watch the double-negative when x₁ is negative.

Common mistakes

Reading center signs wrong: (x − 3)² has h = +3, not −3
Forgetting to take √ for the radius (r² = 25 means r = 5)
Confusing arc length (linear) with sector area (square)
Using diameter when the formula calls for radius
Drawing the tangent line parallel to the radius instead of perpendicular

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