Geometry and Trig is about 15% of SAT Math — smaller than Algebra, but high-leverage, because the questions reward a short, fixed list of formulas and two or three recurring configurations. The trap is assuming everything is on the reference sheet. Several key formulas are not, and those are exactly where points leak.
What's on the reference sheet — and what isn't
The Digital SAT gives a formula sheet on every Math question. Knowing its boundary saves memorization effort and prevents nasty surprises:
| Provided (don't memorize) | NOT provided (memorize cold) |
|---|---|
| Circle area πr² and circumference 2πr | Circle equation (x − h)² + (y − k)² = r² |
| Rectangle & triangle area; Pythagorean theorem | Distance and midpoint formulas |
| 30-60-90 and 45-45-90 side ratios | Slope, and parallel / perpendicular slope rules |
| Volumes of common solids; 360° = 2π; angles sum to 180° | Arc length and sector area setups |
Circles: the equation and its traps
A circle with center (h, k) and radius r has the equation (x − h)² + (y − k)² = r². Two traps recur:
- Sign flip. A center of (3, −2) gives (x − 3)² + (y + 2)² = r². The equation subtracts the center coordinates, so a negative center becomes a plus.
- r² vs. r. The right side is r², not r. If it reads = 25, the radius is √25 = 5, not 25.
- When a circle is given in expanded form, complete the square on the x-terms and y-terms separately to recover the center and radius.
Worked example — complete the square
Find the center and radius of x² + y² − 6x + 4y − 12 = 0.
Group and complete the square: (x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4 → (x − 3)² + (y + 2)² = 25. Center (3, −2), radius 5.
Arc length and sector area
Both are just a fraction of the whole circle, set by the central angle θ. Arc length = (θ/360) × 2πr; sector area = (θ/360) × πr². Set up the fraction first and the rest is plug-in.
Triangles: the ratios that skip the work
Two special right triangles and two Pythagorean triples cover most SAT triangle geometry:
| Triangle | Side ratio | Note |
|---|---|---|
| 45-45-90 | 1 : 1 : √2 | Hypotenuse = leg × √2 |
| 30-60-90 | 1 : √3 : 2 | Short leg opposite 30°; hypotenuse opposite 90° |
| 3-4-5 | 3 : 4 : 5 | Common triple — skip the Pythagorean theorem |
| 5-12-13 | 5 : 12 : 13 | The other triple worth memorizing |
Similar triangles also appear often: when two triangles have equal angles, their sides are proportional, so you can set up a ratio to find a missing length. And any triangle's angles sum to 180°.
Coordinate geometry: three formulas to know cold
These are not on the reference sheet, and they show up constantly. For points (x₁, y₁) and (x₂, y₂):
| Quantity | Formula |
|---|---|
| Distance | √((x₂ − x₁)² + (y₂ − y₁)²) — Pythagoras in disguise |
| Midpoint | ((x₁ + x₂)/2, (y₁ + y₂)/2) — average the coordinates |
| Slope | (y₂ − y₁)/(x₂ − x₁) — rise over run |
Two slope rules finish the topic: parallel lines have equal slopes, and perpendicular lines have slopes that are negative reciprocals (their product is −1). So a line with slope 2/3 is perpendicular to one with slope −3/2.
For Chinese-American families · 华人家庭视角
The most common SAT geometry mistake is not a hard theorem — it's reaching for a formula that isn't on the reference sheet and freezing. Students who have the circle equation, distance/midpoint, slope rules, and the two special triangles memorized rarely lose geometry points. Our diagnostic checks exactly which of these are missing.
SAT 几何最常见的失分不是什么难定理 —— 而是去翻参考公式表却发现某个公式根本不在上面，然后卡住。把圆方程、距离/中点、斜率规则、两个特殊 三角形背熟的学生，几乎不在几何丢分。我们的诊断会精确检查这几项里到底缺哪一个。
Quick reference
- Circle: (x − h)² + (y − k)² = r². Right side is r² (=25 means radius 5); complete the square from expanded form.
- Arc length = (θ/360)·2πr; sector area = (θ/360)·πr².
- Special triangles: 45-45-90 is 1 : 1 : √2; 30-60-90 is 1 : √3 : 2. Triples: 3-4-5 and 5-12-13.
- Distance √((x₂−x₁)²+(y₂−y₁)²), midpoint averages the coordinates, slope is (y₂−y₁)/(x₂−x₁) — none on the reference sheet.
- Parallel lines: equal slopes. Perpendicular lines: negative-reciprocal slopes (product −1).
Frequently asked questions
What is the equation of a circle on the SAT?
A circle with center (h, k) and radius r has the equation (x − h)² + (y − k)² = r². Watch the sign flip: a center of (3, −2) gives (x − 3)² + (y + 2)² = r². When a circle is given in expanded form, you complete the square on the x-terms and the y-terms to recover the center and radius. Note the right side is r², not r — a common trap is reading r² = 25 as radius 25 instead of 5.
What special right triangles do I need to know?
Two ratios cover most SAT geometry. A 45-45-90 triangle has legs in ratio 1 : 1 : √2 (the hypotenuse is a leg times √2). A 30-60-90 triangle has sides in ratio 1 : √3 : 2, with the short leg opposite 30°, the long leg opposite 60°, and the hypotenuse opposite 90°. Also memorize the common Pythagorean triples 3-4-5 and 5-12-13, which appear constantly and let you skip the Pythagorean theorem.
What geometry formulas are on the Digital SAT reference sheet?
The reference sheet gives area and circumference of a circle, area of a rectangle and triangle, the Pythagorean theorem, the special-right-triangle ratios, volumes of common solids, and that a circle has 360° / 2π radians and a triangle's angles sum to 180°. It does NOT give the circle equation, the distance or midpoint formula, or slope — so those you must know cold.
How do I find distance, midpoint, and slope between two points?
For points (x₁, y₁) and (x₂, y₂): distance is √((x₂ − x₁)² + (y₂ − y₁)²) (the Pythagorean theorem in disguise); midpoint is the average of the coordinates, ((x₁ + x₂)/2, (y₁ + y₂)/2); and slope is rise over run, (y₂ − y₁)/(x₂ − x₁). Parallel lines have equal slopes, and perpendicular lines have slopes that are negative reciprocals (their product is −1).
How do arc length and sector area work on the SAT?
Both are fractions of the whole circle set by the central angle. Arc length is (central angle / 360°) × circumference = (θ/360) × 2πr, and sector area is (central angle / 360°) × area = (θ/360) × πr². The key idea the SAT tests: an arc or sector is just its share of the full 360°, so set up the fraction first and the rest is plug-in.
How Lighthouse Math teaches this
Lighthouse Math (灯塔数学) is a diagnostic-driven SAT Math program for Chinese-American families in the Boston / Arlington MA area. We don't assign more volume — we map the two or three specific structural mechanisms silently capping a student's score, then close them. Every concept above is taught bilingually so the underlying math is understood, not just translated.
我们用中英双语讲清机制本身，而不是靠刷题量。先做诊断，找到隐形封顶你孩子分数的 2–3 个结构性卡点，再逐一打破 —— 让孩子真正理解数学，而不只是背题型。