📚 ACT Math Core Formula Handbook | ACT数学核心公式盘点
The ACT Math section tests 60 questions in 60 minutes, covering six major areas: Pre-Algebra, Elementary Algebra, Intermediate Algebra, Coordinate Geometry, Plane Geometry, and Trigonometry. Unlike some other standardized tests, the ACT does not provide a formula sheet — you are expected to memorize all necessary formulas before test day. This guide compiles the most frequently tested formulas and relationships, organized by topic.
ACT数学部分共60道题，限时60分钟，涵盖六大领域：基础代数（Pre-Algebra）、初级代数（Elementary Algebra）、中级代数（Intermediate Algebra）、坐标几何（Coordinate Geometry）、平面几何（Plane Geometry）和三角学（Trigonometry）。与其他一些标准化考试不同，ACT不提供公式表——考生需要在考前牢记所有必要公式。本文按专题整理最常考的公式与关系，助你高效备考。
1. Test Overview & Strategy | 考试概览与策略
Understanding the scoring and pacing of the ACT Math section is your first strategic advantage. The 60 questions are arranged roughly in order of increasing difficulty, but the last 10–15 questions are typically the most challenging. There is no penalty for guessing, so you should never leave a question blank.
了解ACT数学部分的评分与节奏是你第一个策略优势。60道题大致按难度递增排列，最后10–15题通常最具挑战性。猜错不扣分，所以你绝不应留空任何题目。
Key pacing facts:
关键节奏事实：
Exactly 60 minutes for 60 questions → approximately 1 minute per question.
Each correct answer contributes equally to your raw score, regardless of difficulty.
Calculator use is permitted for the entire section, but many questions are faster without one.
60分钟做60题 → 每题约1分钟。
无论难度，每道正确答案对原始分贡献相同。
全程允许使用计算器，但许多题目不用计算器反而更快。
Score = Raw Score (0–60) → Scaled Score (1–36)
得分 = 原始分（0–60）→ 换算分（1–36）
2. Pre-Algebra Essentials | 基础代数核心
Pre-Algebra accounts for approximately 20–24% of the ACT Math test (12–14 questions). Topics include basic operations, percentages, ratios, averages, and number properties. Mastery of these fundamentals is non-negotiable.
基础代数约占ACT数学考试的20–24%（12–14题），涵盖基本运算、百分比、比例、平均数与数的性质。掌握这些基本功是硬性要求。
The average (arithmetic mean) formula appears frequently:
平均数（算术平均值）公式经常出现：
Mean = (x₁ + x₂ + … + xₙ) ÷ n
平均数 = （x₁ + x₂ + … + xₙ）÷ n
Percent change is another must-know:
百分比变化是另一个必考公式：
Percent Change = (New Value − Original Value) ÷ Original Value × 100%
百分比变化 = （新值 − 原值）÷ 原值 × 100%
Weighted average problems require multiplying each value by its weight before summing:
加权平均数问题需要先将每个数值乘以其权重再求和：
Weighted Mean = Σ(wᵢ × xᵢ) ÷ Σwᵢ
加权平均数 = Σ（权重ᵢ × 数值ᵢ）÷ Σ权重ᵢ
3. Elementary Algebra | 初级代数
Elementary Algebra covers approximately 15–20% of the test (9–12 questions). This section tests linear equations, inequalities, systems of equations, and basic expression manipulation.
初级代数约占考试的15–20%（9–12题），考查线性方程、不等式、方程组以及基本表达式化简。
The slope-intercept form of a linear equation is foundational for both algebra and coordinate geometry:
直线方程的斜截式是代数和坐标几何的基础：
y = mx + b, where m = slope, b = y-intercept
y = mx + b，其中m = 斜率，b = 纵截距
For two points (x₁, y₁) and (x₂, y₂), the slope formula is:
对于两点（x₁, y₁）和（x₂, y₂），斜率公式为：
m = (y₂ − y₁) ÷ (x₂ − x₁)
m = （y₂ − y₁）÷（x₂ − x₁）
The quadratic formula is essential for solving ax² + bx + c = 0:
求解ax² + bx + c = 0时，求根公式至关重要：
x = [−b ± √b² − 4ac] ÷ 2a
x = [−b ± √（b² − 4ac）] ÷ 2a
The discriminant, b² − 4ac, determines the nature of the roots:
判别式b² − 4ac决定根的性质：
If b² − 4ac > 0 → two distinct real roots.
If b² − 4ac = 0 → exactly one real root (double root).
If b² − 4ac < 0 → no real roots (two complex roots).
若b² − 4ac > 0 → 两个不等实根。
若b² − 4ac = 0 → 唯一实根（重根）。
若b² − 4ac < 0 → 无实根（两个复数根）。
4. Intermediate Algebra | 中级代数
Intermediate Algebra represents about 15–20% of the test (9–12 questions). Key topics include exponents, radicals, logarithms, functions, matrices, and sequences.
中级代数约占考试的15–20%（9–12题），重点涵盖指数、根式、对数、函数、矩阵和数列。
Exponent rules that appear repeatedly:
反复出现的指数运算法则：
aᵐ × aⁿ = aᵐ⁺ⁿ | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | (aᵐ)ⁿ = aᵐⁿ
aᵐ × aⁿ = aᵐ⁺ⁿ | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | （aᵐ）ⁿ = aᵐⁿ
For radicals and fractional exponents:
根式与分数指数：
√ab = √a × √b | √a/b = √a ÷ √b | am/n = ⁿ√aᵐ
√（ab）= √a × √b | √（a/b）= √a ÷ √b | a^（m/n）= ⁿ√（aᵐ）
Logarithm properties are essential for solving exponential equations:
对数性质是求解指数方程的关键：
logₐ(xy) = logₐx + logₐy | logₐ(x/y) = logₐx − logₐy | logₐ(xⁿ) = n × logₐx
logₐ（xy）= logₐx + logₐy | logₐ（x/y）= logₐx − logₐy | logₐ（xⁿ）= n × logₐx
Arithmetic and geometric sequences appear regularly:
等差数列与等比数列经常出现：
Arithmetic: aₙ = a₁ + (n − 1)d | Geometric: aₙ = a₁ × rⁿ⁻¹
等差数列：aₙ = a₁ +（n − 1）d | 等比数列：aₙ = a₁ × rⁿ⁻¹
5. Coordinate Geometry | 坐标几何
Coordinate geometry constitutes approximately 15–20% of the test (9–12 questions). In addition to lines, you must know the distance formula, midpoint formula, and circle equations.
坐标几何约占考试的15–20%（9–12题）。除直线外，你还需要掌握距离公式、中点公式和圆的方程。
Distance between two points (x₁, y₁) and (x₂, y₂):
两点（x₁, y₁）和（x₂, y₂）间的距离：
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
d = √[（x₂ − x₁）² +（y₂ − y₁）²]
Midpoint of a segment:
线段中点：
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
M = （（x₁ + x₂）/2，（y₁ + y₂）/2）
Standard form of a circle centered at (h, k) with radius r:
圆心为（h, k）、半径为r的圆的标准方程：
(x − h)² + (y − k)² = r²
（x − h）² +（y − k）² = r²
Parallel lines have equal slopes (m₁ = m₂); perpendicular lines have slopes satisfying m₁ × m₂ = −1. You also need the point-slope form:
平行直线斜率相等（m₁ = m₂）；垂直直线斜率满足m₁ × m₂ = −1。你还需要掌握点斜式：
y − y₁ = m(x − x₁)
y − y₁ = m（x − x₁）
6. Plane Geometry: Triangles | 平面几何：三角形
Plane geometry is the largest category on the ACT, representing approximately 25–30% of the test (15–18 questions). Triangles deserve special attention because they appear in nearly every geometry problem.
平面几何是ACT考试中占比最大的一类，约占总题量的25–30%（15–18题）。三角形值得特别关注，因为几乎每道几何题中都会出现三角形。
Sum of interior angles of any triangle:
任意三角形内角和：
∠A + ∠B + ∠C = 180°
∠A + ∠B + ∠C = 180°
Area and perimeter:
面积与周长：
Area = ½ × base × height | Perimeter = sum of all sides
面积 = ½ × 底 × 高 | 周长 = 各边长度之和
The Pythagorean theorem applies to right triangles:
勾股定理适用于直角三角形：
a² + b² = c² (c is the hypotenuse)
a² + b² = c²（c为斜边）
Special right triangles save precious time:
特殊直角三角形可以节省宝贵时间：
| Type | 类型 | Side Ratio | 边长比 | Key Property | 关键性质 |
| 45°-45°-90° | 1 : 1 : √2 | Hypotenuse = leg × √2 |
| 30°-60°-90° | 1 : √3 : 2 | Side opposite 30° is half the hypotenuse |
| 类型 | 边长比 | 关键性质 |
| 45°-45°-90° | 1 : 1 : √2 | 斜边 = 直角边 × √2 |
| 30°-60°-90° | 1 : √3 : 2 | 30°角的对边是斜边的一半 |
7. Plane Geometry: Circles & Polygons | 平面几何：圆与多边形
Circle formulas are heavily tested in the ACT. You must know circumference, area, arc length, and sector area — often in the same question.
圆的相关公式在ACT中考查频率很高。你必须掌握圆周长、面积、弧长和扇形面积——这些常常出现在同一道题中。
Circumference and area:
圆周长与面积：
C = 2πr = πd | A = πr²
周长C = 2πr = πd | 面积A = πr²
For a sector with central angle θ (in degrees):
对于圆心角为θ（以度为单位）的扇形：
Arc Length = (θ / 360°) × 2πr | Sector Area = (θ / 360°) × πr²
弧长 = （θ / 360°）× 2πr | 扇形面积 = （θ / 360°）× πr²
For polygons, the sum of interior angles of an n-sided polygon is:
对于多边形，n边形的内角和为：
(n − 2) × 180°
（n − 2）× 180°
For a regular n-gon, each interior angle equals [(n − 2) × 180°] ÷ n, and each exterior angle equals 360° ÷ n.
对于正n边形，每个内角等于[（n − 2）× 180°] ÷ n，每个外角等于360° ÷ n。
8. Plane Geometry: Areas & Volumes | 平面几何：面积与体积
Beyond circles and triangles, ACT frequently tests areas of other 2D shapes and volumes of 3D solids. These questions often combine multiple formulas in a single problem.
除圆和三角形外，ACT经常考查其他二维图形的面积和三维立体图形的体积。这些问题通常将多个公式组合在同一道题中。
Common 2D formulas:
常见二维图形公式：
Rectangle: A = l × w (length × width)
Parallelogram: A = base × height
Trapezoid: A = ½ × (b₁ + b₂) × h
矩形：A = 长 × 宽
平行四边形：A = 底 × 高
梯形：A = ½ ×（上底 + 下底）× 高
Volumes that appear regularly:
常考的体积公式：
Rectangular prism: V = l × w × h
Cylinder: V = πr²h
Sphere: V = (4/3)πr³ | Surface Area = 4πr²
Cone: V = (1/3)πr²h
Rectangular pyramid: V = (1/3) × base area × height
长方体：V = 长 × 宽 × 高
圆柱体：V = πr²h
球体：V = (4/3)πr³ | 表面积 = 4πr²
圆锥体：V = (1/3)πr²h
四棱锥：V = (1/3) × 底面积 × 高
1 cubic unit = 1 unit × 1 unit × 1 unit | 1立方单位 = 1单位 × 1单位 × 1单位
9. Trigonometry | 三角学
Trigonometry makes up approximately 5–7% of the ACT (about 4 questions). The scope is narrower than in SAT Math, focusing on right-triangle ratios, basic identities, and the unit circle.
三角学约占ACT考试的5–7%（约4题）。其考查范围比SAT数学更窄，集中在直角三角形比值、基本恒等式和单位圆。
The three fundamental ratios in a right triangle (given angle θ):
直角三角形中的三个基本比值（对于角θ）：
sin θ = opposite / hypotenuse | cos θ = adjacent / hypotenuse | tan θ = opposite / adjacent
sin θ = 对边 / 斜边 | cos θ = 邻边 / 斜边 | tan θ = 对边 / 邻边
The key trigonometric identities to memorize:
必须牢记的关键三角恒等式：
sin²θ + cos²θ = 1 | tan θ = sin θ ÷ cos θ
sin²θ + cos²θ = 1 | tan θ = sin θ ÷ cos θ
For special angles, memorize the sine and cosine values derived from the special right triangles:
对于特殊角，记住从特殊直角三角形推导出的正弦和余弦值：
| θ | 0° | 30° | 45° | 60° | 90° |
| sin θ | 0 | ½ | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | ½ | 0 |
10. Statistics, Probability & Data | 统计、概率与数据
Statistics and probability questions overlap with pre-algebra and typically involve means, medians, modes, probabilities, and data interpretation from tables or graphs. Expect around 4–6 such questions.
统计与概率题与基础代数有所重叠，通常涉及平均数、中位数、众数、概率以及表格或图表的数据解读。预计会出现4–6道此类题目。
Basic probability formula:
基本概率公式：
P(event) = number of favorable outcomes ÷ total number of possible outcomes
P（事件）= 有利结果数 ÷ 所有可能结果总数
For independent events A and B (both occurring):
对于独立事件A和B（同时发生）：
P(A and B) = P(A) × P(B)
P（A且B）= P（A）× P（B）
For mutually exclusive events A or B:
对于互斥事件A或B：
P(A or B) = P(A) + P(B)
P（A或B）= P（A）+ P（B）
The median of an even number of data points is the average of the two middle values. The mode is the most frequently occurring value. The range is maximum − minimum.
偶数个数据的中位数是中间两个数的平均值。众数是出现频率最高的数值。极差 = 最大值 − 最小值。
11. Complex Numbers & Vectors | 复数与向量
The ACT includes a limited number of complex number and vector questions (usually 1–2). While not numerous, these points are easy to secure if you know the basic operations.
ACT包含少量复数和向量问题（通常1–2道）。虽然数量不多，但如果你掌握基本运算，这些分数很容易拿到。
The imaginary unit is defined as i = √−1, giving i² = −1. Powers of i cycle:
虚数单位定义为i = √（−1），因此i² = −1。i的幂呈循环：
i¹ = i | i² = −1 | i³ = −i | i⁴ = 1 (then repeats)
i¹ = i | i² = −1 | i³ = −i | i⁴ = 1（然后循环）
To divide complex numbers, multiply numerator and denominator by the conjugate:
要除以复数，将分子和分母同时乘以共轭复数：
(a + bi) ÷ (c + di) = [(a + bi)(c − di)] ÷ (c² + d²)
（a + bi）÷（c + di）= [（a + bi）（c − di）] ÷（c² + d²）
For vectors in component form ⟨x, y⟩, addition and subtraction are component-wise, and the magnitude is:
对于分量形式⟨x, y⟩的向量，加减法按分量进行，模长为：
|v| = √x² + y²
|v| = √（x² + y²）
12. Final Review Tips | 考前复习建议
As you approach test day, prioritize active recall — cover the formula sheet and recite each formula from memory. Then, apply formulas in timed practice sets rather than reading through notes repeatedly. Consistency matters more than intensity.
在临近考试日时，优先进行主动回忆——遮住公式表，凭记忆复述每个公式。然后，在限时练习中运用公式，而不是反复阅读笔记。持续性比强度更重要。
Three final recommendations:
三条最终建议：
Create a personal error log: record every formula you misapplied during practice and review that list weekly.
Know the calculator policy: a TI-84 or equivalent is sufficient; graphing calculators are allowed but not required.
Practice with real ACT tests from official booklets to build stamina and familiarity with question style.
建立个人错题本：记录练习中每次用错的公式，每周复习这份清单。
了解计算器政策：TI-84或同等型号足够；允许使用图形计算器，但不是必须。
使用官方真题册练习，以增强持久力并熟悉题型风格。
Master these formulas, and you will walk into the ACT Math section with the confidence of a well-prepared test-taker. Remember: the ACT does not give you a formula sheet, so your preparation is your formula sheet.
掌握这些公式，你将带着充分准备的自信心走进ACT数学考场。请记住：ACT不会给你公式表，所以你的备考本身就是你的公式表。
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