📚 ACT Math Core Formulas Pre-Test Self-Check | ACT数学核心公式考前自检
As your ACT test date approaches, one of the most effective final preparations is to quickly run through the essential math formulas you are expected to know. Unlike the SAT, the ACT does not provide a formula sheet, so having these equations at your fingertips can save valuable time and boost your confidence. Use this self-check guide to spot any gaps in your memory and reinforce the key relationships you will rely on during the exam.
随着ACT考试日临近，最后阶段最有效的准备之一就是快速梳理你必须掌握的核心数学公式。与SAT不同，ACT不提供公式表，因此把这些公式牢记于心可以节省宝贵时间并增强信心。用这份自检指南找出记忆中的漏洞，巩固你在考场上必须依仗的关键关系式。
1. Linear Equations and Slope | 线性方程与斜率
You will encounter many questions involving straight lines, so be sure you can move fluidly between the slope-intercept form, point-slope form, and standard form. Check that you recall how to find the slope from two points and how to interpret the meaning of m and b in y = mx + b.
你会遇到许多涉及直线的问题，因此要确保能在斜截式、点斜式和一般式之间灵活转换。检查你是否记得如何通过两点求斜率，以及如何理解 y = mx + b 中 m 与 b 的含义。
m = (y₂ – y₁) / (x₂ – x₁)
The slope m tells you the steepness and direction of a line. A positive slope rises from left to right, while a negative slope falls. A horizontal line has m = 0, and a vertical line has an undefined slope.
斜率 m 反映直线的倾斜程度和方向。正斜率从左到右逐渐上升，负斜率则下降。水平线的斜率为 0，竖直线的斜率无定义。
Point-slope form is incredibly useful when you know a point (x₁, y₁) on the line and the slope: y – y₁ = m(x – x₁). From here you can easily rearrange to slope-intercept or standard form. Make sure you can also find the x-intercept by setting y = 0 and the y-intercept by setting x = 0.
点斜式在已知直线上一点 (x₁, y₁) 和斜率时特别有用：y – y₁ = m(x – x₁)。由此你可以轻松整理成斜截式或一般式。确保你也会通过令 y = 0 求 x 轴截距，令 x = 0 求 y 轴截距。
2. Quadratic Equations and the Discriminant | 二次方程与判别式
Quadratics appear in various forms on the ACT, from factoring to graphing parabolas. The single most important formula to memorize is the quadratic formula, which gives the solutions of ax² + bx + c = 0 for a ≠ 0.
二次方程在ACT中以多种形式出现，从因式分解到抛物线图像。最需要记住的公式就是二次方程求根公式，它给出了 ax² + bx + c = 0（a ≠ 0）的解。
x = (-b ± √(b² – 4ac)) / (2a)
The expression inside the square root, b² – 4ac, is called the discriminant Δ. It determines the number and type of solutions: if Δ > 0 there are two distinct real roots; if Δ = 0 there is exactly one real root (a repeated root); if Δ < 0 there are no real roots (two complex roots).
根号内的表达式 b² – 4ac 称为判别式 Δ。它决定了根的个数和类型：若 Δ > 0，有两个不等实根；若 Δ = 0，有一个实根（重根）；若 Δ < 0，无实根（两个共轭复根）。
You should also be comfortable with the factored form y = a(x – p)(x – q) where p and q are the x-intercepts, and the vertex form y = a(x – h)² + k where (h, k) is the vertex. The vertex’s x-coordinate can be found quickly using x = -b / (2a).
你还应熟悉因式分解式 y = a(x – p)(x – q)，其中 p 和 q 是 x 轴截距，以及顶点式 y = a(x – h)² + k，其中 (h, k) 为顶点。顶点的 x 坐标可通过 x = -b / (2a) 快速求得。
3. Exponent Rules and Radicals | 指数运算法则与根式
Manipulating exponents and radicals is a fundamental skill tested through simplifying expressions, solving equations, and comparing values. Check that you can instantly apply these core rules without hesitation.
指数与根式的运算是基本技能，常在化简表达式、解方程和比较数值时考查。检查自己能否毫不犹豫地运用以下核心法则。
xᵃ · xᵇ = xᵃ⁺ᵇ
(xᵃ)ᵇ = xᵃᵇ
x⁻ⁿ = 1 / xⁿ (x ≠ 0)
xᵃ / xᵇ = xᵃ⁻ᵇ
When working with radicals, recall that the square root can be written as a fractional exponent: √x = x^(½), and ∛x = x^(⅓). This connection allows you to combine radicals with exponent rules seamlessly. For example, multiplying √x by ∛x can be tackled by rewriting as x^(½) · x^(⅓) = x^(½⁺⅓) = x^(⁵⁄₆).
处理根式时，记住平方根可写为分数指数：√x = x^(½)，立方根 ∛x = x^(⅓)。这种联系让你能无缝地将根式与指数法则结合。比如，√x 乘以 ∛x 可改写为 x^(½) · x^(⅓) = x^(½⁺⅓) = x^(⁵⁄₆)。
Also be aware of common traps, such as adding exponents incorrectly when terms are not like bases, and remember that x² times x³ is x⁵, but (x²)³ is x⁶, and x² + x³ cannot be combined into a single power.
还要警惕常见陷阱，比如底数不同时错误地相加指数；记住 x² 乘以 x³ 等于 x⁵，但 (x²)³ 等于 x⁶，而 x² + x³ 不能合并为单一幂次。
4. Geometry: Triangles, Area, and Perimeter | 几何：三角形、面积与周长
ACT geometry requires you to know area and perimeter formulas for common shapes and be able to apply triangle properties, including the Pythagorean theorem. Many questions can be solved quickly by pulling out these formulas.
ACT几何要求你掌握常见图形的面积与周长公式，并能运用三角形性质，包括勾股定理。很多问题通过调用这些公式可以快速解决。
| Shape | Area Formula | Perimeter / Remarks |
|---|---|---|
| Rectangle | A = l × w | P = 2l + 2w |
| Square | A = s² | P = 4s |
| Triangle | A = ½ bh | Sum of all sides |
| Trapezoid | A = ½ (b₁ + b₂)h | Sum of all sides |
| Parallelogram | A = b × h | h is perpendicular height |
The Pythagorean theorem a² + b² = c² applies exclusively to right triangles, where c is the hypotenuse. Also remember commonly used Pythagorean triples like 3-4-5 and 5-12-13, as they can save time.
勾股定理 a² + b² = c² 仅适用于直角三角形，其中 c 是斜边。同时记住常用的勾股数如 3-4-5 和 5-12-13，它们可以节省时间。
For any triangle, the sum of interior angles is 180°. The area can also be found using Heron’s formula if three sides are known, but the basic ½ bh is more frequent. Don’t forget that the height h must be perpendicular to the chosen base.
对任意三角形，内角和为 180°。若已知三边也可用海伦公式求面积，但更常见的是基本公式 ½ bh。别忘了高 h 必须垂直于所选底边。
5. Circles: Circumference, Area, Arc Length, and Sectors | 圆：周长、面积、弧长与扇形
Circle problems on the ACT frequently test whether you can distinguish between radius and diameter, and whether you can correctly apply the formulas for circumference and area. A common mistake is to confuse the two—circumference is 2πr (or πd), while area is πr².
ACT中的圆问题常测试你能否区分半径与直径，以及能否正确运用周长和面积公式。常见错误是将两者混淆——周长是 2πr（或 πd），而面积是 πr²。
C = 2πr or C = πd
A = πr²
An arc length is a fraction of the circumference. If a central angle measures θ degrees, the arc length L = (θ/360) × 2πr. Similarly, the area of a sector with central angle θ is (θ/360) × πr². Check that you can convert these ideas to radians if needed—though ACT seldom requires radian measure, it can appear in trigonometry contexts.
弧长是周长的一部分。若圆心角为 θ 度，弧长 L = (θ/360) × 2πr。同理，圆心角为 θ 的扇形面积是 (θ/360) × πr²。尽管ACT很少要求弧度制，但能在三角题中用到，确认你能灵活转换。
If the question gives you the diameter, always halve it to find r before plugging into the area formula—squaring the diameter by mistake is one of the most common errors.
如果题目给出直径，代入面积公式前一定要先除以2得到半径——误将直径平方是一个极常见的错误。
6. Volume and Surface Area of 3D Figures | 立体图形的体积与表面积
You are expected to know volume formulas for basic solids: rectangular prisms, cylinders, cones, spheres, and pyramids. While some of these formulas may be given in a question if the context is unusual, you should have the most common ones memorized.
你需要掌握基本几何体的体积公式：长方体、圆柱、圆锥、球体和棱锥。虽然有些不常见的公式可能在题目中给出，但最常见的你必须牢记。
| Solid | Volume (V) | Surface Area (SA) |
|---|---|---|
| Rectangular Prism | V = l × w × h | SA = 2lw + 2lh + 2wh |
| Cube | V = s³ | SA = 6s² |
| Cylinder | V = πr²h | SA = 2πr² + 2πrh |
| Sphere | V = ⁴⁄₃ πr³ | SA = 4πr² |
| Cone | V = ⅓ πr²h | SA = πr² + πrl (l = slant height) |
| Square Pyramid | V = ⅓ (base area) × h | SA = base area + ½ × perimeter × slant height |
Notice that many volume formulas involve a factor related to the base area times height, with cones and pyramids requiring the ⅓ factor. Surface area problems often ask you to find a missing dimension when the volume and other lengths are known—practice rearranging these equations.
注意许多体积公式都涉及“底面积 × 高”的因子，而圆锥和棱锥还需要乘上 ⅓。表面积问题经常要求你在已知体积和其他长度时求缺失的维度——练习对这些公式进行变形。
7. Trigonometry: Right Triangle Ratios and Special Angles | 三角学：直角三角形比与特殊角
ACT trigonometry focuses heavily on right triangle relationships: sine, cosine, and tangent. The mnemonic SOHCAHTOA is indispensable: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
ACT三角学重点考查直角三角形中的关系：正弦、余弦和正切。记忆口诀 SOHCAHTOA 不可或缺：Sine = 对边/斜边，Cosine = 邻边/斜边，Tangent = 对边/邻边。
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
You must also know the exact values for the special angles 30°, 45°, and 60°. These values originate from the 30°-60°-90° and 45°-45°-90° triangles. For instance, sin 30° = ½, cos 45° = √2/2, tan 60° = √3. Quick recall of these can help you solve problems without a calculator delay.
你还必须记住特殊角 30°、45° 和 60° 的精确值。这些值源于 30°-60°-90° 和 45°-45°-90° 三角形。例如 sin 30° = ½，cos 45° = √2/2，tan 60° = √3。快速回想这些值能帮你无需犹豫地用计算器解题。
Reciprocal functions—cosecant, secant, cotangent—are less common but may appear: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ. Be prepared to recognize them and relate them back to sine, cosine, and tangent.
倒数函数——余割、正割、余切——出现较少但可能考到：csc θ = 1/sin θ，sec θ = 1/cos θ，cot θ = 1/tan θ。做好准备能认出它们并与正弦、余弦和正切建立联系。
8. Coordinate Geometry: Distance, Midpoint, and Slope | 坐标几何：距离、中点与斜率
Coordinate geometry combines algebra and geometry. The three formulas you absolutely need are the distance formula, the midpoint formula, and the slope formula. They are essentially applications of the Pythagorean theorem and averaging.
坐标几何结合了代数与几何。你必须掌握的三个公式是距离公式、中点公式和斜率公式。它们本质上分别是勾股定理和取平均值的应用。
Distance between (x₁, y₁) and (x₂, y₂): d = √[(x₂ – x₁)² + (y₂ – y₁)²]
Midpoint M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Slope m = (y₂ – y₁) / (x₂ – x₁)
The distance formula is directly derived from a right triangle’s hypotenuse, so visualizing the horizontal and vertical legs can help you avoid sign errors. Midpoint is simply the average of the x-coordinates and the average of the y-coordinates.
距离公式直接由直角三角形的斜边推导而来，可视化水平直角边和竖直直角边有助于避免符号错误。中点就是 x 坐标的平均值与 y 坐标的平均值。
You should also be able to determine if three points are collinear by checking whether the slope between any two pairs is the same, or if a point lies on a line given by an equation.
你还应能通过检验任意两点间斜率是否相等来判断三点是否共线，或者判断一个点是否在给定方程所表示的直线上。
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