ACT Math Formula Sheet and Reference Guide

Unlike the SAT, the ACT Math test does not provide you with a formula reference sheet. Success depends on your ability to recall and correctly apply a core set of mathematical formulas under timed pressure. Mastering these formulas is not about rote memorization but about understanding their components and practicing their application in the varied contexts the ACT presents. This guide organizes the essential formulas you must know, explains their use, and provides strategies to make them second nature.

The Geometry You Must Memorize

Geometry constitutes a significant portion of the ACT Math section, and the formulas for area and volume are its non-negotiable foundation. You must know these from memory.

Start with plane shapes. The area of a rectangle is $A=lw$ (length times width). For a triangle, it's $A=\frac{1}{2}bh$, where $b$ is the base and $h$ is the perpendicular height. The area of a circle is $A=\pi r^2$, and its circumference is $C=2\pi r$ or $C=\pi d$. For a trapezoid, remember $A=\frac{1}{2}h(b_1+b_2)$, averaging the two parallel bases and multiplying by the height.

Moving to three dimensions, volume formulas are critical. For a rectangular prism (box), it's $V=lwh$. For a cylinder, $V=\pi r^{2}h$—think of it as the circle's area stretched through the height. The volume of a sphere is $V=\frac{4}{3}\pi r^3$, and its surface area is $SA=4\pi r^2$. For a cone, $V=\frac{1}{3}\pi r^{2}h$; note the $\frac{1}{3}$, which it shares with the pyramid formula $V=\frac{1}{3}lwh$. A common test trick involves combining these shapes, like a cylinder topped with a hemisphere, so practice identifying components.

Special Right Triangles are a huge time-saver. You must know the side ratios for the 45°-45°-90° and 30°-60°-90° triangles. For a 45°-45°-90° triangle, if each leg has length $x$, the hypotenuse is $x\sqrt{2}$. For a 30°-60°-90° triangle, if the side opposite the 30° angle is $x$, the side opposite the 60° angle is $x\sqrt{3}$, and the hypotenuse is $2x$. Recognizing these ratios can solve a problem in seconds without trigonometry.

Essential Algebra and Coordinate Geometry Formulas

Algebra forms the backbone of the test, and certain formulas are indispensable. The quadratic formula, used to find the roots of any equation in the form $a{x}^2+bx+c=0$, is: $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ You should also recognize that the expression under the radical, $b^2-4ac$, is the discriminant. It tells you about the nature of the roots: if positive, two real roots; if zero, one real root; if negative, two complex roots.

Coordinate geometry links algebra to visual graphs. The slope formula for a line through points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2-y_1}{x_2-x_1}$. Remember, "rise over run." From this, you can write the equation of a line in slope-intercept form: $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept.

The distance formula finds the length between two points: $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$. It is directly derived from the Pythagorean Theorem. The midpoint formula finds the point halfway between them: $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. For circles, the standard equation is $(x-h{)}^2+(y-k{)}^2=r^2$, where $(h,k)$ is the center and $r$ is the radius. You may need to complete the square to get an equation into this useful form.

Trigonometry and Circle Relationships

While ACT trigonometry is less extensive than pre-calculus, a few key relationships are guaranteed to appear. In a right triangle, you must know SOH CAH TOA:

• Sine of an angle = Opposite / Hypotenuse
• Cosine of an angle = Adjacent / Hypotenuse
• Tangent of an angle = Opposite / Adjacent

These ratios allow you to solve for unknown sides or angles. Also, understand the reciprocal functions: cosecant is the reciprocal of sine, secant of cosine, and cotangent of tangent. You should know the sine and cosine values for the key angles from the special right triangles (e.g., $\sin 30^{\circ }=\frac{1}{2}$, $\cos 45^{\circ }=\frac{\sqrt{2}}{2}$).

For all circles, remember the relationship between arcs, central angles, and sectors. The length of an arc is a fraction of the circumference: $\frac{\text{central angle}}{360^{\circ }}\times 2\pi r$. Similarly, the area of a sector is a fraction of the total area: $\frac{\text{central angle}}{360^{\circ }}\times \pi r^2$. Think of it as a pie slice—the angle tells you what fraction of the pie you have.

Common Pitfalls

1. Misapplying the Area Formulas: The most frequent error is using the wrong height, especially in triangles and trapezoids. The height must be perpendicular to the base you are using. In a parallelogram, do not use the slanted side as the height. Always double-check this relationship in the diagram.
2. Mixing Up Slope and Distance Formulas: Students often invert rise and run in the slope formula or forget the square root in the distance formula. A helpful mnemonic: Slope is a ratio (subtraction), Distance is a root (Pythagorean Theorem with a square root). Write them side-by-side on your practice sheet to contrast them.
3. Forgetting the "Plus/Minus" in the Quadratic Formula: When you take the square root of both sides while solving or use the quadratic formula, there are usually two solutions ($\pm$). The ACT often includes trap answers that give only the positive root. If the problem context allows for two solutions, check both.
4. Overcomplicating with Trigonometry: If you see a problem with angles of 30°, 60°, or 45°, check for special right triangles first. Using SOH CAH TOA is often slower and introduces more rounding error than simply applying the known side ratios $x,x\sqrt{3},2x$ or $x,x,x\sqrt{2}$. Use trigonometry as a tool, but not your only tool.

Summary

• The ACT provides no formula sheet. You are responsible for memorizing all area, volume, coordinate geometry, and key trigonometry formulas. Create organized flashcards and practice daily.
• Geometry is formula-intensive. Prioritize memorizing area formulas for rectangles, triangles, circles, and trapezoids, and volume formulas for prisms, cylinders, spheres, and cones. The 45°-45°-90° and 30°-60°-90° triangle ratios are essential shortcuts.
• Algebra and coordinate geometry are interconnected. The quadratic formula, distance formula, midpoint formula, slope formula, and circle equation are all frequently tested. Understand how the distance formula derives from the Pythagorean Theorem.
• Know SOH CAH TOA and circle sector formulas. Basic right-triangle trigonometry is tested, as is the relationship between a central angle and the arc length or sector area it creates.
• Practice application, not just recall. The ACT tests your ability to select and use the correct formula in novel situations. Work through many practice problems, identifying which formula is required for each step. Your goal is automatic, accurate recall under timed conditions.