SAT Math Essential Formulas to Memorize

While the SAT Math section provides a reference box of formulas, stopping to search for them wastes precious seconds. Strategic memorization of key relationships transforms them from looked-up facts to mental tools, allowing you to solve problems faster and with greater confidence. This guide details the essential formulas you must know cold, organized by the concepts they empower you to conquer.

Algebra Foundations: The Language of Lines and Equations

Algebra forms the backbone of the SAT Math section. Mastering a few core formulas allows you to efficiently interpret graphs, solve for variables, and model real-world scenarios.

The most critical is the slope-intercept form of a linear equation: $y=mx+b$. Here, $m$ represents the slope (the rate of change), and $b$ represents the y-intercept (where the line crosses the y-axis). Understanding this formula lets you instantly graph a line or extract its key features from an equation. For example, in the equation $y=-3x+5$, the slope is $-3$ (meaning the line falls 3 units vertically for every 1 unit it moves horizontally), and the y-intercept is $5$.

When you encounter quadratic equations—those with an $x^2$ term—the quadratic formula is your ultimate weapon. For any equation in the form $a{x}^2+bx+c=0$, the solutions for $x$ are given by: $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ Memorizing this formula ensures you can always find the roots (x-intercepts) of a parabola, even when it cannot be easily factored. On the test, a question might ask for the solutions to $2x^2-4x-6=0$. Plugging $a=2$, $b=-4$, and $c=-6$ into the formula will reliably yield the correct answers.

Geometry and Trigonometry: Conquering Shapes and Angles

Geometry questions test your spatial reasoning and knowledge of fixed relationships. The Pythagorean theorem is non-negotiable: for any right triangle, the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the lengths of the other two sides ($a$ and $b$). The formula is $a^2+b^2=c^2$. This theorem is used constantly, from finding a diagonal distance to verifying a triangle is a right triangle.

To save even more time, you must memorize the side length ratios for special right triangles.

• $45^{\circ }$-$45^{\circ }$-$90^{\circ }$ Triangle: The legs are equal. If a leg has length $x$, the hypotenuse has length $x\sqrt{2}$.
• $30^{\circ }$-$60^{\circ }$-$90^{\circ }$ Triangle: The side opposite the $30^{\circ }$ angle is the shortest. If its length is $x$, the side opposite the $60^{\circ }$ angle is $x\sqrt{3}$, and the hypotenuse is $2x$.

For circles, know that circumference is $C=2\pi r$ or $C=\pi d$, and area is $A=\pi r^2$. A common SAT task is to relate a part of the circle (an arc or sector) to the whole. The length of an arc is a fraction of the circumference: $\frac{\text{central angle}}{360^{\circ }}\times 2\pi r$. The area of a sector is the same fraction of the total area: $\frac{\text{central angle}}{360^{\circ }}\times \pi r^2$.

The SAT also tests basic trigonometric identities within right triangles. Recall SOH CAH TOA:

• Sine of an angle = $\frac{\text{Opposite}}{\text{Hypotenuse}}$
• Cosine of an angle = $\frac{\text{Adjacent}}{\text{Hypotenuse}}$
• Tangent of an angle = $\frac{\text{Opposite}}{\text{Adjacent}}$

You should also know the complementary relationship: $\sin (x^{\circ })=\cos (90^{\circ }-x^{\circ })$.

Advanced Algebra & Coordinate Geometry: Mapping Relationships

This domain connects algebra to geometric figures on the coordinate plane. The distance formula calculates the length between two points $(x_1,y_1)$ and $(x_2,y_2)$. It is essentially the Pythagorean theorem applied to the coordinate plane: $$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$$ For instance, the distance between $(1,2)$ and $(4,6)$ is $\sqrt{(4-1{)}^2+(6-2{)}^2}=\sqrt{9+16}=\sqrt{25}=5$.

Closely related is the midpoint formula, which finds the point exactly halfway between two endpoints: $$\text{Midpoint}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$ Using the same points, the midpoint is $\left(\frac{1+4}{2},\frac{2+6}{2}\right)=(2.5,4)$. These formulas are frequently tested in questions about line segments, diagonals of shapes, or finding centers.

Common Pitfalls

Even with memorized formulas, careless errors can cost points. Here are the most common mistakes and how to avoid them.

1. Mixing Up Slope-Intercept and Standard Form: The formula $y=mx+b$ is for slope-intercept form. Do not confuse it with standard form ($Ax+By=C$). If an equation is given in standard form, like $3x+2y=12$, you must solve for $y$ to find the slope and intercept, revealing $y=-\frac{3}{2}x+6$.

1. Misapplying Special Right Triangle Ratios: The ratios are fixed. A common error is to assign the $x\sqrt{3}$ side to the $45^{\circ }$-$45^{\circ }$-$90^{\circ }$ triangle. Remember: the $45^{\circ }$ triangle has sides $x,x,x\sqrt{2}$; the $30^{\circ }$-$60^{\circ }$-$90^{\circ }$ triangle has sides $x,x\sqrt{3},2x$. Always start by identifying which type of triangle you have.

1. Confusing Distance and Midpoint Formulas: The distance formula requires squaring the differences and taking a square root, resulting in a single number. The midpoint formula involves simple averages of the coordinates, resulting in an ordered pair $(x,y)$. Saying the "midpoint is $\sqrt{...}$" is a clear sign you've used the wrong formula.

1. Forgetting the Quadratic Formula's Structure: Under time pressure, students might misremember the order of operations, especially the "$-b$" part and the "$\pm$" before the square root. Drill the formula: "$-b$ plus-or-minus the square root of $b$ squared minus $4ac$, all over $2a$." Write it down immediately when you start the test section to cement it.

Summary

• Master the core algebra trio: The slope-intercept form ($y=mx+b$) for lines, the quadratic formula for solving parabolas, and the Pythagorean theorem ($a^2+b^2=c^2$) for right triangles.
• Internalize geometry shortcuts: Know the side ratios for $45^{\circ }$-$45^{\circ }$-$90^{\circ }$ and $30^{\circ }$-$60^{\circ }$-$90^{\circ }$ triangles, and the formulas for circle area ($A=\pi r^2$) and circumference ($C=2\pi r$).
• Navigate the coordinate plane with ease: Use the distance formula to find lengths and the midpoint formula to find centers between two points.
• Leverage SOH CAH TOA: Apply the basic trigonometric ratios (sine, cosine, tangent) to solve right triangle problems involving side lengths and angles.
• Practice recall, not just recognition: The goal is to summon these formulas instantly without hesitation, turning memorized knowledge into actionable problem-solving speed.