PSAT Math Formulas

Why This Matters

The PSAT Math section isn't just testing whether you memorized a formula—it's testing whether you understand when and why to use it. The formulas below cover the core mathematical relationships you'll encounter: linear relationships, quadratic behavior, geometric measurement, and proportional reasoning. When you recognize which concept a problem is really asking about, the right formula becomes obvious.

Here's the thing: the PSAT gives you some formulas in the reference box, but not all of them. More importantly, knowing a formula means nothing if you can't identify the problem type. As you study, don't just memorize—ask yourself what each formula measures and when you'd reach for it. That's how you turn formula knowledge into points.

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Linear Relationships and Coordinate Geometry

These formulas describe how points and lines behave on the coordinate plane. The underlying principle is that linear relationships have constant rates of change, and we can measure position, distance, and direction using coordinates.

Slope Formula

• $m = \frac{y_2 - y_1}{x_2 - x_1}$—calculates the rate of change between any two points on a line
• Rise over run tells you how much $y$ changes for every unit change in $x$
• Parallel lines share the same slope; perpendicular lines have slopes that are negative reciprocals

Midpoint Formula

• $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$—finds the exact center between two coordinate points
• Average the coordinates separately—$x$-values together, $y$-values together
• Common in geometry problems asking about bisectors, centers, or dividing segments equally

Distance Formula

• $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$—calculates straight-line distance between two points
• Derived from the Pythagorean theorem—the distance is the hypotenuse of a right triangle
• Watch for problems that set up coordinates just so you can find a length

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Quadratic and Polynomial Relationships

Quadratics show up constantly on the PSAT—in word problems, graph analysis, and pure algebra. These formulas help you find solutions (roots) and understand the curved behavior of parabolas.

Quadratic Formula

• $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$—solves any equation in the form $ax^2 + bx + c = 0$
• The discriminant ($b^2 - 4ac$) tells you how many real solutions exist: positive = 2, zero = 1, negative = 0
• Use when factoring isn't obvious—this formula always works for quadratics

Pythagorean Theorem

• $a^2 + b^2 = c^2$—relates the three sides of any right triangle
• $c$ is always the hypotenuse (longest side, opposite the right angle)
• Memorize common triples: 3-4-5, 5-12-13, 8-15-17, and their multiples

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Area and Perimeter (2D Geometry)

These formulas measure flat space. The key principle: area counts square units inside a shape, while perimeter/circumference measures the boundary length.

Area of a Rectangle

• $A = l \times w$—multiply length by width for total square units
• Perimeter is $2l + 2w$ if you need the boundary instead
• Foundation for composite shapes—many irregular figures break into rectangles

Area of a Triangle

• $A = \frac{1}{2}bh$—base and height must be perpendicular to each other
• Height isn't always a side—look for the altitude drawn to the base
• Works for all triangles, not just right triangles

Area of a Circle

• $A = \pi r^2$—radius squared, then multiplied by $\pi$
• Given diameter? Divide by 2 first to get radius
• Sector area is a fraction of this: $\frac{\theta}{360} \times \pi r^2$ for degree measure

Circumference of a Circle

• $C = 2\pi r$ (or equivalently $C = \pi d$)—the distance around the circle
• Arc length is a fraction of circumference based on the central angle
• Connects to unit circle concepts if you continue to SAT or precalculus

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Volume (3D Geometry)

Volume formulas measure three-dimensional space. The pattern: most volumes equal the base area multiplied by height.

Volume of a Rectangular Prism

• $V = lwh$—length × width × height gives cubic units
• Think of it as base area ($lw$) times height—this pattern repeats in other solids
• Common in word problems about boxes, rooms, or containers

Volume of a Cylinder

• $V = \pi r^2 h$—circular base area ($\pi r^2$) times height
• Same logic as the prism: base area × height
• Watch units—if radius is in inches and height in feet, convert first

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Sequences and Growth Patterns

These formulas describe how values change in predictable patterns. Linear sequences add a constant; exponential sequences multiply by a constant.

Arithmetic Sequence

• $a_n = a_1 + (n-1)d$—each term adds the common difference $d$
• Linear growth: the graph of an arithmetic sequence is a straight line
• To find $d$, subtract any term from the next: $d = a_2 - a_1$

Geometric Sequence

• $a_n = a_1 \cdot r^{n-1}$—each term multiplies by the common ratio $r$
• Exponential growth ($r > 1$) or decay ($0 < r < 1$)
• To find $r$, divide any term by the previous: $r = \frac{a_2}{a_1}$

Exponential Growth/Decay

• $A = P(1 + r)^t$—models real-world increase or decrease over time
• $P$ is the starting amount, $r$ is the rate (as a decimal), $t$ is time
• For decay, use $(1 - r)$ instead of $(1 + r)$

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Probability and Statistics

Probability measures likelihood, and it appears throughout the PSAT in data analysis questions.

Basic Probability

• $P(A) = \frac{\text{favorable outcomes}}{\text{total possible outcomes}}$—always a value between 0 and 1
• Probability of 0 means impossible; probability of 1 means certain
• For "or" questions, add probabilities (but subtract overlap if events aren't mutually exclusive)

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Quick Reference Table

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Self-Check Questions

1. Which two formulas are both derived from the Pythagorean theorem, and how do their applications differ?

2. A problem gives you two points and asks for "the point exactly halfway between them." Which formula do you use, and what's the key operation involved?

3. Compare and contrast arithmetic and geometric sequences: What type of real-world scenario would each model? Give one example of each.

4. You're given a quadratic equation and asked how many real solutions it has. What part of the quadratic formula answers this without solving completely?

5. A cylinder and a rectangular prism have the same height. Explain how their volume formulas follow the same underlying pattern, and identify what differs between them.