SAT Math Formulas

Why This Matters

The SAT Math section isn't just testing whether you can plug numbers into formulas—it's testing whether you understand when and why to use each formula. The test writers design questions that require you to recognize which mathematical relationship applies to a given situation, often disguising familiar concepts in unfamiliar contexts. You're being tested on your ability to connect geometry, algebra, coordinate relationships, and quantitative reasoning into a coherent problem-solving toolkit.

Here's the good news: the SAT actually provides many formulas in a reference box at the start of each math section. But here's what they don't tell you—students who've internalized these formulas solve problems faster and more accurately than those who have to look them up mid-problem. Don't just memorize these formulas; know what each one measures, when it applies, and how it connects to other mathematical concepts. That's what separates a good score from a great one.

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Coordinate Geometry Fundamentals

These formulas describe relationships between points on the coordinate plane. They all derive from the same underlying principle: using horizontal and vertical distances to calculate geometric properties.

Slope Formula

• $m = \frac{y_2 - y_1}{x_2 - x_1}$—measures the steepness of a line as rise over run
• Positive slopes go upward left-to-right; negative slopes go downward; zero slope means horizontal
• Essential for writing linear equations and determining if lines are parallel (same slope) or perpendicular (negative reciprocal slopes)

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 formed by horizontal and vertical legs
• Appears in problems asking for lengths of segments, perimeters on coordinate planes, or radius calculations

Midpoint Formula

• $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$—finds the exact center between two points
• Averages the x-coordinates and y-coordinates separately—think of it as finding the "middle" in each dimension
• Useful for bisecting segments, finding centers of circles, or verifying that a point lies equidistant from two others

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Triangle Relationships

Triangles appear constantly on the SAT because they're the building blocks of more complex shapes. Master these formulas, and you'll unlock solutions to problems involving polygons, circles, and coordinate geometry.

Area of a Triangle

• $A = \frac{1}{2}bh$—base and height must be perpendicular to each other
• Works for all triangles, not just right triangles—find or draw the perpendicular height if needed
• Often combined with other formulas when triangles are embedded in larger shapes or coordinate problems

Pythagorean Theorem

• $a^2 + b^2 = c^2$—applies only to right triangles, where $c$ is always the hypotenuse (longest side, opposite the right angle)
• Use it to find a missing side length when you know two sides of a right triangle
• The foundation for the distance formula—recognizing this connection helps you see when each applies

Special Right Triangle Ratios

• 30-60-90 triangle: sides in ratio $1 : \sqrt{3} : 2$—the side opposite 30° is shortest, opposite 90° is longest
• 45-45-90 triangle: sides in ratio $1 : 1 : \sqrt{2}$—the two legs are equal, hypotenuse is $\sqrt{2}$ times a leg
• Memorize these ratios—they let you find missing sides instantly without using the Pythagorean theorem

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Circle Measurements

Circle formulas connect a circle's radius to its area and perimeter. The radius is your key—if you can find it, you can find everything else.

Area of a Circle

• $A = \pi r^2$—measures the space inside the circle
• Radius is squared, so doubling the radius quadruples the area—watch for this relationship in comparison problems
• Often tested alongside sector area, which is a fraction of the full circle: $\frac{\theta}{360} \times \pi r^2$

Circumference of a Circle

• $C = 2\pi r$ (or equivalently $C = \pi d$, where $d$ is diameter)
• Measures the distance around the circle—think perimeter
• Connects to arc length problems, where you find a fraction of the circumference based on the central angle

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Quadrilateral and 3D Shapes

These formulas extend basic principles to more complex shapes. Notice how they build on simpler concepts—trapezoid area uses triangle logic, and prism volume extends rectangle area into three dimensions.

Area of a Trapezoid

• $A = \frac{1}{2}(b_1 + b_2)h$—averages the two parallel bases, then multiplies by height
• Think of it as the area of a rectangle with the average base length—this helps you remember the formula
• Height must be perpendicular to both bases, not the length of the slanted sides

Volume of a Rectangular Prism

• $V = lwh$—length times width times height, or equivalently, base area times height
• Extends to any prism: $V = Bh$, where $B$ is the area of the base
• Essential for word problems involving boxes, rooms, tanks, or any rectangular container

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Algebraic Problem-Solving

These formulas help you solve equations and work with algebraic expressions. They're your tools for finding unknown values when direct calculation isn't possible.

Quadratic Formula

• $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$—finds solutions to any quadratic equation in form $ax^2 + bx + c = 0$
• The discriminant ($b^2 - 4ac$) reveals the nature of solutions: positive = two real solutions, zero = one solution, negative = no real solutions
• Use when factoring is difficult—but try factoring first, as it's often faster on the SAT

Probability Formula

• $P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$—gives probability as a value between 0 and 1
• Identify the sample space carefully—many errors come from miscounting total outcomes
• Often combined with counting principles or presented in table/chart format on the SAT

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Sequences and Growth Models

These formulas describe patterns and changes over time. Arithmetic sequences grow by addition; geometric sequences grow by multiplication—this fundamental difference determines which formula applies.

Arithmetic Sequence

• $a_n = a_1 + (n-1)d$—finds the $n$th term using first term ($a_1$) and common difference ($d$)
• Linear growth—each term increases by the same amount; graphs as a straight line
• Watch for problems describing constant increases like "adds $50 each month" or "grows by 3 each year"

Geometric Sequence

• $a_n = a_1 \cdot r^{n-1}$—finds the $n$th term using first term ($a_1$) and common ratio ($r$)
• Exponential growth—each term is multiplied by the same factor; graphs as a curve
• Look for language like "doubles each day" or "increases by 10% each year"

Exponential Growth/Decay

• $A = P(1 \pm r)^n$—$P$ is initial amount, $r$ is rate (as decimal), $n$ is time periods
• Use + for growth, - for decay—population growth uses +, depreciation uses -
• The most common real-world application on SAT: compound interest, population models, and half-life problems

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Quick Reference Table

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Self-Check Questions

1. Both the Distance Formula and the Pythagorean Theorem involve squared terms and square roots. How are they mathematically related, and when would you use each one?

2. You're given a problem about a circle where the circumference is 10π. How would you find the area? Which formulas connect, and what's the key value you need to find first?

3. A problem states that a population "increases by 15% each year." Would you use the arithmetic sequence formula or the exponential growth formula? What specific value would $r$ be?

4. Compare the discriminant in the quadratic formula to the process of factoring. If $b^2 - 4ac = 0$, what does that tell you about the quadratic, and how would this appear if you tried to factor it?

5. An SAT question shows two points on a coordinate plane and asks three things: the distance between them, the midpoint, and the slope of the line connecting them. Which formula would you use for each, and which two formulas use the exact same input values in different ways?