SAT Math Formulas

Why This Matters

The SAT Math section tests whether you understand when and why to use each formula, not just whether you can plug in numbers. The test writers design questions that require you to recognize which mathematical relationship applies, often disguising familiar concepts in unfamiliar contexts.

The SAT provides many formulas in a reference box at the start of each math section. But students who've internalized these formulas solve problems faster and more accurately than those who 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 concepts. That's what separates a good score from a great one.

---

Coordinate Geometry Fundamentals

These formulas describe relationships between points on the coordinate plane. They all build on the same idea: 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; undefined slope means vertical
• Essential for writing linear equations and determining if lines are parallel (same slope) or perpendicular (negative reciprocal slopes, meaning $m_1 \cdot m_2 = -1$)

Distance Formula

• $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ calculates straight-line distance between two points
• This is 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
• You're just averaging the x-coordinates and y-coordinates separately
• Useful for bisecting segments, finding centers of circles, or verifying that a point lies equidistant from two others

---

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

• $A = \frac{1}{2}bh$ where the base and height must be perpendicular to each other
• Works for all triangles, not just right triangles. If the height isn't obvious, draw or find the perpendicular height from the base to the opposite vertex.
• 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.
• Common Pythagorean triples to memorize: 3-4-5, 5-12-13, 8-15-17, and 7-24-25 (plus their multiples, like 6-8-10)

Special Right Triangle Ratios

• 30-60-90 triangle: sides in ratio $1 : \sqrt{3} : 2$. The side opposite 30° is shortest, opposite 60° is $\sqrt{3}$ times the shortest, and opposite 90° is twice the shortest.
• 45-45-90 triangle: sides in ratio $1 : 1 : \sqrt{2}$. The two legs are equal, and the hypotenuse is $\sqrt{2}$ times a leg.
• These ratios let you find missing sides instantly without the Pythagorean theorem. For example, if the hypotenuse of a 30-60-90 triangle is 10, the short leg is 5 and the longer leg is $5\sqrt{3}$.

---

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: $A_{\text{sector}} = \frac{\theta}{360} \times \pi r^2$ (where $\theta$ is the central angle in degrees)

Circumference of a Circle

• $C = 2\pi r$ (or equivalently $C = \pi d$, where $d$ is diameter)
• Measures the distance around the circle, like a perimeter
• Connects to arc length problems: $\text{arc length} = \frac{\theta}{360} \times 2\pi r$

---

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 whose width is the average of the two bases
• 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
• This extends to any prism: $V = Bh$, where $B$ is the area of the base (works for cylinders too, where $B = \pi r^2$)
• Essential for word problems involving boxes, rooms, tanks, or any rectangular container

---

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 the form $ax^2 + bx + c = 0$
• The discriminant ($b^2 - 4ac$) reveals the nature of solutions:
  • Positive → two real solutions
  • Zero → exactly one real solution (a repeated root)
  • Negative → no real solutions
• Use when factoring is difficult, but try factoring first since it's often faster on a timed test

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 or forgetting to exclude cases.
• On the SAT, probability is often presented in two-way frequency tables. Read the row and column headers carefully to make sure you're pulling the right numbers.

---

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 the first term ($a_1$) and common difference ($d$)
• This produces linear growth: each term increases by the same amount, and the graph is 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 the first term ($a_1$) and common ratio ($r$)
• This produces exponential growth (or decay): each term is multiplied by the same factor, and the graph is a curve
• Look for language like "doubles each day" or "increases by 10% each year"

Exponential Growth/Decay

• $A = P(1 \pm r)^t$ where $P$ is the initial amount, $r$ is the rate (as a decimal), and $t$ is the number of time periods
• Use + for growth, - for decay. Population growth uses +; depreciation uses -.
• For example, $1,000 growing at 5% per year for 3 years: $A = 1000(1.05)^3 = \$1,157.63$
• Common SAT applications: compound interest, population models, and half-life problems

---

Quick Reference Table

---

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 where the circumference of a circle is $10\pi$. 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. If the discriminant $b^2 - 4ac = 0$, what does that tell you about the quadratic's solutions, 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?