What Formulas Are Given On The Sat

The SAT, a standardized test widely used for college admissions in the United States, assesses a student's skills in reading, writing, and mathematics. While the reading and writing sections primarily test comprehension and language proficiency, the math section requires a strong understanding of various mathematical concepts and formulas. Fortunately, the College Board, the organization that administers the SAT, provides a reference sheet at the beginning of each math section that includes some essential formulas. However, this sheet is not exhaustive, and students are expected to know additional formulas and concepts to succeed on the test. This article will provide a comprehensive overview of the formulas provided on the SAT reference sheet, as well as other crucial formulas and concepts that students should master to excel on the math section.

Formulas Provided on the SAT Reference Sheet

The SAT reference sheet includes formulas from the following areas of mathematics:

Area and Circumference of a Circle
Area of a Rectangle
Area of a Triangle
The Pythagorean Theorem
Special Right Triangles
Volume of a Rectangular Solid
Volume of a Cylinder

Let's examine each of these formulas in detail.

1. Area and Circumference of a Circle

Area of a Circle: ( A = \pi r^2 )
- Where ( A ) represents the area of the circle, and ( r ) is the radius.
- The area is the space enclosed within the circle.
Circumference of a Circle: ( C = 2 \pi r ) or ( C = \pi d )
- Where ( C ) is the circumference, ( r ) is the radius, and ( d ) is the diameter.
- The circumference is the distance around the circle.

2. Area of a Rectangle

Area of a Rectangle: ( A = lw )
- Where ( A ) is the area, ( l ) is the length, and ( w ) is the width.
- The area is the space enclosed within the rectangle.

3. Area of a Triangle

Area of a Triangle: ( A = \frac{1}{2} bh )
- Where ( A ) is the area, ( b ) is the base, and ( h ) is the height.
- The height is the perpendicular distance from the base to the opposite vertex.

4. The Pythagorean Theorem

Pythagorean Theorem: ( a^2 + b^2 = c^2 )
- In a right triangle, ( a ) and ( b ) are the lengths of the legs, and ( c ) is the length of the hypotenuse (the side opposite the right angle).
- This theorem is used to find the length of one side of a right triangle if the lengths of the other two sides are known.

5. Special Right Triangles

The SAT reference sheet includes information about two special right triangles:

45-45-90 Triangle:
- In a 45-45-90 triangle, the sides are in the ratio ( x : x : x\sqrt{2} ), where ( x ) is the length of each leg, and ( x\sqrt{2} ) is the length of the hypotenuse.
30-60-90 Triangle:
- In a 30-60-90 triangle, the sides are in the ratio ( x : x\sqrt{3} : 2x ), where ( x ) is the length of the side opposite the 30-degree angle, ( x\sqrt{3} ) is the length of the side opposite the 60-degree angle, and ( 2x ) is the length of the hypotenuse.

6. Volume of a Rectangular Solid

Volume of a Rectangular Solid: ( V = lwh )
- Where ( V ) is the volume, ( l ) is the length, ( w ) is the width, and ( h ) is the height.
- The volume is the amount of space inside the rectangular solid.

7. Volume of a Cylinder

Volume of a Cylinder: ( V = \pi r^2 h )
- Where ( V ) is the volume, ( r ) is the radius of the base, and ( h ) is the height.
- The volume is the amount of space inside the cylinder.

Additional Formulas and Concepts to Know for the SAT

While the SAT reference sheet provides some essential formulas, students are expected to know many additional formulas and concepts to perform well on the math section. These include:

Basic Algebra
Linear Equations and Functions
Quadratic Equations and Functions
Systems of Equations
Exponents and Radicals
Geometry
Trigonometry
Data Analysis and Statistics
Complex Numbers

Let's explore each of these areas further.

1. Basic Algebra

Slope-Intercept Form: ( y = mx + b )
- Where ( m ) is the slope and ( b ) is the y-intercept.
Point-Slope Form: ( y - y_1 = m(x - x_1) )
- Where ( (x_1, y_1) ) is a point on the line and ( m ) is the slope.
Standard Form of a Linear Equation: ( Ax + By = C )
- Where ( A ), ( B ), and ( C ) are constants.
Average/Mean: ( \text{Average} = \frac{\text{Sum of terms}}{\text{Number of terms}} )
Percentage Change: ( \text{Percentage Change} = \frac{\text{New Value - Old Value}}{\text{Old Value}} \times 100% )
Simple Interest: ( I = PRT )
- Where ( I ) is the interest, ( P ) is the principal, ( R ) is the rate, and ( T ) is the time.
Compound Interest: ( A = P(1 + \frac{r}{n})^{nt} )
- Where ( A ) is the amount after ( t ) years, ( P ) is the principal, ( r ) is the annual interest rate, ( n ) is the number of times interest is compounded per year, and ( t ) is the number of years.

2. Linear Equations and Functions

Slope of a Line: ( m = \frac{y_2 - y_1}{x_2 - x_1} )
- Given two points ( (x_1, y_1) ) and ( (x_2, y_2) ) on the line.
Parallel Lines:
- Parallel lines have the same slope.
Perpendicular Lines:
- The slopes of perpendicular lines are negative reciprocals of each other (i.e., ( m_1 \cdot m_2 = -1 )).
Linear Function: ( f(x) = mx + b )
- Where ( m ) is the slope and ( b ) is the y-intercept.

3. Quadratic Equations and Functions

Standard Form of a Quadratic Equation: ( ax^2 + bx + c = 0 )
Quadratic Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )
- Used to find the roots (solutions) of a quadratic equation.
Vertex Form of a Quadratic Equation: ( y = a(x - h)^2 + k )
- Where ( (h, k) ) is the vertex of the parabola.
Factoring Quadratic Equations:
- Expressing a quadratic equation as a product of two binomials.
Discriminant: ( \Delta = b^2 - 4ac )
- If ( \Delta > 0 ), the equation has two real solutions.
- If ( \Delta = 0 ), the equation has one real solution.
- If ( \Delta < 0 ), the equation has no real solutions (two complex solutions).

4. Systems of Equations

Substitution Method:
- Solving one equation for one variable and substituting that expression into the other equation.
Elimination Method:
- Adding or subtracting the equations to eliminate one variable.
Graphical Method:
- Graphing both equations and finding the point of intersection.

5. Exponents and Radicals

Exponent Rules:
- ( a^m \cdot a^n = a^{m+n} )
- ( \frac{a^m}{a^n} = a^{m-n} )
- ( (a^m)^n = a^{mn} )
- ( a^0 = 1 )
- ( a^{-n} = \frac{1}{a^n} )
Radical Rules:
- ( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} )
- ( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} )
Rationalizing the Denominator:
- Removing radicals from the denominator of a fraction.

6. Geometry

Area of a Trapezoid: ( A = \frac{1}{2}(b_1 + b_2)h )
- Where ( b_1 ) and ( b_2 ) are the lengths of the parallel sides (bases), and ( h ) is the height.
Area of a Parallelogram: ( A = bh )
- Where ( b ) is the base and ( h ) is the height.
Sum of Angles in a Triangle: ( 180^\circ )
Sum of Angles in a Quadrilateral: ( 360^\circ )
Interior Angle of a Regular Polygon: ( \frac{(n-2) \times 180^\circ}{n} )
- Where ( n ) is the number of sides.
Exterior Angle of a Regular Polygon: ( \frac{360^\circ}{n} )
- Where ( n ) is the number of sides.
Arc Length: ( \text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r )
- Where ( \theta ) is the central angle in degrees, and ( r ) is the radius.
Area of a Sector: ( \text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2 )
- Where ( \theta ) is the central angle in degrees, and ( r ) is the radius.
Volume of a Sphere: ( V = \frac{4}{3}\pi r^3 )
- Where ( r ) is the radius.
Surface Area of a Sphere: ( A = 4\pi r^2 )
- Where ( r ) is the radius.
Volume of a Cone: ( V = \frac{1}{3}\pi r^2 h )
- Where ( r ) is the radius and ( h ) is the height.
Distance Formula: ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )
- Given two points ( (x_1, y_1) ) and ( (x_2, y_2) ) in the coordinate plane.
Midpoint Formula: ( \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) )
- Given two points ( (x_1, y_1) ) and ( (x_2, y_2) ) in the coordinate plane.

7. Trigonometry

Basic Trigonometric Ratios:
- ( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} )
- ( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} )
- ( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} )
Pythagorean Identity: ( \sin^2(\theta) + \cos^2(\theta) = 1 )
Values of Trigonometric Functions for Special Angles:
- ( \sin(30^\circ) = \frac{1}{2}, \cos(30^\circ) = \frac{\sqrt{3}}{2}, \tan(30^\circ) = \frac{\sqrt{3}}{3} )
- ( \sin(45^\circ) = \frac{\sqrt{2}}{2}, \cos(45^\circ) = \frac{\sqrt{2}}{2}, \tan(45^\circ) = 1 )
- ( \sin(60^\circ) = \frac{\sqrt{3}}{2}, \cos(60^\circ) = \frac{1}{2}, \tan(60^\circ) = \sqrt{3} )

8. Data Analysis and Statistics

Range: ( \text{Range} = \text{Maximum value} - \text{Minimum value} )
Median:
- The middle value in a sorted list of numbers.
Mode:
- The value that appears most frequently in a dataset.
Standard Deviation:
- A measure of the spread of a dataset around its mean.
Probability: ( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} )

9. Complex Numbers

Imaginary Unit: ( i = \sqrt{-1} )
Complex Number: ( a + bi )
- Where ( a ) is the real part and ( b ) is the imaginary part.
Addition and Subtraction of Complex Numbers:
- ( (a + bi) + (c + di) = (a + c) + (b + d)i )
- ( (a + bi) - (c + di) = (a - c) + (b - d)i )
Multiplication of Complex Numbers:
- ( (a + bi)(c + di) = (ac - bd) + (ad + bc)i )

Tips for Remembering and Applying Formulas

Understand the Concepts: Don't just memorize formulas. Understand the underlying concepts and how the formulas are derived.
Practice Regularly: Practice solving problems using the formulas to become comfortable with them.
Create Flashcards: Use flashcards to review formulas and their applications.
Use Mnemonics: Create mnemonics or memory aids to help you remember formulas.
Review Regularly: Review formulas regularly to keep them fresh in your mind.
Apply Formulas in Different Contexts: Practice applying formulas in different types of problems to develop a deeper understanding.
Take Practice Tests: Take practice tests to identify areas where you need to improve.
Stay Calm During the Test: Stay calm and focused during the test to avoid making careless mistakes.

Conclusion

The SAT math section requires a solid understanding of various mathematical concepts and formulas. While the SAT reference sheet provides some essential formulas, students must know additional formulas and concepts to succeed on the test. This article has provided a comprehensive overview of the formulas provided on the SAT reference sheet, as well as other crucial formulas and concepts that students should master. By understanding these formulas, practicing regularly, and staying calm during the test, students can improve their performance on the SAT math section and increase their chances of getting into their dream college. Remember, the key to success is not just memorizing formulas, but understanding the underlying concepts and applying them effectively to solve problems.