What Formulas Does The Sat Give You

On the SAT, a collection of key formulas is provided at the beginning of each math section. This reference sheet is designed to support test-takers by offering quick access to essential mathematical equations and concepts. Understanding these formulas and knowing how to apply them effectively is critical for maximizing your score on the SAT math sections.

What Formulas Are Provided on the SAT?

The SAT provides a concise list of formulas that cover areas such as geometry and measurement. These formulas are included to ensure that the test assesses mathematical reasoning rather than rote memorization. Familiarizing yourself with these formulas before the test day can save you time and reduce anxiety during the exam.

Here’s a breakdown of the formulas you can expect to see:

1. Area and Circumference of a Circle

• Area: ( A = \pi r^2 )
• Circumference: ( C = 2 \pi r )

Where ( r ) is the radius of the circle. These formulas are fundamental for solving problems involving circles, sectors, and other related geometric figures.

2. Area of a Rectangle

• Area: ( A = lw )

Where ( l ) is the length and ( w ) is the width. This is one of the most basic formulas, used extensively in geometry and algebra questions.

3. Area of a Triangle

• Area: ( A = \frac{1}{2} bh )

Where ( b ) is the base and ( h ) is the height. Knowing this formula is essential for solving problems involving triangles, including those within more complex shapes.

4. Pythagorean Theorem

• Equation: ( a^2 + b^2 = c^2 )

Where ( a ) and ( b ) are the lengths of the legs of a right triangle, and ( c ) is the length of the hypotenuse. This theorem is crucial for finding unknown side lengths in right triangles and is often used in conjunction with other geometric principles.

5. Properties of Special Right Triangles

The SAT provides the side length ratios for two special right triangles:

• 30°-60°-90° Triangle: The sides are in the ratio ( 1:\sqrt{3}:2 ).
• 45°-45°-90° Triangle: The sides are in the ratio ( 1:1:\sqrt{2} ).

These ratios are invaluable for quickly determining side lengths without needing to use trigonometric functions.

6. Volume Formulas

• Volume of a Cube: ( V = lwh ) (where length, width, and height are the dimensions of the cube)
• Volume of a Cylinder: ( V = \pi r^2 h ) (where ( r ) is the radius of the base and ( h ) is the height)

These formulas help in solving problems that involve three-dimensional shapes and their volumes.

7. Volume of a Cone

• Volume: ( V = \frac{1}{3} \pi r^2 h )

Where ( r ) is the radius of the base and ( h ) is the height. This formula is necessary for calculating the volume of cones, which may appear in various geometry problems.

8. Volume of a Pyramid

• Volume: ( V = \frac{1}{3} lwh )

Where ( l ) is the length, ( w ) is the width, and ( h ) is the height. This is used for finding the volume of pyramids, which are less common but still important to understand.

9. Relationships of Angles and Arcs in a Circle

• Angle Measure: The number of degrees of arc in a circle is 360.
• Radian Measure: The number of radians of arc in a circle is ( 2\pi ).

These relationships are important for problems involving circles, arcs, and angles, particularly in trigonometry and advanced geometry questions.

How to Effectively Use the Provided Formulas

Having these formulas at your fingertips is only part of the battle. The real key is understanding how to apply them effectively. Here’s how to maximize their use:

1. Familiarize Yourself with the Formulas

Before the test, take the time to memorize the formulas provided. Knowing them well will allow you to quickly recall and apply them without wasting precious time looking them up.

2. Understand the Variables

Make sure you know what each variable in the formula represents. For example, in the area of a triangle formula ( A = \frac{1}{2} bh ), understand that ( b ) is the base and ( h ) is the height. Misidentifying variables can lead to incorrect answers.

3. Practice Applying the Formulas

Work through a variety of practice problems that require you to use the formulas. This will help you become comfortable with applying them in different contexts. Pay attention to how the formulas can be manipulated to solve for different variables.

4. Recognize When to Use a Formula

Develop the ability to recognize when a particular formula is needed. This comes with practice and a solid understanding of the underlying mathematical concepts. Look for keywords or clues in the problem that indicate which formula to use.

5. Don't Over-Rely on Formulas

While the formulas are helpful, don't rely on them exclusively. Sometimes, understanding the underlying concept is more important than simply plugging numbers into a formula. Use your problem-solving skills and logical reasoning to approach problems from different angles.

Formulas Not Provided on the SAT

While the SAT provides a helpful list of formulas, there are several important formulas and concepts that are not included. Knowing these can give you an edge and improve your problem-solving speed and accuracy.

1. Distance Formula

• Formula: ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )

This formula calculates the distance between two points ((x_1, y_1)) and ((x_2, y_2)) in the coordinate plane. It is derived from the Pythagorean theorem and is useful in geometry and coordinate geometry problems.

2. Midpoint Formula

• Formula: ( M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) )

This formula finds the midpoint of a line segment with endpoints ((x_1, y_1)) and ((x_2, y_2)). It is essential for coordinate geometry problems involving line segments and their midpoints.

3. Slope Formula

• Formula: ( m = \frac{y_2 - y_1}{x_2 - x_1} )

This formula calculates the slope of a line passing through two points ((x_1, y_1)) and ((x_2, y_2)). The slope indicates the steepness and direction of a line and is fundamental in linear equations and graphing.

4. Slope-Intercept Form of a Line

• Equation: ( y = mx + b )

Where ( m ) is the slope and ( b ) is the y-intercept. This form is widely used to represent linear equations and is helpful for graphing lines and solving related problems.

5. Point-Slope Form of a Line

• Equation: ( y - y_1 = m(x - x_1) )

Where ( m ) is the slope and ((x_1, y_1)) is a point on the line. This form is useful when you know the slope and a point on the line and need to find the equation of the line.

6. Quadratic Formula

• Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )

This formula solves for the roots of a quadratic equation in the form ( ax^2 + bx + c = 0 ). It is essential for solving quadratic equations that cannot be easily factored.

7. Vertex Form of a Quadratic Equation

• Equation: ( y = a(x - h)^2 + k )

Where ((h, k)) is the vertex of the parabola. This form is useful for identifying the vertex of a parabola and understanding the transformations of quadratic functions.

8. Average Speed Formula

• Formula: ( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} )

This formula calculates the average speed of an object over a given distance and time. It is commonly used in word problems involving rates and distances.

9. Simple Interest Formula

• Formula: ( I = PRT )

Where ( I ) is the interest, ( P ) is the principal amount, ( R ) is the interest rate, and ( T ) is the time. This formula calculates the simple interest earned on an investment or loan.

10. Compound Interest Formula

• Formula: ( A = P\left(1 + \frac{r}{n}\right)^{nt} )

Where ( A ) is the final amount, ( P ) is the principal amount, ( r ) is the interest rate, ( n ) is the number of times interest is compounded per year, and ( t ) is the number of years. This formula calculates the compound interest earned on an investment.

11. Trigonometric Functions

• Sine: ( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} )
• Cosine: ( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} )
• Tangent: ( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} )

These functions relate the angles of a right triangle to the ratios of its sides. They are fundamental in trigonometry and are used in various problems involving angles and triangles.

12. Pythagorean Identity

• Identity: ( \sin^2(\theta) + \cos^2(\theta) = 1 )

This identity relates the sine and cosine of an angle and is useful in simplifying trigonometric expressions and solving equations.

13. Equation of a Circle

• Equation: ( (x - h)^2 + (y - k)^2 = r^2 )

Where ((h, k)) is the center of the circle and ( r ) is the radius. This equation is used to represent circles in the coordinate plane and is helpful for solving problems involving circles and their properties.

14. Arithmetic Sequence Formula

• Formula: ( a_n = a_1 + (n - 1)d )

Where ( a_n ) is the nth term, ( a_1 ) is the first term, ( n ) is the term number, and ( d ) is the common difference. This formula is used to find the nth term of an arithmetic sequence.

15. Geometric Sequence Formula

• Formula: ( a_n = a_1 \cdot r^{(n - 1)} )

Where ( a_n ) is the nth term, ( a_1 ) is the first term, ( r ) is the common ratio, and ( n ) is the term number. This formula is used to find the nth term of a geometric sequence.

Strategies for Remembering Additional Formulas

Memorizing formulas not provided on the SAT can seem daunting, but there are several effective strategies to help you retain them:

1. Flashcards

Create flashcards with the formula on one side and its name and application on the other. Review these flashcards regularly to reinforce your memory.

2. Practice Problems

The best way to remember formulas is to use them. Work through a variety of practice problems that require you to apply the formulas. This will help you understand how the formulas work and when to use them.

3. Mnemonics

Use mnemonics or memory aids to help you remember the formulas. For example, you can create a catchy phrase or acronym that reminds you of the formula.

4. Teach Someone Else

Teaching someone else a formula is a great way to reinforce your own understanding. Explain the formula and its application to a friend or family member.

5. Regular Review

Set aside time each week to review the formulas. Regular review will help you keep the formulas fresh in your mind and prevent you from forgetting them.

Tips for Test Day

On the day of the test, keep these tips in mind to make the most of the formulas provided and the ones you've memorized:

1. Stay Calm

If you encounter a problem that seems difficult, don't panic. Take a deep breath and try to identify the relevant formulas and concepts.

2. Read Carefully

Read each problem carefully to ensure you understand what is being asked. Pay attention to keywords and clues that can help you identify the correct formula to use.

3. Show Your Work

Show your work step-by-step. This will help you catch any errors you may make and can also earn you partial credit even if you don't arrive at the correct answer.

4. Check Your Answers

If you have time, check your answers. Plug your answer back into the original equation or problem to see if it makes sense.

5. Manage Your Time

Keep an eye on the clock and manage your time wisely. Don't spend too much time on any one problem. If you're stuck, move on and come back to it later if you have time.

Conclusion

Mastering the formulas provided on the SAT, as well as those you need to memorize, is essential for achieving a high score on the math sections. By familiarizing yourself with the formulas, understanding how to apply them, and practicing regularly, you can build confidence and improve your performance on the test. Remember to stay calm, read carefully, and manage your time effectively on test day. With the right preparation and mindset, you can succeed on the SAT math sections and achieve your academic goals.