- Arithmetic and Number Theory:
- Basic properties of numbers (odd, even, prime, etc.)
- Factors and multiples
- Average: \( \text{Average} = \frac{\text{sum of terms}}{\text{number of terms}} \)
- Percent change: \( \text{Percent Change} = \frac{\text{new value} - \text{original value}}{\text{original value}} \times 100\% \)
- Algebra:
- Simplifying expressions
- Factoring quadratic expressions: \( ax^2 + bx + c \)
- Quadratic formula: \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \)
- Systems of equations
- Exponents and exponential growth/decay
- Geometry:
- Area formulas:
- Rectangle: \( A = l \times w \)
- Triangle: \( A = \frac{1}{2} \times b \times h \)
- Circle: \( A = \pi r^2 \)
- Perimeter/Circumference:
- Rectangle: \( P = 2l + 2w \)
- Circle: \( C = 2\pi r \)
- Pythagorean theorem: \( a^2 + b^2 = c^2 \)
- Volume formulas:
- Rectangular prism: \( V = l \times w \times h \)
- Cylinder: \( V = \pi r^2 h \)
- Area formulas:
- Trigonometry:
- Basic trig ratios (for right triangles): \( \sin \), \( \cos \), \( \tan \)
- \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
- Probability and Statistics:
- Mean, median, mode, and range
- Basic probability: \( P(A) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}} \)
- Combinations and permutations
- Functions:
- Function notation: \( f(x) \)
- Linear function slope: \( y = mx + b \)
- Slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
- Coordinate Geometry:
- Distance formula: \( D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)
- Midpoint formula: \( \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \)
This is a general list and may not be exhaustive. While the SAT provides some of these formulas, it's beneficial for students to be familiar with them for quicker problem-solving during the test.
You are only given geometry formulas, so prioritize memorizing your algebra and trigonometry formulas before test day (we’ll cover these in the next section). You should focus most of your study effort on algebra anyways, because geometry has been de-emphasized on the new SAT and now makes up just 10% (or less) of the questions on each test.
Nonetheless, you do need to know what the given geometry formulas mean. The explanations of those formulas are as follows:
Area of a Circle
- π is a constant that can, for the purposes of the SAT, be written as 3.14 (or 3.14159)
- r is the radius of the circle (any line drawn from the center point straight to the edge of the circle)
Circumference of a Circle
- d is the diameter of the circle. It is a line that bisects the circle through the midpoint and touches two ends of the circle on opposite sides. It is twice the radius.
Area of a Rectangle
- l is the length of the rectangle
- w is the width of the rectangle
Area of a Triangle
- b is the length of the base of triangle (the edge of one side)
- h is the height of the triangle
- In a right triangle, the height is the same as a side of the 90-degree angle. For non-right triangles, the height will drop down through the interior of the triangle, as shown above (unless otherwise given).
The Pythagorean Theorem
- In a right triangle, the two smaller sides (a and b) are each squared. Their sum is the equal to the square of the hypotenuse (c, longest side of the triangle).
Properties of Special Right Triangle: Isosceles Triangle
- An isosceles triangle has two sides that are equal in length and two equal angles opposite those sides.
- An isosceles right triangle always has a 90-degree angle and two 45 degree angles.
- The side lengths are determined by the formula: x, x, x square root(2), with the hypotenuse (side opposite 90 degrees) having a length of one of the smaller sides *2.
- E.g., An isosceles right triangle may have side lengths of 12, 12, and 122.
Properties of Special Right Triangle: 30, 60, 90 Degree Triangle
- A 30, 60, 90 triangle describes the degree measures of the triangle’s three angles.
- The side lengths are determined by the formula: x, x square root of(3), and 2x
- The side opposite 30 degrees is the smallest, with a measurement of x.
- The side opposite 60 degrees is the middle length, with a measurement of x square root of(3).
- The side opposite 90 degree is the hypotenuse (longest side), with a length of 2x.
- For example, a 30-60-90 triangle may have side lengths of 5, 53, and 10.
Volume of a Rectangular Solid
- l is the length of one of the sides.
- h is the height of the figure.
- w is the width of one of the sides.
Volume of a Cylinder
- r is the radius of the circular side of the cylinder.
- ℎ is the height of the cylinder.
Volume of a Sphere
- r is the radius of the sphere.
Volume of a Cone
- r is the radius of the circular side of the cone.
- ℎ is the height of the pointed part of the cone (as measured from the center of the circular part of the cone).
Volume of a Pyramid
- l is the length of one of the edges of the rectangular part of the pyramid.
- ℎ is the height of the figure at its peak (as measured from the center of the rectangular part of the pyramid).
- w is the width of one of the edges of the rectangular part of the pyramid.
Law: the number of degrees in a circle is 360
Law: the number of radians in a circle is 2*pi
Law: the number of degrees in a triangle is 180
- More Algebra:
- Properties of exponents:
- \(a^m \times a^n = a^{m+n}\)
- \(a^m \div a^n = a^{m-n}\)
- \((a^m)^n = a^{mn}\)
- \(a^0 = 1\)
- \(a^{-n} = \frac{1}{a^n}\)
- Properties of radicals:
- \(\sqrt[m]{a^n} = a^{\frac{n}{m}}\)
- \(\sqrt[m]{a \times b} = \sqrt[m]{a} \times \sqrt[m]{b}\)
- Direct and inverse variation:
- Direct: \(y = kx\)
- Inverse: \(xy = k\) or \(y = \frac{k}{x}\)
- Properties of exponents:
- More Geometry:
- Surface area of a cylinder: \(2\pi r^2 + 2\pi rh\)
- Total surface area of a rectangular prism: \(2lw + 2lh + 2wh\)
- Volume of a cone: \(V = \frac{1}{3}\pi r^2 h\)
- Volume of a pyramid: \(V = \frac{1}{3}Bh\) (where \(B\) is the base area)
- Volume of a sphere: \(V = \frac{4}{3}\pi r^3\)
- Surface area of a sphere: \(4\pi r^2\)
- Angle sum of a polygon: \(180(n-2)\) (where \(n\) is the number of sides)
- More Trigonometry:
- Reciprocal identities:
- \(\csc(\theta) = \frac{1}{\sin(\theta)}\)
- \(\sec(\theta) = \frac{1}{\cos(\theta)}\)
- \(\cot(\theta) = \frac{1}{\tan(\theta)}\)
- Reciprocal identities:
- Complex Numbers:
- Standard form: \(a + bi\)
- Magnitude/modulus: \(|a + bi| = \sqrt{a^2 + b^2}\)
- Multiplication: \((a + bi)(c + di) = ac - bd + (ad + bc)i\)
- Sequences and Series:
- Arithmetic sequence: \(a_n = a_1 + (n-1)d\)
- Sum of arithmetic series: \(S_n = \frac{n}{2}(a_1 + a_n)\)
- Geometric sequence: \(a_n = a_1 \times r^{(n-1)}\)
- Sum of geometric series (when \(|r| < 1\)): \(S = \frac{a_1}{1 - r}\)
- Coordinate Geometry:
- Equation of a circle: \((x - h)^2 + (y - k)^2 = r^2\)
- Functions:
- Absolute value function: \(f(x) = |x|\)
- Exponential function: \(f(x) = a \times b^x\)
- Logarithmic function: \(f(x) = \log_b x\)
- Properties of logarithms:
- \(\log_b(xy) = \log_b x + \log_b y\)
- \(\log_b \frac{x}{y} = \log_b x - \log_b y\)
- \(\log_b x^n = n \times \log_b x\)
