# Formula Sheets

Below are equations you need to know for various tests. Contact our tutors to learn why and for more thorough resources. It is more important to know how to use these equations but memorizing helps.
Test for Admission to Catholic High Schools (TACHS):
Perimeter = sum of the outside parts; what you would measure if you made a fence around it
of a square = 4 times the length of a side (P = 4s)
of rectangle = twice the length plus twice the width (P = 2L + 2W) or twice the base plus twice the height (P = 2b + 2h)

Circumference = Pi times the diameter (C = πD) or Pi times twice the radius (C = r)

Pythagorean Theorem: a2 + b2 = c2

Area = the stuff inside a flat 2-dimensional object; what you would measure if you painted it; square units (ft2, m2, etc)
of a square = 4 times the length of a side (P = 4s)
of rectangle = length * width (A = LW) or base times height (A = bh)
of a triangle = half the base times the height (A = 0.5bh)

of a circle = the radius squared, then times Pi (C = πr2)

Volume = the stuff inside a 3-dimensional object; what you would measure if you filled it with liquid; cubic units (ft3, m3, etc)
=the area of a face * the height

for rectangular solids = length * width * height (V = L W h)

Average = the sum of the numbers divided by the how many (Average = sum / #)

The sum of the angles in a triangle is 180O.

The sum of the angles in a rectangle is 360O.

The sum of the angles in a circle is 360O.
Common measurements should also be included, such as:
4 quarts in a gallon, 3 feet in a yard, 100 centimeters in a meter, 1000 millimeters in a meter, 1000 meter in a kilometer

Distance = rate * time can also be included but is a specific version of a proportion. Anything that can be solved as d = rt can also be solved as a proportion. There are many types of problems that can be solved as proportions so there is no need to memorize a new formula, but it would be a good idea to understand how this formula relates to both proportions and graphing (y = mx + b) for the SHSAT.

Additionally, the student should be fluent in common operations:
Like signs: add the numbers together and keep the same sign. Example: -1 + -2 = -3
Unlike signs: subtract the numbers and take the sign of the larger number. Example: -1 + 5 = 4. This is the same as saying 5 – 1 = 4 since addition is commutative.
Subtracting Integers: Subtracting a number is the same as adding a negative version of that number. Example: 5 – 1 = 5 + -1
“Keep, Change, Change”

Multiplying and Dividing Integers:   Multiply or divide normally but answer is negative if unlike signs and positive if like signs. Example: -2 * -5 = +10

Adding and Subtracting Fractions: Find common denominator, then add or subtract numerators. Example: 1/2 + 1/4 = 3/4

Multiplying Fractions:    Multiply across. Example: 1/2 * 1/2 = 1/4 (Half of a half is a quarter)

Dividing Fractions:    Dividing by a fraction is the same as multiplying by the reciprocal. Example: 2/5  divided by 2 = 1/5
“Keep, Change, Flip”

The order of operations always applies. Parentheses come first, then exponents, then multiplication or division (as you should see from above, they can be expressed as the same operation), then addition or subtraction (similarly, they can be expressed as the same operation). When in doubt about the order between multiplication and division or addition and subtraction, go left to right.

Division can be expressed as the division symbol, “/”, a fraction (and therefore a decimal less than 1 or percentage less than 100%), “per”, or raised to a negative exponent like 5-2 = 1/25

SHSAT, HSPT Formula Sheet:

All TACHS Formula and the following:
The sum of the angles in any shape is 2 less than the number of sides, then multiply by 180  (Sum = (n-2)*180O)
You know that a triangle (3 sides) has 180 degrees and a quadrilateral (4 sides) has 360, and so on. If you forget the formula, you can use that pattern.

Volume of  Sphere = (4/3)πr3

Multiplying Monomials: Add the exponents. Example: x3 *  x2 = x5
Dividing Monomials: Subtract the exponents. Example: x3 /  x2 = x1 = x
Earth Science Regents: Earth Science Regents:(pdf)
Chemistry Regents:    Chemistry Regents Reference Table (pdf)

Physics Regents:    Physics Regents Reference Table (pdf)

AlgebraII / Trigonometry Regents, Algebra 1 Common Core, Geometry Common Core, AP, SAT 2 Subject Tests Coming Soon 