Math Flashcards
Binomial formulas
(x+y)² = x²+2xy+y²
(x-y)² = x²-2xy+y²
(x+y)*(x-y) = x²-y²
Perfect squares from 11² to 20² and 25²
121, 144, 169, 196, 225, 256, 289, 324, 361, 400
625
Fractions to decimal of 1/6, 1/8, 1/12
- 167
- 125
- 0833
5 exponent rules
x^-n=1/x^n x^m * x^n=x^(m+n) (x^m)^n=x^(m*n) (xy)^n= x^n * y^n m*sqrt(x^n)=x^(n/m)
Formulas for variance and standard deviation
variance V² = [∑n=1 (avg(x)-xn)²] / n
standard deviation S = sqrt(V²)
Sum of interior angles of a polygon
180(n-2)
with n = numer of sides
How to determine, whether a fraction in decimal form terminates
The fraction’s denominator in fully reduced form must have a prime factorization that consists of only 2’s and/or 5’s.
Square root of 2, 3, and 5
sqrt(2) = 1.4
sqrt(3) = 1.7
sqrt(5) = 2.25
How to factor a quadratic equation x²+px+q
Find two numbers whose product is q and whose sum is p. Multiply the factors by (-1) to get the results for x
Formula for inverse proportionality
y1x1=y2x2
How to simplify a fraction with an expression similiar to 3-sqr(2) in the denominator
Multiply it with the conjugation of the denominator (e.g. 3+sqr(2))
Area of a trapezoid
(Base1 + Base2) x Height x 0.5
Restrictions of a triangle’s side
The sum of any two sides must be greater than the third side
The difference of any two sides must be smaller than the third side
Common right triangles
3-4-5
5-12-13
8-15-17
Multiples of those
Sides of the 45-45-90 triangle
leg; 45°; x
hypotenuse; 90°; x*sqrt(2)
Sides of the 30-60-90 triangle
Leg; 30°; x
Leg; 60°; xsqrt(3)
Hypotenuse; 90°; 2x
What is the value of an exterior angle of a triangle?
The sum of the opposite interior angles
3 statements about circles
If you know any one of the values C, r, d, A, you can determine the rest
An inscribed angle is equal to half of the arc it intercepts
If one of the sides of an inscribed triangle is a diameter of the circle, then the triangle must be a right triangle
Area of a rhombus
(Diagonal1 x Diagonal2) / 2
Area of an equilateral triangle
(S²*sqrt(3))/4
What’s the ratio of the area of two similiar triangles/polygons etc. with corresponding side lengths in ratio a:b?
a²:b²
Fomulas for the diagonal of a square and a cube
Square: d=ssqrt(2)
Cube: d=ssqrt(3)
Key to solving perpendicular bisector problems
Perpendicular lines have negative reciprocal slopes
How to maximize the size of a parallelogram or triangle with two given sides
place the two given sides perpendicular to each other
First six factorials
1!: 1 2!: 2 3!: 6 4!: 24 5!: 120 6!: 720
The number of ways of arranging n distinct objects, if x occurs 3 times and y 5 times
n! / (3!5!)
Which method should you use, if a probability question contains “at least” or “at most” language?
1-P(Not A)
How to calculate the GCF with the prime column method
take the lowest power of each column and multiply the whole numbers
How to calculate the LCM with the prime column method
take the highest power of each column and multiply the whole numbers
How many factors does a number with prime factorization ax x by x cz (where a, b, c are primes) have?
(x+1)(y+1)(z+1)
A perfect square has an … number of total factors
odd
The prime factorization of a perfect square/cube contains only powers which are multiples of
2/3
Gaußsche Summenformel
[n*(n+1)]/2
Number of k-element subsets of a set with n elements
(n über k) = n! / (k!(n-k)!)
Formula for average speed
Average Speed = Total Distance / Total Time
prime numbers up to 101
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101
What is FOIL?
(a + b) x (c + d)
F = firsts (a x c) O = outers (a x d) I = inners (b x c) L = lasts (b x d)
(a + b) x (c + d) = ac + ad + bc + bd
What is the volume of an cube with edge-length?
V = s^3
That’s the reason we say a number to the third power is a “cube” – it comes from the volume for a geometric cube!
Approcimately, how big is pi
3,14
That’s the reason we say a number to the third power is a “cube” – it comes from the volume for a geometric cube!
What is the surface area of a cube with edge-length s?
surface area = 6 s^2
Name three different sets of three lengths that satisfy the Pythagorean Theorem
These sets are called Pythagorean Triplets, numbers that satisfy a2+b2=c2. {3, 4, 5} and {5, 12, 13} and {8, 15, 17}, as well as multiples of these, are good triplets to know.
What’s true about the slopes of two perpendicular lines in the x-y plane?
The slopes of perpendicular lines are opposite reciprocals.
Opposite = change the ± sign. Reciprocal = take 1 over it.
Examples of opposite reciprocals:
m1 = 3 and m2= -1/3 m1 = -5 and m2=1/5 m1= -2/7 and m2=7/2
Any of those pairs could be slopes of two perpendicular lines.
What is a trapezoid?
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Length of a circular arc?
To find an arclength, you need the radius and the angle of the arc. Set up a proportion of part-to-whole: the arc angle is part of 360°, which is the whole angle, and the arclength is part of the circumference, which is the whole length.
arclength/2pir = arcangle/360
Slope between two points
Rise = the change in vertical distance = the difference of y-coordinates
Run = the change in horizontal distance = the difference of x-coordinates
What does the Pythagorean Theorem say? What do the symbols in the formula mean? For what types of triangles is it true?
a^2+b^2=c^2
where a and b are legs and c is the hypotenuse; this only is true for right triangles.
What is the length of the diagonal between opposite vertices of a rectangular solid with dimensions h by w by d?
diagonal^2 = h^2 + w^2 + d^2
diagonal = squrt(h^2 + w^2 + d^2)
True or False:
A square is a rectangle.
True. A square is a special case in the category “rectangle.” A square is a rectangle that also happens to be a rhombus.
What does it mean to say a triangle is isosceles?
If a triangle is isosceles then it has (a) at least two equal sides, and (b) the angles opposite those sides are also equal. These two, fact (a) & fact (b), always go together: you can’ t have one without the other.
If a line has a negative slope, it must pass through which two quadrants?
II & IV
What is the slope of a vertical line?
A vertical line has a big rise and no run, so its slope would be something divided by zero, which is mathematically problematic. The way we say this is:
slope of a vertical line = undefined
In other words, in the process of trying to find it, we inevitably break the mathematical law, so we can’t give a sensible numerical answer.
True or False:
If AB = CD, and opposite sides are parallel, then ABCD must be a square. (Diagram is not drawn to scale)
False. Opposite sides parallel makes ABCD a parallelogram. Parallelograms always have opposite sides that are equal. ABCD is a parallelogram but not necessarily a rhombus, rectangle, or square.
What is the area of a triangle? What do the symbols mean in the formula?
b is the base (the length of any side), and h is the length of the altitude perpendicular to that base
area = (1/2)bh
What is the total surface area of a rectangular solid with dimensions h by w by d?
surface area = 2hw + 2wd + 2hd
What does it mean to say that a polygon is regular?
A regular polygon has
(1) all equal side lengths
and
(2) all equal angle measures
Definition prime number
A prime number is divisible by itself and 1, where “itself” is a different number than 1
How can you express the following term:
20^5
20^5 = 2^5 * 10^5
How can you express the following term:
2^5 * 2^6
2^5 * 2^6 = 2^11
Area of a circle
(pi) *r^2
pi) *((d^2)/4
Sector of Circle:
Area of Sector
[(Degress of central angle)/360 ] * area circle
Rectangular Solids
Area of Front & Back Faces
2 * (Length * Height)
Rectangular Solids
Area of Top & Bottom Faces
2 * (Length * Width)
Rectangular Solids
Area of Front & Back Faces
2 * (Length * Width)
Cylinder:
Area of the top & bottom circular bases
(pi)r^2 + (pi)r^2 = 2(pi)r^2
Cylinder:
Lateral surface area
2pir*h
Cylinder:
Total surface area
2(pi)r^2 + 2pir*h
Cylinder:
Volume
(pi)(r^2)h
Cone:
Surface area
pirl + pi*r^2
Cone:
Volume
V=1/3 (area of Cylinder)
V=(1/3)(pi)(r^2)*h
Sphere:
Surface area
4(pi)r^2
Sphere:
Volume
(4/3)(pi)r^3
Coordinate geometry:
Distance between two points
sqrt[ (x1-x2)^2 + (y1-y2)^2 ]
Coordinate geometry:
Mid-point between two points
Coordinates = [ (x1+x2)/2 , (y1+y2)/2 ]
Coordinate geometry:
general form
y = m*x + b
Triangle:
Area of an isosceles triangle
(1/2) * (leg^2)
Triangle:
Area of an right triangle
(1/2) * (L1 * L2)
Square:
Perimeter
4*s
Square:
Area (#1 and #2)
s^2
1/2)*(d^2
Rhombus:
Perimeter
4*s
Circle:
Circumference (#2)
(pi)*d
2(pi)r
Circle:
Diameter
Diameter = Circumference/pi
Circle:
Radius
Diameter = Circumference/(2*pi)
Arc of Circle:
Arc Length (Central)
Arc length (central) = [(Degress of central angle)/360] * C
with C = Circumference
Arc of Circle:
Arc Length (Inscribed)
Arc length (inscribed) = [ 2 * (Degress of inscribed angle)/360] * C
