MATHCOUNTS Formulas

An octahedron is two square pyramids stacked on top of each other base on base.

What is an octahedron?

A relationship between two numbers in which one is a multiple of the other.

What is direct variation?

e ≈ 2.71828182846... Remember, e is the asymptote of the function (1 + 1/x) raised to the power of x.

What is e and what does it represent?

A list, table, or graph that displays the frequency of various outcomes in a sample.

What is frequency distribution?

A relationship between two variables in which the product is a constant.

What is inverse variation?

Number of Success/Number Total or Number Total - Number of Failure/Number Total

What is probability?

Two numbers that have none of the same factors such as 14 (2 × 7) and 15 (3 × 5).

What is relatively prime?

Area of Rhombus = (d₁ × d₂)/2 Remember: d₁ and d₂ are diagonals of the rhombus.

What is the area of a rhombus given its diagonals?

distance = √(x₁ - x₂)² + (y₁ - y₂)² (x₁, y₁) and (x₂, y₂) are the points you are trying to find the distance between.

What is the distance formula?

The first eight digits of π is 3.1415926... Remember, π is the ratio of a circle's circumference to its diameter.

What is the first eight digits of π?

Surface Area = πr² + πrs Remember: r = radius of your circular base. s = slant height, not the regular height. s could also be equivalent to √r² + h² where h is the regular height of the cone.

What is the formula for the surface area of a cone?

Just plug in the numbers for the following: (x₂ + r[x - x₂], y₂ + r[y - y₂])

How do you dilate point (x, y) by a scale factor of r with center of dilation (x₂, y₂)?

degrees in exterior angle = 360 / n n is the number of sides of the regular polygon.

How do you find the number of degrees in an exterior angle in a regular polygon with n sides?

degrees in interior angle = 180 - 360 / n n is the number of sides of the regular polygon.

How do you find the number of degrees in an interior angle in a regular polygon with n sides?

Number of diagonals: (n² - 3n) / 2 n is the number of sides of the polygon.

How do you find the number of diagonals of a polygon with n sides?

total number of degrees = 180(n - 2) n is the number of sides of the polygon.

How do you find the total number of degrees in a polygon with n sides?

First: You find a line perpendicular to the line you're reflecting over that runs through the point you're reflecting. Second: You find the intersection of the two perpendicular lines (the line you're reflecting over and the line perpendicular to the line you're reflecting over that runs through the point you're reflecting). Third: Label the intersection as the midpoint of a segment and then say one of the endpoints is the point you're reflecting over a line. Using that you find the other endpoint and that's your reflected point.

How do you reflect a point over a line?

Just plug in the numbers to the following: (y + [x₂ -y₂], -x + [x₂ + y₂])

How do you rotate (x, y) ninety degrees clockwise about (x₂, y₂)?

Just plug in the numbers to the following: (-y + [y₂ + x₂], x + [y₂ -x₂])

How do you rotate (x, y) ninety degrees counterclockwise about (x₂, y₂)?

The first eight powers of three are: 1, 3, 9, 27, 81, 243, 729, and 2187

What are the first eight powers of three?

The first fifteen perfect cubes are: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, and 3375

What are the first fifteen perfect cubes?

The first fifty perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, and 2500

What are the first fifty perfect squares?

The first twelve powers of two are: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, and 2048

What are the first twelve powers of two?

The twenty-five primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97

What are the twenty-five prime numbers from one to one-hundred?

Twin primes are primes that differ by only two. An example would be three and five, or five and seven.

What are twin primes?

It means distance = rate multiplied by time Remember: If distance was measured in miles and the time was measured in hours, the rate would have to be measured in miles per hour.

What does d = rt mean?

area = √s(s - x)(s - y)(s - z) s is the semi-perimeter which is half the perimeter, or in this case, half the sum of x, y, and z. x, y, and z are just the different side lengths of the triangle.

What is Heron's Formula?

A special right triangle named by its angles. Has side lengths in the ratio of 1 : √3 : 2.

What is a 30-60-90 triangle?

A special right triangle named by its angles. Has side lengths in the ratio of 1 : 1 : √2.

What is a 45-45-90 triangle?

A decagon is a polygon with ten sides and its angles total to 1,440 degrees.

What is a decagon?

A dodecagon is a polygon with twelve side and its angles total to 1,800 degrees.

What is a dodecagon?

A frustum is basically a large cone, with a smaller cone chopped off the top of it.

What is a frustum?

A hendecagon is a polygon with eleven sides and its angles total to 1,620 degrees.

What is a hendecagon?

A heptagon is a polygon with seven sides and its angles total to 900 degrees.

What is a heptagon?

A hexagon is a polygon with six sides and its angles total to 720 degrees.

What is a hexagon?

The positive integers and sometimes but rarely zero is included.

What is a natural number?

A nonagon is a polygon with nine sides and its angles total to 1,260 degrees.

What is a nonagon?

A parallelogram is a quadrilateral with opposite sides of equal measure. Adjacent angles are supplementary and opposite angles are congruent. Two pairs of parallel sides.

What is a parallelogram?

A pentagon is a polygon with five sides and its angles total to 540 degrees.

What is a pentagon?

A polyhedron is a three-dimensional figure which is composed of faces as two-dimensional shapes.

What is a polyhedron?

The divisors or factors of a number except for the number itself.

What is a proper divisor or proper factor?

A quadrilateral is a polygon with four side and its angles total to 360 degrees.

What is a quadrilateral?

A rectangle is a quadrilateral with opposites sides having equal measure, and all angles of measures ninety degrees. Has two pairs of parallel sides.

What is a rectangle?

If a bike were to make one full rotation that would be one revolution.

What is a revolution?

A rhombus is a quadrilateral with sides of all equal measure. Adjacent angles are supplementary and opposite angles are equal in measure. Has two pairs of parallel sides. Diagonals of a rhombus are perpendicular bisectors.

What is a rhombus?

A triangle with one angle measure ninety degrees. The other two angles are complementary.

What is a right triangle?

A scalene triangle is a triangle with no sides of equal length, and no angles of equal measure.

What is a scalene triangle?

A square is a quadrilateral with all sides of equal length, and all angles of measures ninety degrees. Has two pairs of parallel sides. When split in half by diagonal, two 45-45-90 triangles formed.

What is a square?

It's the same as a triangular prism.

What is a tetrahedron?

A triangle is a polygon with three sides and its angles total to 180 degrees.

What is a triangle?

A number that comes out when you plug an integer n into (n² + n)/2 The first ten triangular numbers are: 1, 3, 6, 10, 15, 21, 28, 36, 45, and 55 These can also be represented as the first numbers where they can be the sum of the first n positive integers.

What is a triangular number?

The same thing as a bisect except it's split into three angles of equal measure.

What is a trisect?

A unit fraction is a fraction with a numerator of one, and a denominator that is a positive integer.

What is a unit fraction?

A triangle with all angles of measures less than ninety degrees.

What is an acute triangle?

The apex is the angle on an isosceles triangle that's different than the other two angles that are equal. The common polygon vertex at the top of a pyramid or cone is also known as an apex.

What is an apex?

An equilateral triangle is a triangle with three sides of equal length, and three angles each of measure sixty degrees. When you draw an altitude in it, you make a pair of 30-60-90 triangles.

What is an equilateral triangle?

An isosceles triangle is a triangle with two sides of equal length, and the base angles of equal measure.

What is an isosceles triangle?

An n-gon is a polygon with n sides, and its angles total to 180(n - 2) degrees. n is a value greater than or equal to thirteen.

What is an n-gon?

A triangle with one angle measuring more than ninety degrees. The other two each acute.

What is an obtuse triangle?

An octagon is a polygon with eight sides and its angles total to 1,080 degrees.

What is an octagon?

Area = πr²h/3 Remember: r = radius of your circular base h = height of the cone but not the slant height, from the base to the highest point on the cone straight up.

What is the formula for the volume of a cone?

Area = lwh/3 Remember: l = length of the base of the pyramid. w = width of the base of the pyramid. h = height of the pyramid but not the slant height, from the base to the highest point on the pyramid straight up.

What is the formula for the volume of a pyramid?

The geometric mean of a and b is √ab Remember: You multiply all the terms, and then you find the nth root where n is the number of terms you multiplied together.

What is the geometric mean of a and b?

Golden Ratio: (1 +/- √5)/2 It represents two things- a. It represents the positive and negative solution of the quadratic x² - x - 1 b. The positive solution represents the ratio of a diagonal on a regular pentagon to its side length. c. It's also the most irrational real number. REMEMBER: If the Golden Ratio = z Then 1/z = z - 1

What is the golden ratio and what does it represent?

If x is the space diagonal, then a² + b² + c² = x² or Space Diagonal = √a² + b² + c²

What is the length of the space diagonal of a rectangular prism with side lengths a, b, and c?

y - y₁ = m(x - x₁) Finds line that has a slope of m, and runs through the point (x₁, y₁).

What is the point-slope formula?

R = xyz/4a Remember: R: Radius x, y, and z: Side lengths of the triangle. a: The area of the triangle.

What is the radius of a circle when it's circumscribed about a triangle?

area = |(ad + cf + be) - (bc + de + af)| / 2 Finds area of any shape where a, b, c, d, e, and f are coordinates from the points (a, b), (c, d), and (e, f).

What is the shoelace theorem with points (a, b), (c, d), and (e, f)?

Space diagonal = x√3 Remember: x = The edge length of the cube.

What is the space diagonal of a cube?

Sum = a/(1-r) Remember: a = first term of the infinite sequence. r = ratio of a term to the term before it. This only works if the numbers absolute value are decreasing like 8, 4, 2, 1, 1/2... but doesn't work if the sequence remains the same or is increasing, that sum is just infinite.

What is the sum of an infinite geometric sequence?

Sum = n² + n

What is the sum of the first n positive even integers?

Sum = (n² + n) / 2

What is the sum of the first n positive integers?

Sum = n²

What is the sum of the first n positive odd integers?

Sum = n(a + z)/2 Remember: n = number of terms in the sequence (a + z)/2 = mean or average of set of numbers which is calculated by adding the first and last numbers of the series, and then diving by two.

Whats is the sum of an arithmetic series?