Mathematics Cheat-Sheet Review

Question-and-answer flashcards summarizing key formulas and concepts from the math lecture notes.

What is the formula for the area of a square with side length s?

A = s²

What is the perimeter of a square with side length s?

P = 4s

What is the formula for the area of a rectangle with length ℓ and width w?

A = ℓw

How do you find the perimeter of a rectangle?

P = 2ℓ + 2w

What is the area formula for a triangle with base b and height h?

A = ½bh

How is the perimeter of a triangle calculated?

P = a + b + c (sum of all three sides)

What is the area of a parallelogram with base b and height h?

A = bh

What is the perimeter of a parallelogram with adjacent sides a and b?

P = 2a + 2b

State the area formula for a trapezoid with bases a, b and height h.

A = ½(a + b)h

What is the volume of a cube with edge length s?

V = s³

How do you find the surface area of a cube?

SA = 6s²

Give the volume formula for a right circular cylinder of radius r and height h.

V = πr²h

Give the surface-area formula for a right circular cylinder.

SA = 2πrh + 2πr² (often written SA = 2πrh + 2πr²)

What formula gives the distance between points P₁(x₁,y₁) and P₂(x₂,y₂)?

√[(x₂ − x₁)² + (y₂ − y₁)²]

State the midpoint formula for the segment joining (x₁,y₁) and (x₂,y₂).

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

Write the standard equation of a circle with center (h,k) and radius r.

(x − h)² + (y − k)² = r²

How is the slope of a line through (x₁,y₁) and (x₂,y₂) calculated?

m = (y₂ − y₁)/(x₂ − x₁)

State the point–slope form of a line.

y − y₁ = m(x − x₁)

State the slope-intercept form of a line.

y = mx + b

Write the nth term of an arithmetic sequence with first term a₁ and common difference d.

a_n = a₁ + (n − 1)d

Give the sum of the first n terms of an arithmetic series.

S_n = n/2 [2a₁ + (n − 1)d]

Write the nth term of a geometric sequence with first term a₁ and ratio r.

a_n = a₁ r^{n−1}

Give the sum of the first n terms of a finite geometric series (r ≠ 1).

S_n = a₁(1 − r^n)/(1 − r)

What is the sum of an infinite geometric series when |r| < 1?

S = a₁/(1 − r)

What is the sum of the interior angles of any triangle?

180°

State the Pythagorean Theorem.

In a right triangle, c² = a² + b² (c is hypotenuse).

What property relates corresponding sides of similar triangles?

They are proportional (ratios of corresponding sides are equal).

Define sin θ in a right triangle using SOHCAHTOA.

sin θ = opposite / hypotenuse

Define cos θ in a right triangle.

cos θ = adjacent / hypotenuse

Define tan θ in a right triangle.

tan θ = opposite / adjacent

How is a central angle’s measure related to its intercepted arc?

The central angle (in degrees) equals the degree measure of its intercepted arc.

How is an inscribed angle’s measure related to its intercepted arc?

The inscribed angle equals half the measure of its intercepted arc.

Give the formula for arc length s of a circle (θ in radians).

s = rθ

Give the formula for the area of a sector of radius r and angle θ (radians).

Area = ½r²θ

State the Law of Sines for triangle ABC.

a/sin A = b/sin B = c/sin C

Write the Law of Cosines for side a.

a² = b² + c² − 2bc cos A

State the fundamental Pythagorean trigonometric identity.

sin²θ + cos²θ = 1

How do you convert degrees to radians?

Multiply degrees by π/180

What is the reciprocal identity for csc θ?

csc θ = 1/sin θ

Write the quadratic formula for ax² + bx + c = 0.

x = [−b ± √(b² − 4ac)] / (2a)

How does the discriminant Δ = b² − 4ac determine the number of real solutions?

Δ > 0 two real solutions; Δ = 0 one real solution; Δ < 0 no real (two imaginary) solutions.

State the distributive property.

a(b + c) = ab + ac

State the associative property of addition.

(a + b) + c = a + (b + c)

State the commutative property of multiplication.

ab = ba

What is the multiplicative identity property?

a × 1 = a

What is the zero-exponent rule?

x⁰ = 1 (for x ≠ 0)

State the product rule for exponents.

a^m · a^n = a^{m+n}

State the logarithm product property.

logb(mn) = logb m + log_b n

State the change-of-base formula for logarithms.

logb a = logk a / log_k b (k is any positive base)

Factor the perfect-square trinomial a² + 2ab + b².

(a + b)²

Factor the difference of squares a² − b².

(a + b)(a − b)

Factor the sum of cubes a³ + b³.

(a + b)(a² − ab + b²)

What is the formula for the number of permutations of n items taken r at a time?

P(n,r) = n! / (n − r)!

What is the formula for the number of combinations of n items taken r at a time?

C(n,r) = n! / [r!(n − r)!]

State the Fundamental Counting Principle.

If one event can occur m ways and another n ways, the pair can occur m×n ways.

Define probability of event A.

P(A) = (number of favorable outcomes) / (total possible outcomes)

How many meters are in 1 kilometer?

1 km = 1000 m

What is the relationship between liters and milliliters?

1 L = 1000 mL

How do you write 9 235 in scientific notation?

9.235 × 10³

State the divisibility test for 3.

A number is divisible by 3 if the sum of its digits is divisible by 3.

What is the measure of complementary angles?

They add up to 90°.

What is the measure of supplementary angles?

They add up to 180°.