Pythagorean theorem
Answer: a² + b² = c²
Explanation
c is the hypotenuse of a right triangle.
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Algebra and Geometry: Essential Formulas (High School)
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Geometry
Similar triangles: what is preserved?
Answer: Corresponding angles are equal and corresponding sides are proportional.
Explanation
Area scales by the square of the side ratio.
Triangle inequality theorem
Answer: The sum of any two sides of a triangle must be greater than the third side.
Explanation
Used to test whether three lengths can form a triangle.
What are complementary and supplementary angles?
Answer: Complementary angles add to 90°; supplementary add to 180°.
Explanation
C comes before S, and 90 comes before 180.
Equation of a circle
Answer: (x − h)² + (y − k)² = r²
Explanation
Center (h, k), radius r.
SOH-CAH-TOA
Answer: sin = opposite/hypotenuse; cos = adjacent/hypotenuse; tan = opposite/adjacent
Explanation
Right-triangle trig ratios.
Special right triangle 30-60-90 side ratios
Answer: 1 : √3 : 2
Explanation
Short leg opposite 30°, long leg = short × √3, hypotenuse = 2 × short.
Special right triangle 45-45-90 side ratios
Answer: 1 : 1 : √2
Explanation
Legs equal; hypotenuse = leg × √2.
Surface area of a sphere and a cylinder
Answer: Sphere: 4πr²; cylinder: 2πr² + 2πrh
Explanation
Cylinder = two circles plus the rectangle that wraps around.
Volume of a cone, pyramid, and sphere
Answer: Cone: ⅓πr²h; pyramid: ⅓ · base area · h; sphere: 4/3 πr³
Explanation
Cone and pyramid are one third of the matching prism/cylinder.
Volume of a rectangular prism and a cylinder
Answer: V = l·w·h; V = πr²h
Explanation
Both are base area × height.
Sum of interior angles of a polygon
Answer: (n − 2) · 180°
Explanation
Each interior angle of a regular polygon: (n − 2)·180° / n.
Area of a trapezoid
Answer: A = ½ (b1 + b2) · h
Explanation
Average of the parallel sides times the height.
Area and circumference of a circle
Answer: A = πr²; C = 2πr = πd
Explanation
r is the radius, d the diameter.
Area of a triangle
Answer: A = ½ · base · height
Explanation
Height must be perpendicular to the base.
Algebra
Slope formula between two points
Answer: m = (y2 − y1) / (x2 − x1)
Explanation
Rise over run. Undefined for vertical lines.
Simple interest and compound interest
Answer: I = Prt; A = P(1 + r/n)^(nt)
Explanation
P principal, r annual rate, t years, n compounding periods per year.
Geometric sequence nth term
Answer: a_n = a_1 · r^(n−1)
Explanation
r is the common ratio.
Arithmetic sequence nth term
Answer: a_n = a_1 + (n − 1)d
Explanation
d is the common difference.
Midpoint formula
Answer: M = ((x1 + x2)/2, (y1 + y2)/2)
Explanation
Average the x's and the y's.
Distance formula
Answer: d = √((x2 − x1)² + (y2 − y1)²)
Explanation
Comes from the Pythagorean theorem.
Exponent rules: power and negative exponent
Answer: (a^m)^n = a^(mn); a^(−n) = 1 / a^n
Explanation
Any nonzero number to the 0 power is 1.
Exponent rules: product and quotient
Answer: a^m · a^n = a^(m+n); a^m / a^n = a^(m−n)
Explanation
Same base required.
Perfect square trinomials
Answer: a² + 2ab + b² = (a + b)²; a² − 2ab + b² = (a − b)²
Explanation
Check: the middle term is twice the product of the roots of the outer terms.
Difference of squares
Answer: a² − b² = (a + b)(a − b)
Explanation
Example: x² − 9 = (x + 3)(x − 3).
Vertex of a parabola y = ax² + bx + c
Answer: x = −b / (2a); plug in to find y.
Explanation
Vertex form: y = a(x − h)² + k with vertex (h, k).
What does the discriminant b² − 4ac tell you?
Answer: Positive: two real solutions; zero: one real solution; negative: no real solutions (two complex).
Explanation
It is the part under the square root in the quadratic formula.
Quadratic formula
Answer: x = (−b ± √(b² − 4ac)) / (2a)
Explanation
Solves ax² + bx + c = 0.
Point-slope form of a line
Answer: y − y1 = m(x − x1)
Explanation
Useful when you know one point and the slope.
Slope-intercept form of a line
Answer: y = mx + b
Explanation
m is the slope, b is the y-intercept.
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