Source: SYMKC Series, Calculus Prep (Texas A&M University)
Tags: geometry, pythagorean theorem, trigonometry, triangle congruence, perimeter, area, volume, surface area, similarity, angles, parallel lines, transformations, logic, polygon, circle, arc, sector, special right triangles
This covers every geometry formula and concept you are expected to recall instantly: the Pythagorean theorem, trig ratios, triangle congruence criteria, area/volume/surface area formulas for all standard shapes, angle relationships with parallel lines, polygon properties, and basic logic (conditionals and their converses). If you are heading into Pre-Cal or Calculus, these need to be second nature.
Pythagorean theorem
In a right triangle with legs a and b and hypotenuse c: a² + b² = c².
SOH-CAH-TOA
The mnemonic for right-triangle trig ratios. Sin = opposite/hypotenuse, Cos = adjacent/hypotenuse, Tan = opposite/adjacent.
Congruent
Two figures are congruent if they have the same shape and size. All corresponding sides and angles are equal.
Similar
Two figures are similar if they have the same shape but not necessarily the same size. Corresponding angles are equal; corresponding sides are proportional.
Complementary angles
Two angles that add up to 90°.
Supplementary angles
Two angles that add up to 180°.
Conditional statement
A logical statement of the form "if p, then q."
Contrapositive
The statement "if not q, then not p." Always has the same truth value as the original conditional.
a² + b² = c²
Only applies to right triangles. The hypotenuse (c) is always the side opposite the right angle.
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
In a right triangle with vertices A (at the top), B (at the right angle), and C:
Side a (opposite A) is the side across from angle A
Side b (opposite B, the base) is across from the right angle
Side c (opposite C) is across from angle C
a: the altitude (height drawn perpendicular to the base)
b: the base
c: the hypotenuse (in a right triangle)
Five ways to prove two triangles are congruent:
SSS (Side-Side-Side)
SAS (Side-Angle-Side)
ASA (Angle-Side-Angle)
AAS (Angle-Angle-Side)
HL (Hypotenuse-Leg, for right triangles only)
Note: AAA proves similarity, not congruence. SSA (or "the ambiguous case") does not prove congruence.
If two similar figures have a ratio of sides a:b, then:
Ratio of perimeters = a : b
Ratio of areas = a² : b²
Ratio of volumes = a³ : b³
45-45-90 triangle
Legs are equal: x and x
Hypotenuse: x√2
30-60-90 triangle
Short leg (opposite 30°): x
Long leg (opposite 60°): x√3
Hypotenuse (opposite 90°): 2x
Square: P = 4s
Rectangle: P = 2l + 2w
Circumference of a circle: C = 2πr = πd
Square: A = s²
Rectangle: A = lw
Parallelogram: A = bh
Trapezoid: A = ½(b₁ + b₂)h
Circle: A = πr²
Right triangle: A = ½bh
Any triangle (Heron's formula): A = √(s(s – a)(s – b)(s – c)), where s = (a + b + c)/2
Equilateral triangle: A = (s²√3)/4
Regular polygon: A = ½ · apothem · perimeter
Cube: SA = 6s²
Sphere: SA = 4πr²
Cylinder: SA = 2πr² + 2πrh
Cube: V = s³
Rectangular prism: V = lwh
Cylinder: V = πr²h
Pyramid: V = (1/3)Bh, where B is the area of the base
Cone: V = (1/3)πr²h
Sphere: V = (4/3)πr³
Volume is measured in cubic units
An acute angle measures less than 90°
An obtuse angle measures more than 90° but less than 180°
Complementary angles add up to 90°
Supplementary angles add up to 180°
When two parallel lines are cut by a transversal, the following angle relationships hold (using the standard labelling where a, b, c, d are angles at the first intersection and e, f, g, h at the second):
Corresponding angles are congruent: a = e, b = f, c = g, d = h
Alternate interior angles are congruent: c = f, d = e
Alternate exterior angles are congruent: a = h, b = g
Co-interior (same-side interior) angles are supplementary: c + e = 180°, d + f = 180°
Translation: a slide; every point moves the same distance in the same direction
Reflection: a flip over a line of reflection; the figure is a mirror image
Rotation: a turn about a fixed point by a given angle
Conditional statement: if p, then q
Converse: if q, then p
Inverse: if not p, then not q
Contrapositive: if not q, then not p
The contrapositive always shares the truth value of the original conditional. The converse and inverse share a truth value with each other, but not necessarily with the original.
3: Triangle
4: Quadrilateral
5: Pentagon
6: Hexagon
7: Heptagon
8: Octagon
9: Nonagon
10: Decagon
Triangle: 180°
Quadrilateral: 360°
Any regular polygon with n sides: (n – 2) × 180°
Know these: radius, diameter, chord, secant, tangent, arc (major and minor), sector, central angle, inscribed angle.
Arc length: L = (θ/360°) × 2πr (or L = rθ when θ is in radians)
Sector area: A = (θ/360°) × πr² (or A = ½r²θ in radians)
√2 ≈ 1.414
√3 ≈ 1.732
A line extends infinitely in both directions
A ray has one endpoint and extends infinitely in one direction
A segment has two endpoints
Pythagorean theorem: a² + b² = c²
Heron's formula: A = √(s(s – a)(s – b)(s – c)), s = (a + b + c)/2
Interior angle sum: (n – 2) × 180°
Arc length: L = (θ/360) × 2πr
Sector area: A = (θ/360) × πr²
⚠️ SSA does not prove congruence. This is the "ambiguous case" and a very common trick question.
⚠️ For similarity ratios: areas scale by the square, volumes by the cube. Students frequently apply the linear ratio to area problems.
⚠️ In a 30-60-90 triangle, the hypotenuse is 2x (not x√3). The long leg is x√3.
⚠️ Volume formulas for pyramids and cones include the factor of 1/3. Forgetting it is one of the most common errors in geometry.
⚠️ The contrapositive is logically equivalent to the original conditional. The converse is not. This comes up in proof-based questions.
⚠️ √2 ≈ 1.414 and √3 ≈ 1.732 show up in special-triangle problems constantly. Worth memorising.
Q: What is the area of a triangle with sides 5, 12, and 13?
A: It is a right triangle (5² + 12² = 13²), so A = ½(5)(12) = 30.
Q: Find the interior angle sum of a regular octagon.
A: (8 – 2) × 180° = 1,080°.
Q: In a 45-45-90 triangle with leg length 7, what is the hypotenuse?
A: 7√2.
Q: Two similar triangles have side ratios of 3:5. What is the ratio of their areas?
A: 9:25.
Q: State the contrapositive of "if it rains, then the ground is wet."
A: "If the ground is not wet, then it did not rain."
Q: What is the volume of a cone with radius 3 and height 10?
A: V = (1/3)π(3²)(10) = 30π.
geometry review, SYMKC, pythagorean theorem, SOH-CAH-TOA, trig ratios, triangle congruence, SSS, SAS, ASA, AAS, HL, similarity, scale factor, special right triangles, 30-60-90, 45-45-90, perimeter, area, volume, surface area, circle, circumference, arc length, sector area, Heron's formula, polygon, interior angles, parallel lines, transversal, corresponding angles, alternate interior angles, transformations, translation, reflection, rotation, conditional, converse, inverse, contrapositive, complementary, supplementary