Geometry Regents Study Guide

11 Units · Beginner-Friendly Notes · Step-by-Step Worked Examples · Interactive Quiz + Flashcards · Authentic Jan 2026 Regents Exam · Saved Progress

Term

Definition

Click card to flip · Rate yourself to track weak cards

Hard mode

Quick ways to lock in the facts you keep forgetting. Read the big trick, then the small note tells you what it unlocks. Say them out loud — silly is memorable.

📐 Trigonometry

SOH-CAH-TOA

Sin=Opp/Hyp · Cos=Adj/Hyp · Tan=Opp/Adj. Say: “Some Old Hippie Caught Another Hippie Tripping On Acid.”

“Co” = Complementary

Cosine & sine are cofunctions: sin(x)=cos(90−x). When you see cos = sin, the two angles ADD to 90.

Side? sin/cos/tan. Angle? hit the −1

Looking for a missing SIDE → use sin/cos/tan. Looking for a missing ANGLE → use sin⁻¹/cos⁻¹/tan⁻¹.

Hyp is always across from the 90°

The hypotenuse faces the right angle and is the longest side — never plug it in as a leg.

⭕ Circle Angles — “Where’s the vertex?”

Center = Whole, On = Half

Vertex at the center → angle = the whole arc. Vertex on the circle (inscribed) → HALF the arc.

IN you ADD, OUT you SUBTRACT

Vertex inside (two chords) → ½(arc + arc). Vertex outside (secants/tangents) → ½(big arc − small arc).

Diameter = right angle

An angle inscribed in a semicircle (its arc is a diameter) is always 90°.

Tangent & radius hold hands at 90°

A tangent line is perpendicular to the radius at the point it touches.

🥧 Area, Circumference & Volume

“Cherry Pie Delicious, Apple Pies Are 2”

C=πd (Cherry Pie Delicious) and A=πr² (Apple Pies Are r-squared).

Pointy solids get a THIRD

Cones & pyramids come to a point → V = ⅓Bh. Flat-topped prisms & cylinders → V = Bh (no third).

“Four-thirds pi r cubed”

Sphere volume = 4⁄3 πr³. Sphere surface = 4πr². (Volume is the one with the cube.)

Volume is CUBIC, area is SQUARE

Answer in cm³ for volume, cm² for surface area — match the unit to the dimension.

🔄 Transformations

Reflect over an axis → that letter STAYS

Over the x-axis: x stays, y flips → (x, −y). Over the y-axis: y stays, x flips → (−x, y).

y = x → just SWAP

Reflecting over y = x swaps the coordinates: (x, y) → (y, x).

180° → negate BOTH

Rotate 180°: (x, y) → (−x, −y). For 90° CCW: swap then negate the new first number → (−y, x).

Slide–Flip–Turn keep size; Dilation changes it

Translations, reflections, rotations are rigid (congruent). Only a dilation resizes (similar).

📊 Lines & Coordinate Geometry

Slope = “rise over run”

m = change in y ÷ change in x. The y’s go on top.

Perpendicular? “Flip it & switch the sign”

Negative reciprocal: 2 → −½, and ¾ → −4/3. Parallel lines just keep the SAME slope.

Midpoint = “average the points”

Average the x’s and average the y’s. (Distance = secretly the Pythagorean theorem.)

Circle: opposite signs inside

(x−h)²+(y−k)²=r². The center flips the signs: (x−2)²+(y+3)² → center (2, −3).

✅ Triangle Congruence & Similarity

No “Donkey Theorem” (no ASS)

Valid: SSS, SAS, ASA, AAS, HL. SSA doesn’t work — and reversed it spells the donkey. Avoid it.

AAA = “All Angles → just Similar”

Three equal angles prove SIMILAR (same shape), not congruent (same size).

CPCTC = the “after” move

Only use CPCTC AFTER you’ve proven two triangles congruent, to grab one more equal part.

Add a dimension, add a power

Scale factor k: lengths ×k, area ×k², volume ×k³.

⬛ Quadrilaterals & Polygons

A square is BOTH

A square is a rectangle AND a rhombus, so it has every property of both.

Rhombus = X (perpendicular)

Rhombus diagonals cross at 90° (and bisect the angles). Rectangle diagonals are equal in length.

Exteriors always lap the track = 360°

The exterior angles of ANY polygon add to 360°. Interior sum = (n − 2)·180°.

Triangle inequality: shorts beat the long

The two shortest sides must add to MORE than the longest, or the triangle can’t close.

⚡ Fast Facts to Memorize Cold

30-60-90 = 1 : √3 : 2

Short : long : hyp. Hyp is double the short leg. 45-45-90 = 1 : 1 : √2 (legs equal).

Triples: 3-4-5, 5-12-13, 8-15-17

Memorize these right-triangle sets (and their multiples like 6-8-10) to skip the Pythagorean work.

Centroid is 2/3 from the vertex

It cuts each median 2:1 (the bigger piece touches the vertex).

D = M / V triangle

Density = Mass ÷ Volume. Cover the one you want: M = D·V, V = M ÷ D.

Carry onto itself = 360 ÷ n

A regular n-gon maps onto itself every 360/n°. Hexagon = 60°, octagon = 45°.

Trapezoid median = average of bases

Midsegment = ½(base₁ + base₂).

⭐ Facts You Must Know Cold

Carry onto itself

regular n-gon: 360 ⁄ n°

Trapezoid median

½(base₁ + base₂)

Cofunctions

sin x = cos(90° − x)

Leg geom. mean

leg² = (hyp)(adjacent segment)

Altitude geom. mean

alt = √(seg₁ · seg₂)

Cavalieri

same height + cross-sections ⇒ same volume

Dilate a line

slope kept; off-center ⇒ parallel image

Centroid

divides each median 2 : 1

SAS Area

½ · a · b · sin(C)

⭕ Circle Segment & Angle Rules (must-know)

Situation	Rule
Central angle	= intercepted arc
Inscribed angle	= ½ intercepted arc
Inscribed in a semicircle	= 90° (subtends a diameter)
Tangent–chord angle	= ½ intercepted arc
Cyclic quadrilateral	opposite angles are supplementary
Two chords meet inside	angle = ½(sum of arcs); (part)(part) = (part)(part)
Two secants meet outside	angle = ½(difference of arcs); (whole)(ext) = (whole)(ext)
Tangent + secant	tangent² = (whole secant)(external part)
Tangent & radius	meet at 90° at the point of tangency
⊥ from center to a chord	bisects the chord and its arc

✏️ Constructions You Must Know

Copy a segment / angle — reproduce a length or angle with arcs of equal radius.
Perpendicular bisector — equal arcs from both endpoints; the line through the crossings is ⊥ at the midpoint.
Angle bisector — arc from the vertex, then equal arcs from the two crossings.
Perpendicular through a point (on or off the line) — equal arcs to set up a perpendicular bisector.
Equilateral triangle — two same-radius arcs (radius = the side) from each endpoint.
Square / hexagon inscribed in a circle, and parallel line through a point.
The Geometry Regents has one or two constructions — leave ALL construction marks.

📐 Area & Perimeter Formulas

Triangle Area

A = ½ b h

Rectangle Area

A = l w

Parallelogram

A = b h

Trapezoid

A = ½ (b₁ + b₂) h

Circle Area

A = π r²

Circumference

C = 2π r = π d

Regular Polygon

A = ½ a P (apothem × perimeter)

Equilateral △

A = (√3 ⁄ 4) s²

🧊 Surface Area & Volume

Prism / Cylinder V

V = B h (cyl: πr²h)

Pyramid / Cone V

V = ⅓ B h (cone: ⅓πr²h)

Sphere Volume

V = 4⁄3 π r³

Cube Volume

V = s³

Cylinder SA

2πr² + 2πr h

Sphere SA

4π r²

Density

D = mass ⁄ volume

Pop. Density

people ⁄ area

📊 Coordinate Geometry (memorize these 3)

Distance

√[(x₂−x₁)² + (y₂−y₁)²]

Midpoint

((x₁+x₂)⁄2 , (y₁+y₂)⁄2)

Slope

(y₂−y₁) ⁄ (x₂−x₁)

Slope-Intercept

y = m x + b

Point-Slope

y − y₁ = m(x − x₁)

Parallel slopes

equal (m₁ = m₂)

Perpendicular slopes

negative reciprocals (m₁·m₂ = −1)

Circle

(x−h)² + (y−k)² = r²

📏 Right Triangle Trig

Pythagorean

a² + b² = c²

sin (SOH)

Opposite ⁄ Hypotenuse

cos (CAH)

Adjacent ⁄ Hypotenuse

tan (TOA)

Opposite ⁄ Adjacent

Find an angle

use sin⁻¹, cos⁻¹, tan⁻¹

Co-function

sin x = cos(90° − x)

📐 Special Right Triangles

Triangle	Side Ratio	Rule
45-45-90	1 : 1 : √2	hypotenuse = leg · √2
30-60-90	1 : √3 : 2	short leg : long leg : hypotenuse

⭕ Circle Angle Rules (where is the vertex?)

Vertex location	Angle equals…
At the CENTER (central angle)	the whole intercepted arc
ON the circle (inscribed angle)	½ the intercepted arc
INSIDE (two chords cross)	½ the SUM of the two arcs
OUTSIDE (secants/tangents)	½ the DIFFERENCE of the arcs
Inscribed in a semicircle	90° (right angle)

🔄 Transformation Rules (about the origin)

Transformation	Rule (x, y) →	Size?
Translation	(x + a, y + b)	same
Reflect over x-axis	(x, −y)	same
Reflect over y-axis	(−x, y)	same
Reflect over y = x	(y, x)	same
Rotate 90° CCW	(−y, x)	same
Rotate 180°	(−x, −y)	same
Rotate 270° CCW	(y, −x)	same
Dilation (factor k)	(kx, ky)	CHANGES

⬡ Polygon Angle Sums

Polygon	Sides (n)	Interior Sum (n−2)·180°	Each angle if regular
Triangle	3	180°	60°
Quadrilateral	4	360°	90°
Pentagon	5	540°	108°
Hexagon	6	720°	120°
Octagon	8	1080°	135°

Exterior angles of ANY polygon always add to 360°. Each exterior of a regular n-gon = 360 ⁄ n.

✅ Triangle Congruence vs. Similarity

Congruent (same size): SSS, SAS, ASA, AAS, HL.
Similar (same shape): AA, SAS~, SSS~.
NOT valid: SSA and AAA (AAA only proves similar).
CPCTC: use only AFTER proving triangles congruent.
Scale factor k: length × k, area × k², volume × k³.

📋 Quadrilateral Family Properties

Shape	Key diagonal / side facts
Parallelogram	opp sides ∥ & ≅; diagonals bisect each other
Rectangle	parallelogram + 4 right angles + ≅ diagonals
Rhombus	parallelogram + 4 ≅ sides + ⊥ diagonals
Square	rectangle AND rhombus (all properties)
Isosceles Trapezoid	1 pair ∥ sides; ≅ legs; ≅ diagonals; ≅ base angles

Geometry Regents Study Guide — EXPANDED EDITION

See what your study plan could look like

📊 Session Progress by Unit

📐 Trigonometry

⭕ Circle Angles — “Where’s the vertex?”

🥧 Area, Circumference & Volume

🔄 Transformations

📊 Lines & Coordinate Geometry

✅ Triangle Congruence & Similarity

⬛ Quadrilaterals & Polygons

⚡ Fast Facts to Memorize Cold

⭐ Facts You Must Know Cold

⭕ Circle Segment & Angle Rules (must-know)

✏️ Constructions You Must Know

📐 Area & Perimeter Formulas

🧊 Surface Area & Volume

📊 Coordinate Geometry (memorize these 3)

📏 Right Triangle Trig

📐 Special Right Triangles

⭕ Circle Angle Rules (where is the vertex?)

🔄 Transformation Rules (about the origin)

⬡ Polygon Angle Sums

✅ Triangle Congruence vs. Similarity

📋 Quadrilateral Family Properties