bythreedu · Math Tracks · Curriculum v1.0 · 2026

Shape, space,
and the logic of
proof.

A full geometry course built around reasoning and spatial thinking, not formula memorization. From points and lines to congruence proofs to circle theorems — every concept is grounded in why it's true, not just what to apply.

Duration~24weeks
Total Lessons24at your pace
Units8progressive phases
Assignments8+problem sets
DestinationTrig & Precalc
Ready
00Approach
Course Philosophy

Geometry Is Logic, Not Formulas

Most students experience geometry as a collection of formulas to memorize. That's not geometry — that's arithmetic with shapes. Real geometry is about reasoning: if these things are true, what must follow? This course teaches you to think in that chain of logic before ever using a formula.

Spatial Thinking as a Skill

The ability to visualize and reason about shapes in space is something that develops with practice. Every unit includes diagram work, construction practice, and problems that require you to build a picture before writing an equation. The diagram always comes first.

01Curriculum
Learning Path
01
Unit 1 · Weeks 1–2
Foundations: Points, Lines, and Angles
3 lessons · Basic definitions · Angle relationships · Geometric notation
undefined termsangle pairsgeometric notation
+
L1
Points, Lines, and Planes
The three undefined terms of geometry and why they're undefined. Segments, rays, and angles defined from them. Geometric notation read and written correctly.
L2
Angle Pairs
Complementary, supplementary, vertical, and linear pairs. Each relationship derived from the definitions, not memorized as a list. Setting up equations to find missing angles.
L3
Parallel Lines and Transversals
Corresponding, alternate interior, alternate exterior, and co-interior angles. Which pairs are congruent and which are supplementary, and why. Proving lines parallel from angle relationships.
Assignment 1
Find all missing angles in 5 parallel line diagrams. For each, write the angle relationship you used and whether the angles are congruent or supplementary before solving.
You read any angle diagram and find missing measures using relationships, not guessing.
02
Unit 2 · Weeks 3–5
Triangles: Congruence and Properties
3 lessons · Triangle classification · Congruence shortcuts · Triangle inequalities
congruence shortcutstriangle sumtriangle inequality
+
L4
Triangle Classification and Properties
Classifying by sides and angles. The triangle angle sum theorem derived from parallel lines. Exterior angle theorem. Isoceles triangle properties from the symmetry of the figure.
L5
Congruence: SSS, SAS, ASA, AAS, HL
What congruence means and why CPCTC follows from it. Each shortcut explained — why SSA is not a valid shortcut (the ambiguous case). Identifying which shortcut applies in a given diagram.
L6
Triangle Inequalities
The triangle inequality theorem: why the sum of any two sides must exceed the third. The relationship between side lengths and opposite angle measures. Checking whether three lengths form a valid triangle.
Assignment 2
Identify the congruence shortcut (or lack thereof) for 8 triangle pairs. Write one sentence justifying each answer. Solve for missing sides and angles in 4 congruent triangle problems using CPCTC.
You identify congruent triangles, apply CPCTC correctly, and reason about triangle inequalities.
03
Unit 3 · Weeks 6–8
Similarity and Proportional Reasoning
3 lessons · Similar figures · AA, SAS, SSS similarity · Proportional sides
similarity shortcutsscale factorproportional sides
+
L7
Similar Figures and Scale Factors
Similarity defined: same shape, proportional sides, equal angles. Scale factor as the ratio of corresponding sides. Finding missing sides using proportions.
L8
Triangle Similarity: AA, SAS, SSS
Why AA is sufficient for similarity when SSA is not for congruence. Identifying which similarity shortcut applies. Indirect measurement using similar triangles.
L9
Proportional Parts and Midsegments
The triangle midsegment theorem. The side-splitter theorem. Using proportional parts to find lengths without knowing the full triangle.
Assignment 3
Solve 5 similarity problems finding missing side lengths. Then solve 2 indirect measurement problems — each must include a labeled diagram showing which triangles are similar and why.
You identify similar triangles, apply proportional reasoning, and use similarity for indirect measurement.
04
Unit 4 · Weeks 9–11
The Pythagorean Theorem and Right Triangles
3 lessons · Pythagorean theorem · Special right triangles · Applications
Pythagorean theorem30-60-9045-45-90
+
L10
The Pythagorean Theorem
Proof by area rearrangement — the theorem derived, not just stated. Finding missing sides. The converse: determining whether a triangle is right, acute, or obtuse from its side lengths.
L11
Special Right Triangles
30-60-90 and 45-45-90 triangles derived from equilateral and square constructions. Finding all sides from one given side without the Pythagorean theorem every time.
L12
Right Triangle Applications
Distance in the coordinate plane via the Pythagorean theorem. Real-world right triangle problems: ladders, ramps, diagonals. Setting up diagrams before writing any equation.
Assignment 4
Solve 10 right triangle problems — 4 using the Pythagorean theorem, 3 using 30-60-90 ratios, 3 using 45-45-90 ratios. Each solution must start with a labeled diagram.
You solve any right triangle problem and select the fastest method before calculating.
05
Unit 5 · Weeks 12–14
Quadrilaterals and Polygons
3 lessons · Quadrilateral properties · Polygon angle sums · Coordinate geometry
parallelogram propertiespolygon angle sumcoordinate proof
+
L13
Parallelogram Properties and Special Quadrilaterals
Properties of parallelograms, rectangles, rhombuses, squares, and trapezoids — each derived from the definition, not memorized as a list. The hierarchy of quadrilaterals and what it means.
L14
Polygon Angle Sums
Interior and exterior angle sums derived by triangulating the polygon. Finding the measure of each interior angle in a regular polygon. Using angle sums to find missing angles in irregular polygons.
L15
Coordinate Geometry of Quadrilaterals
Proving a quadrilateral is a specific type using slope, distance, and midpoint. Classifying a quadrilateral from its vertices. The midpoint quadrilateral theorem.
Assignment 5
Given 4 vertices, classify the quadrilateral and justify using slope and distance. Do this for 3 quadrilaterals. Then find all missing angles in 4 polygon diagrams showing every equation.
You classify any quadrilateral from its properties and prove geometric claims using coordinates.
06
Unit 6 · Weeks 15–18
Circles
3 lessons · Circle parts · Arc and angle relationships · Equations of circles
central anglesinscribed anglescircle equation
+
L16
Circle Parts and Arc Relationships
Radius, diameter, chord, secant, tangent. Arc measure vs. arc length. Central angles and their relationship to intercepted arcs. Arc addition postulate.
L17
Inscribed Angles and Angle-Arc Theorems
The inscribed angle theorem derived from the central angle theorem. Angles formed by chords, secants, and tangents inside and outside the circle. Each formula understood, not memorized.
L18
Equations of Circles
The standard form equation derived from the Pythagorean theorem. Writing the equation from center and radius. Finding center and radius from general form by completing the square.
Assignment 6
Find missing arc measures and angles in 6 circle diagrams. Label which theorem applies for each. Then write the equation of 3 circles from given information and find center/radius from 2 general form equations.
You solve any circle angle or arc problem and connect circle geometry to coordinate geometry via the standard form equation.
07
Unit 7 · Weeks 19–21
Area, Surface Area, and Volume
3 lessons · 2D area · 3D surface area · Volume of solids
area formulas derivedsurface areavolume
+
L19
Area of 2D Figures
Area formulas for triangles, parallelograms, trapezoids, and circles — each derived geometrically. Composite figures broken into parts. Area on the coordinate plane using the shoelace formula.
L20
Surface Area of 3D Solids
Nets as the tool for understanding surface area. Prisms, pyramids, cylinders, and cones. Lateral surface area vs. total surface area and when each is relevant.
L21
Volume of 3D Solids
Volume formulas derived from the base-times-height idea. Cavalieri's principle connecting prisms and cylinders to pyramids and cones. Sphere volume and surface area. Comparing volumes of similar figures.
Assignment 7
Find area, surface area, and volume for 8 figures — 3 composite 2D shapes and 5 3D solids. Each solution must include a net or decomposition sketch before any calculation.
You compute area, surface area, and volume for any standard figure and understand where every formula comes from.
08
Unit 8 · Weeks 22–24
Transformations and Capstone
3 lessons · Rigid motions · Dilations · Capstone review
translationsreflectionsdilations
+
L22
Rigid Motions: Translations, Reflections, Rotations
Each transformation defined precisely. Composition of transformations. Why rigid motions preserve congruence. Using coordinates to describe and perform each transformation.
L23
Dilations and Similarity
Dilation as a non-rigid transformation. Scale factor and center of dilation. Connecting dilations to similar figures — similarity as a dilation followed by a rigid motion.
L24
Capstone Review and Gap Work
Student-directed review of the two weakest units. Full problem set across all 8 units reviewed together, with written corrections on every missed item.
Capstone
A comprehensive problem set covering all 8 units. Every missed problem gets a written correction with reasoning — where the logic broke down, and how to rebuild it.
You complete geometry with a full command of reasoning, measurement, and proof — ready for trigonometry and beyond.
02Reference
Core Concepts
∠
Angle Pairs
Vertical angles are congruent. Supplementary angles sum to 180°. Complementary angles sum to 90°. Parallel lines cut by a transversal create corresponding, alternate, and co-interior angle pairs.
Parallel lines · Transversals · Triangle sum
△≅△
Congruence
Two figures are congruent if one can be mapped onto the other by a rigid motion. For triangles: SSS, SAS, ASA, AAS, and HL are sufficient shortcuts. SSA is not.
CPCTC · Rigid motions · Proofs
a²+b²
Pythagorean Theorem
In a right triangle, the square of the hypotenuse equals the sum of squares of the legs. The converse is also true: if a²+b²=c², the triangle is right.
30-60-90 · 45-45-90 · Distance formula
∼
Similarity
Two figures are similar if corresponding angles are congruent and corresponding sides are proportional. For triangles: AA, SAS, and SSS similarity are the shortcuts.
Scale factor · Proportional sides · Indirect measurement
○
Circle Relationships
Central angle = intercepted arc. Inscribed angle = half the intercepted arc. Tangent-radius angle = 90°. Every circle theorem connects an angle to an arc.
Arc length · Sector area · Circle equation
V
Volume vs. Surface Area
Surface area is the total area of all faces — measured in square units. Volume is the amount of space inside — measured in cubic units. They scale differently: double the linear dimensions, volume multiplies by 8.
Prisms · Pyramids · Spheres
03Capstone
Final Review and Proof Readiness

The capstone covers every unit with emphasis on reasoning and proof — not just computation. Students who complete this are ready for trigonometry, precalculus, and any standardized geometry assessment.

○
Gap Identification Session
You pick the two units that felt hardest. We review them together before the comprehensive problem set.
○
Comprehensive Problem Set
24 problems covering all 8 units, including proof problems. Completed independently.
○
Written Corrections
Every missed problem corrected with full reasoning — not just the right answer.
○
Next Track Readiness Review
A session mapping what you've built to what trigonometry and precalculus will require.
04Investment
Pricing Options
per Hour
$85
1-hr lesson
Flexible · No commitment
Online only
  • Flexible scheduling
  • 1-on-1 focused attention
  • No long-term commitment

Best for trying the curriculum before committing to a plan.

Book a session
Popular
per Month
$650
4 lessons · 1.5 hours/lesson
1 lesson/week · book more to move faster
Online + In-Person available
  • Session notes delivered after every lesson
  • Async DM support between sessions
  • Priority scheduling

Booking 2x per week draws down lessons faster — no extra charge.

Get started
Full Track
$3,120
24 lessons · 1.5 hours/lesson
~6 months · save vs. monthly
Online + In-Person available
  • Full course arc — all 8 units
  • Session notes after every lesson
  • Capstone review + written corrections
  • Next track readiness session

The complete Geometry arc in one commitment.

Enroll now
In-person sessions are available on Monthly and Full Track plans at $800/mo and $4,320 respectively. In-person is available within NYC. Per-hour sessions are online only.
05Contact
Let's Talk

Whether you're working through a geometry course right now or building toward something later, one session is enough to map out the path.

Do I need algebra before geometry? +
Yes — you'll be solving equations throughout this course. Comfortable with one and two-step equations is the minimum. If algebra feels shaky, the Algebra track is the right starting point.
Is proof-writing part of the course? +
Yes, but we build to it. The first units focus on reasoning and angle relationships. By Unit 2 we're writing formal congruence proofs. By the end you're writing two-column and paragraph proofs fluently.
Are lessons online or in-person? +
Per-hour sessions are online only. Monthly and Full Track plans include the option for in-person sessions in NYC. In-person is priced slightly higher — see pricing above.
What if I need to reschedule? +
Give 24 hours notice and we'll find another time. Monthly and Full Track students get priority rebooking.