"Geometry curriculum" is searched by parents whose child has just been placed in the course, by students who want to know what is coming, and by families arriving from another system who need to know where their child stands. The course is the odd one out in US high school math: it sits between Algebra 1 and Algebra 2, it introduces proof, and it is the one math course whose content barely changed in a century. This page sets out the units in the order most schools teach them, what each one assumes, how a standard year is paced, which states test it, where students lose marks, how it maps to CBSE Class 9 and 10, and a readiness check on the algebra the course uses every day. The printable checklist puts the thirteen units and their skills on one sheet.
Download the geometry curriculum checklist (PDF) (the thirteen units with the skills in each as tick boxes, the month-by-month pace, the state tests, and the ten-question readiness check with answers; two pages, free).
The thirteen units, in order
The Common Core geometry standards (HSG: congruence, similarity and trigonometry, circles, expressing geometric properties with equations, geometric measurement and dimension, modeling, plus conditional probability) set what is taught. The order below is the one most textbooks and districts follow; integrated-math states spread the same units across Integrated Math 1 to 3.
| Unit | What it covers | What it assumes | Weeks |
|---|---|---|---|
| 1. Foundations | Points, lines, planes; segments, midpoints and distance; angles and angle pairs (complementary, supplementary, vertical, linear pairs); basic constructions with compass and straightedge | Arithmetic; the coordinate plane; solving a linear equation | 2 to 3 |
| 2. Reasoning and proof | Inductive and deductive reasoning; conditional statements, converse, inverse, contrapositive; the algebraic properties as reasons; two-column, paragraph and flow proofs | Unit 1; reading carefully | 2 |
| 3. Parallel and perpendicular lines | Transversals and the angle pairs they make (corresponding, alternate interior, same-side interior); proving lines parallel; slopes of parallel and perpendicular lines; equations of lines | Units 1 and 2; slope and the equation of a line from Algebra 1 | 3 |
| 4. Transformations | Translations, reflections, rotations and their compositions; rigid motions and congruence; symmetry; dilations as the first non-rigid motion | The coordinate plane; function notation helps | 2 to 3 |
| 5. Congruent triangles | Triangle angle sum and exterior angles; SSS, SAS, ASA, AAS and HL; corresponding parts of congruent triangles (CPCTC); isosceles and equilateral triangle theorems; coordinate proofs | Unit 2 (this is where proof is really learned) | 3 to 4 |
| 6. Relationships in triangles | Perpendicular and angle bisectors; medians and altitudes; circumcenter, incenter, centroid, orthocenter; the triangle inequality; indirect proof | Unit 5 | 2 to 3 |
| 7. Quadrilaterals and polygons | Interior and exterior angle sums of polygons; parallelograms and their tests; rectangles, rhombi and squares; trapezoids and kites; coordinate proofs with quadrilaterals | Units 3 and 5; solving systems for two unknown angles | 3 |
| 8. Similarity | Ratios and proportions; similar polygons and scale factor; AA, SSS and SAS similarity; proportional parts (the side-splitter theorem); dilations revisited; indirect measurement | Unit 5; proportions from Grade 7 | 3 |
| 9. Right triangles and trigonometry | The Pythagorean theorem and its converse; simplifying radicals; special right triangles (45-45-90 and 30-60-90); sine, cosine and tangent; angles of elevation and depression; in many schools the laws of sines and cosines | Unit 8; radicals and solving proportions from Algebra 1 | 3 to 4 |
| 10. Circles | Tangents; arcs, chords and central angles; inscribed angles; angles formed by chords, secants and tangents; segment lengths; arc length and sectors; the equation of a circle | Units 5 and 8; completing the square for the circle equation | 3 |
| 11. Area and perimeter | Triangles, parallelograms, trapezoids, rhombi and kites; regular polygons and the apothem; circles and sectors; composite figures; the effect of changing dimensions | Unit 9 for the apothem and sectors | 2 |
| 12. Surface area and volume | Prisms, cylinders, pyramids, cones and spheres; cross-sections and solids of rotation; Cavalieri's principle; density and modeling problems | Unit 11 | 3 |
| 13. Probability | Sample spaces; independent and dependent events; conditional probability and two-way tables; permutations and combinations in some schools | Fractions and percentages | 1 to 2 |
Thirty-two to thirty-six weeks of content for a thirty-six-week year, which is why probability is the unit most often cut and why the SAT's geometry questions (about 15 percent of the math section, and the course that most students have finished by the time they sit it) draw on units 9 to 12.
How the year is paced
| Months | Units | Note |
|---|---|---|
| August to October | 1 to 3 | The vocabulary and the proof format; the first test on proofs is where grades drop |
| November to January | 4 to 6 | Congruent triangles is the heart of the course; the winter break lands in the middle of it |
| February to March | 7 to 9 | Similarity and right-triangle trigonometry: the most SAT-relevant stretch |
| April to May | 10 to 12 | Circles, area, volume; the state test or final sits at the end of this |
| Late May | 13 | Probability, if there is time |
Honors geometry covers the same units with more proof, more constructions and the laws of sines and cosines; it does not add units.
The state tests
| State | Test | What rides on it |
|---|---|---|
| New York | Regents Examination in Geometry | One of the Regents math exams that count toward the Regents diploma (a student needs one; Algebra I is the common one, and Geometry is taken for the Advanced Regents diploma) |
| Florida | Geometry End-of-Course (B.E.S.T.) | 30 percent of the course grade |
| Virginia | Geometry Standards of Learning (SOL) | Verified credit toward the diploma |
| Tennessee | TCAP Geometry end-of-course | Counts in the course grade (the percentage is set by the district) |
| Ohio | Geometry (or Integrated Math II) state test | Part of the graduation competency points |
| Georgia | Georgia Milestones Geometry: Concepts and Connections | 20 percent of the course grade |
| Texas, California, most others | No geometry test | Texas tests Algebra I and not Geometry; the SAT geometry questions are the external test that matters |
Released Regents and Florida exams are the best practice for any state, because the question types (an angle to solve for, a proof to complete, a volume in context) are the same everywhere.
The ten places students lose marks
- Not knowing what counts as a reason in a proof. Every statement needs a definition, a postulate, a theorem or a property; "it looks equal" is not one.
- Confusing congruence with similarity. Congruent is same shape and size; similar is same shape, and the ratio does the rest.
- The special right triangles. Mixing the 45-45-90 ratio (1 : 1 : √2) with the 30-60-90 ratio (1 : √3 : 2), or putting √3 on the wrong side.
- Choosing the trigonometric ratio. Sine, cosine or tangent depends on which sides are known relative to the angle; label the sides before choosing.
- The circle angle rules. An inscribed angle is half its arc; a central angle equals its arc; angles formed outside the circle are half the difference of the arcs.
- Dropping the converse. "If the lines are parallel, the alternate interior angles are equal" and its converse are different statements, used for different jobs.
- Algebra inside geometry. Setting up 3x + 15 = 2x + 40 for two angles and then solving it wrong; most geometry marks lost are algebra marks.
- Units in area and volume. Square units for area, cubic for volume, and converting them (1 ft² is 144 in², 1 ft³ is 1,728 in³).
- Radicals left unsimplified or decimalised. √50 should become 5√2; the exam often wants the exact form.
- Reading the diagram as drawn to scale. Unless the question says so, a diagram is a sketch, and only the marked information can be used.
Geometry against CBSE Class 9 and 10, for families from India
| US geometry unit | Where CBSE teaches it |
|---|---|
| Foundations; parallel lines | Class 9 Lines and Angles |
| Congruent triangles; relationships in triangles | Class 9 Triangles (congruence, inequalities) |
| Quadrilaterals | Class 9 Quadrilaterals |
| Similarity | Class 10 Triangles (similarity, BPT) |
| Right triangles and trigonometry | Class 10 Introduction to Trigonometry and Some Applications of Trigonometry |
| Circles | Class 9 Circles (chords, inscribed angles) and Class 10 Circles (tangents) |
| Area, surface area and volume | Class 9 Heron's Formula and Surface Areas and Volumes; Class 10 Areas Related to Circles and Surface Areas and Volumes |
| Coordinate geometry inside each unit | Class 9 and 10 Coordinate Geometry (distance, section formula) |
| Transformations; two-column proofs; probability | Not in CBSE in this form: constructions and theorem proofs are in CBSE, but not rigid motions, dilations or the two-column format; probability is a separate chapter |
A student who has completed CBSE Class 10 is placed in US geometry with most of the content already known and the proof format entirely new; the first two months are spent learning to write what they already understand. A student arriving after Class 9 is well placed for geometry in 9th grade and ahead on constructions. Ask the school for the placement test rather than the default; students from India are routinely placed a course low.
A ten-question readiness check
The algebra geometry uses every day. No calculator; fifteen minutes.
- Solve 3x + 15 = 2x + 40.
- Two angles are supplementary; one is 4x and the other is x + 30. Find x.
- Simplify √50.
- Simplify √12 × √3.
- Find the distance between (1, 2) and (7, 10).
- Find the midpoint of (−3, 4) and (5, −2).
- Find the slope of the line through (2, 5) and (6, 13), and the slope of a line perpendicular to it.
- Solve the proportion 4/x = 10/15.
- If x² = 49, what is x?
- Solve the system: x + y = 90 and x − y = 20.
Answers: 1) x = 25. 2) 5x + 30 = 180, x = 30. 3) 5√2. 4) 6. 5) 10. 6) (1, 1). 7) 2 and −½. 8) x = 6. 9) 7 or −7 (in geometry, the positive one). 10) x = 55, y = 35.
Nine or ten right: the algebra will not get in the way of the geometry. Six to eight: a fortnight on radicals, proportions and the distance formula before unit 5. Five or fewer: the gap is in Algebra 1, and why am I bad at algebra shows how to find it.
Download the geometry curriculum checklist (PDF)
Where to get help
A 1-on-1 online class with a tutor who holds a master's in mathematics and teaches the US geometry course in the school's order: proofs written together on the shared whiteboard from unit 2, the algebra gaps closed in the first month, the state exam's released questions from February. Before school (7 am Eastern is 4:30 pm for the tutor), after school or weekends, US time; the recording and notes after every class; the same tutor through Algebra 2. US$10 to $13 an hour, billed monthly. The geometry page has the topics; the free 30-minute class is a real lesson on this week's unit.
Questions parents ask
1What is in the high school geometry curriculum?
2What grade is geometry taught in?
3Is there a state test for geometry?
4Is geometry harder than Algebra 1?
5How does US geometry compare with CBSE?
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