Review and reference

Worksheets and ledgers are optional reference records. Write on paper or keep your own notes; these records do not save progress on this website.

On this page

The practice question numbers below are the stable Guide question numbers shown on each question, even when the bank shuffles their order.

How to interpret your result

Use the topic map below to find where you need another pass. The four-question topic totals are diagnostic clues, not reliable SAT score estimates. Rework misses without looking at the answer, then explain the setup in words. If the mistake was only arithmetic, still perform a dimensional or magnitude check.

Section to review	Practice questions
1. Lines and angles	1, 7, 13, 19
2. Triangles, congruence, and similarity	5, 11, 17, 23
3. Right triangles and trigonometry	3, 8, 14, 20
4. Circles and angle measure	6, 12, 18, 24
5. Coordinate geometry	4, 9, 15, 21
6. Perimeter, area, and volume	2, 10, 16, 22

What a strong correction looks like

Instead of “I forgot the formula,” write: “The given ratio described lengths, but I used it directly on areas. I must square the common length factor before scaling area.” A correction should name the relationship, the condition that permits it, and the quantity to which it applies.

Formula and decision reference

What the SAT reference sheet supplies

The published reference sheet includes the relationships summarized below.^[4] Know what each symbol represents and when each formula applies; having a formula available does not identify the correct radius, height, or target for you.

Relationship	Formula or ratio	Watch for
Circle	$A = πr 2,$ $C = 2πr$	Radius versus diameter
Rectangle	$A = lw$	Actual side lengths
Triangle	$\frac{1}{2}$	Perpendicular height
Right triangle	$a 2 + b 2 = c 2$	c is the hypotenuse
Special right triangles	$\sqrt{2}$ $\sqrt{3}$	Match the side to its opposite angle
Rectangular prism	$V = lwh$	Consistent length units
Cylinder	$V = πr 2 h$	Perpendicular height
Sphere	$\frac{4}{3}$	Radius is cubed
Cone	$\frac{1}{3}$	Not slant height
Rectangular-base pyramid	$\frac{1}{3}$	lw is base area
Angle totals	Full turn: $360° = 2π radians;$ triangle: 180°	Match angle units

A fast decision sequence

Start with the target. If it is a side length, look for a right triangle, similar figures, or a radius. If it is a ratio, look for cancellation before calculating actual lengths. If it is an area or volume, identify the dimensions and any missing perpendicular height.

Then use the distinctive condition. Parallel lines suggest equal angles and similarity. A tangent suggests a perpendicular radius. A diameter in an inscribed triangle suggests a right angle. A circle in expanded form suggests completing the square. Similarity suggests a single common length factor. Finish at the right level. A question asking for r² does not require r. A question asking for the coefficient of π does not require a decimal. A question asking for an area ratio may not require either area.

Do not invent a missing condition

A Pythagorean equation needs a right angle. A similarity proportion needs justified corresponding vertices. A tangent-radius right angle needs actual tangency. A scale factor of k³ for volume needs all corresponding lengths scaled by k. A formula is only as valid as the conditions supporting it.

Relationships to know or derive

These additional relationships are not listed on the reference sheet summarized in the formula reference above. The lessons explain their meanings, conditions, and derivations or geometric justification.

Topic	Relationship
Angles	Complementary: sum 90°. Supplementary: sum 180°. Vertical: equal. Parallel-line pairs: determine whether equal or supplementary.
Triangle sides	$\|a - b\| < c < a + b.$ Equal sides have equal opposite angles.
Similarity	Lengths and perimeters scale by k; areas by k²; volumes by k³; angles do not change.
Right-triangle ratios	$sin θ = o/h,$ $cos θ = a/h,$ $tan θ = o/a,$ where h here denotes the hypotenuse.
Complementary angles	$sin θ = cos(90° - θ).$
Trig identities	$sin 2 θ + cos 2 θ = 1;$ $tan θ = sin θ/cos θ$ where defined.
Degree/radian conversion	Multiply degrees by $π/180;$ multiply radians by $180/π.$
Arc and sector, degrees	$s = (θ/360)2πr;$ $A = (θ/360)πr 2 .$
Arc and sector, radians	$s = rθ;$ $\frac{1}{2}$ $Sector perimeter = 2r + s.$
Circle angle facts	$\frac{1}{2}$ Radius perpendicular to tangent. Opposite cyclic-quadrilateral angles supplement.
Unit circle	$(x, y) = (cos θ, sin θ);$ $tan θ = y/x$ when $x \neq 0.$
Distance and midpoint	$\sqrt{(Δ {x)}^{2} + (Δ {y)}^{2}}$ $midpoint = ((x 1 + x 2)/2, (y 1 + y 2)/2).$
Circle equation	$(x - h) 2 + (y - k) 2 = r 2;$ complete the square to recover this form.
Extra plane areas	Parallelogram: bh. Trapezoid: $\frac{1}{2}$ Ring: $π(R 2 - r 2).$
Closed surface areas	Box: $2(lw + l h + wh).$ Cylinder: $2πr 2 + 2πrh.$ Sphere: $4πr 2 .$ Cone: $πr 2 + πrs$ (slant height s).
Hemisphere	Volume: $\frac{2}{3}$ Curved area: $2πr 2 .$ Including circular base: $3πr 2 .$

When a result is impossible

A leg cannot exceed the hypotenuse. An ordinary triangle’s side cannot equal or exceed the sum of the other two. An acute-angle sine or cosine cannot exceed 1. A radius, area, or volume cannot be negative. A circle with standard-form right side $r 2 < 0$ has no real points. Use these checks to detect a wrong setup before chasing arithmetic.

Skills-to-examples map

This map groups worked examples by the relationship you need to practice. Each range links to its first example.

Section	Skill focus	Examples
Lines and angles	Complements, supplements, vertical angles, and parallel lines	1.01–1.07
	Hidden angle sums, sufficient conditions, and bisectors	1.08–1.11
	Exterior angles and polygon angle reasoning	1.12–1.15
Triangles	Angle ratios, equal sides, triangle inequality, correspondence	2.01–2.05
	AA/SAS similarity, proportions, shadows, and perimeters	2.06–2.11
	Area scaling, altitude similarity, congruence evidence, nested areas	2.12–2.15
Right triangles	Pythagorean theorem and special right triangles	3.01–3.06
	Trigonometric ratios, complementary angles, and similarity	3.07–3.11
	Elevation models, perimeter/area constraints, target expressions	3.12–3.15
Circles	Radius, arc length, sector area, radians, and perimeter	4.01–4.07
	Inscribed angles, diameters, tangents, chords, cyclic quadrilaterals	4.08–4.12
	Unit-circle signs, coterminal angles, and eliminating angle variables	4.13–4.15
Coordinates	Distance, midpoint, perpendicular lines, and circle construction	5.01–5.07
	Completing the square and transformations	5.08–5.10
	Tangency, intersections, parameters, and hidden centers	5.11–5.15
Measurement	Plane areas, perpendicular height, and composite boundaries	6.01–6.06
	Solid volumes, slant heights, and exposed surfaces	6.07–6.12
	Percent scaling, area-to-volume ratios, and conservation	6.13–6.15

A review system that turns mistakes into skills

Three passes through the material

Pass 1: Build the relationships. Study one concept block at a time, then attempt its worked examples with the solutions covered. Redraw the figures. Your immediate goal is to justify a setup, not to race through arithmetic.

Pass 2: Remove the cues. Revisit selected examples in a different order and complete the mixed set. Before calculating, name the relationship that connects the givens to the target. Circle any premise you are assuming rather than being given.

Pass 3: Repair and transfer. Re-solve every missed or guessed item from a blank page after a gap. Then vary one feature: change an angle’s unit, ask for a diameter instead of a radius, switch a length ratio to an area ratio, or remove a parallel-line condition. Explain whether the original method still works.

Diagnose the error precisely

Error type	What it looks like	Repair
Missing premise	Using a right-triangle rule without a right angle	Identify the exact statement or theorem that creates the right angle.
Wrong correspondence	Mixing a small side with a nonmatching large side	Write the vertex mapping before any proportion.
Wrong dimension	Scaling area by k or volume by k²	Label the quantity as length, area, or volume; then choose the exponent.
Wrong target	Reporting x, a radius, or an arc when a different quantity is asked	Write the requested quantity at the top of the scratch work.
Algebra or precision	Losing a sign, dropping a root, rounding early	Substitute back, keep exact forms, and check magnitude.

A short reusable error log

For each miss, record the problem number, your first incorrect step, the corrected relationship, and a date for a fresh attempt. A skill is ready for mixed practice when you can solve it without notes and explain why the method works. A memorized answer does not count as a fresh solution.

Final test-day check

Read the target again. Check the right angle or parallel-line condition. Confirm radius versus diameter and height versus slant height. Check degree or radian mode. Keep exact forms until the last step. Verify units and whether a percent change or a total percent is requested. Then enter only the requested answer.

Sources and editorial notes

Source check: September 10, 2026. Official sources were used to verify the exam scope, reference information, and test-day context. The examples, practice questions, explanations, and diagrams were created for this guide; they are not reproduced College Board test items.

[1] College Board. Geometry and Trigonometry.

Official overview of the domain’s four broad skill areas. Used to organize the coverage of this guide.

[2] College Board. The Math Section: Overview.

Official description of Math question formats, content categories, question counts, and practice resources.

[3] College Board. How the SAT Is Structured.

Official timing, module structure, and total question counts.

[4] College Board. Assessment Framework for the Digital SAT Suite, version 3.01, August 2024.

Appendix B specifies Geometry and Trigonometry skills. Appendix D gives Math directions and the reference sheet. The detailed framework supports the inclusion of surface area, unit-circle reasoning, and coordinate-circle equations.

[5] College Board. SAT Suite of Assessments Calculator Policy.

Current official guidance on embedded calculator options and restrictions on handheld calculators. Recheck this source close to your test date because policies can change.

Scope and interpretation

The six sections are a teaching organization, not six separately scored SAT domains. Difficulty labels and study suggestions are editorial choices. The mixed practice set is a learning assessment, not an adaptive test or a score-conversion instrument.

A few derived or supplementary relationships, such as polygon turns and chord/secant products, broaden recognition without requiring a separate advanced geometry course. The main emphasis remains on reasoning from standard angle facts, similarity, right triangles, circle geometry, coordinates, and measurement.

About the mathematical presentation

Illustrations are schematic unless explicitly presented as coordinate constructions. Given labels and stated conditions control the solutions; drawings are not intended as measurement tools. Exact values are retained when useful, and decimal approximations are identified. Formula availability is distinguished from the ability to select and use a formula correctly.

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