SAT Math: Geometry & Trigonometry — Complete Study Guide

Important Note: Reference Sheet

The Digital SAT gives you a math reference sheet in the testing app. You should still memorize these formulas because speed matters, but it helps to know the formulas are available during the test.

These are the key formulas from the official reference sheet:

• Area of circle: $A = \pi r^2$
• Circumference: $C = 2\pi r$
• Area of rectangle: $A = lw$
• Area of triangle: $A = \frac{1}{2}bh$
• Pythagorean theorem: $a^2 + b^2 = c^2$
• Special right triangles: $30$-$60$-$90$ has sides $(x,\ x\sqrt{3},\ 2x)$ and $45$-$45$-$90$ has sides $(x,\ x,\ x\sqrt{2})$
• Volume of rectangular prism: $V = lwh$
• Volume of cylinder: $V = \pi r^2h$
• Volume of sphere: $V = \frac{4}{3}\pi r^3$
• Volume of cone: $V = \frac{1}{3}\pi r^2h$
• Volume of pyramid: $V = \frac{1}{3}lwh$

Geometry and trigonometry usually contribute about 5 to 7 questions, but these are high-leverage points because many students lose easy points to diagram mistakes, unit confusion, and rushed angle logic. This guide focuses on the exact question patterns the SAT repeats.

Strategy tip: For every geometry question, write the target quantity first (area, angle, side length, volume, or expression). That one habit prevents many wrong turns.

1. Area and Volume

Geometry questions often mix basic formulas with context. You may need to break a shape into parts, convert units, or interpret a word problem that hides a familiar formula.

2D area calculations

Common 2D area formulas on SAT:

• Rectangle: $A = lw$
• Triangle: $A = \frac{1}{2}bh$
• Circle: $A = \pi r^2$
• Trapezoid: $$A = \frac{1}{2}(b_1+b_2)h$$

For composite figures, split into simple pieces (rectangles, triangles, semicircles), calculate each area, then add or subtract.

3D volume calculations

Most SAT volume questions use direct substitution:

• Rectangular prism: $V = lwh$
• Cylinder: $V = \pi r^2h$
• Cone: $V = \frac{1}{3}\pi r^2h$
• Sphere: $V = \frac{4}{3}\pi r^3$
• Pyramid: $V = \frac{1}{3}lwh$ (for rectangular base dimensions $l$ and $w$)

Watch whether the problem gives radius or diameter.

Surface area and density

Surface area sometimes appears as context (paint needed, wrapping material, exposed faces). For rectangular prisms:

For cylinders:

Density model:

So you can also use:

Worked Example 1: Composite area

Worked Example 2: Volume with context

Worked Example 3: Density

Strategy tip: In geometry contexts, always include units in intermediate steps. Unit tracking catches many setup errors before they cost points.

2. Lines, Angles, and Polygons

Angle relationships are a frequent SAT geometry target because they test structure recognition more than calculation.

Core angle relationships

• Complementary angles sum to $90^\circ$.
• Supplementary angles sum to $180^\circ$.
• Vertical angles are equal.

When parallel lines are cut by a transversal:

• Corresponding angles are equal.
• Alternate interior angles are equal.
• Same-side interior (co-interior) angles are supplementary.

Polygon angle sums

Interior angle sum of an $n$-gon:

For regular polygons (all equal angles), each interior angle is:

Exterior angle theorem (triangle)

An exterior angle of a triangle equals the sum of the two remote interior angles.

If exterior angle is $E$ and remote interior angles are $a$ and $b$:

Worked Example 4: Parallel lines and transversal

Worked Example 5: Polygon angles

Strategy tip: If you forget a polygon formula, triangulate from one vertex. You can see why an $n$-gon splits into $(n-2)$ triangles.

3. Triangles

Triangles are central in SAT geometry. Most questions use one of a small number of repeatable tools.

Triangle inequality theorem

For side lengths $a$, $b$, and $c$:

Equivalent shortcut: longest side must be less than sum of other two.

Pythagorean theorem and converse

For right triangle with legs $a,b$ and hypotenuse $c$:

Converse: if side lengths satisfy this equation, the triangle is right.

Special right triangles

• $45$-$45$-$90$: sides $x, x, x\sqrt{2}$
• $30$-$60$-$90$: sides $x, x\sqrt{3}, 2x$ (short leg, long leg, hypotenuse)

Similar triangles

Triangles are similar if corresponding angles are equal, and side lengths are proportional.

Common similarity criteria:

• AA similarity
• SAS similarity
• SSS similarity

Proportion setup example:

Congruent triangles

Congruence means same shape and same size (all corresponding sides and angles equal). SAT may test recognition in diagrams.

Triangle area beyond base-height

Standard formula:

Some contexts require finding height from other given values first.

Worked Example 6: Pythagorean theorem application

Worked Example 7: Special right triangle

Worked Example 8: Similar triangle proportions

Strategy tip: For similarity questions, choose one clear corresponding pair first to get the scale factor. Then apply that same factor consistently.

4. Right Triangle Trigonometry

SAT trigonometry is practical and focused. You mainly need SOH-CAH-TOA with degree mode.

SOH-CAH-TOA definitions

For an acute angle $\theta$ in a right triangle:

Complementary angle relationship

In right triangles, acute angles are complementary:

and similarly:

Use cases

• Find missing side given angle and one side.
• Find angle given side ratios.
• Model real contexts (height, slope, distance, elevation).

Angles of elevation and depression

• Elevation: angle measured upward from horizontal.
• Depression: angle measured downward from horizontal.

SAT usually gives enough info to form a right triangle and apply one trig ratio.

Degree mode note

The SAT uses degrees, not radians, for trig in standard questions. Ensure calculator is in degree mode.

Worked Example 9: Find a side using trig

Worked Example 10: Angle of elevation

Worked Example 11: Complementary angle relationship

Strategy tip: Many SAT trig questions become one-step problems if you choose the ratio that uses known and target sides immediately.

5. Circles

Circle questions combine algebra and geometry and appear in both modules.

Circle equation

Standard form of a circle:

• Center: $(h,k)$
• Radius: $r$

Converting from general form

General quadratic form in $x$ and $y$:

Convert by grouping $x$ and $y$ terms and completing the square.

Arc length and sector area

For central angle $\theta$ in degrees:

Central and inscribed angles

• Central angle intercepts arc with same measure as arc.
• Inscribed angle intercepting same arc is half the central angle.

Tangent lines

A tangent line is perpendicular to radius at point of tangency.

If radius to tangent point forms one side and tangent line forms another, the angle is $90^\circ$.

Radian basics

SAT is mostly degree-based, but conversion may appear:

So:

Worked Example 12: Circle equation from conditions

Worked Example 13: Sector area in context

Worked Example 14: Tangent line problem

Strategy tip: In tangent problems, explicitly mark the right angle at tangency. That mark often unlocks the whole question.

Quick Reference

Desmos Tips (Graphing Circles)

Desmos can save time on coordinate-geometry circle questions.

1. To graph a circle directly, type standard form:

   $$(x-h)^2 + (y-k)^2 = r^2$$

2. To inspect center and radius quickly:

   • Rewrite equation into standard form first.
   • Then graph it and verify geometry visually.

3. For intersection problems (line and circle):

   • Enter both equations.
   • Click intersection points to read coordinates.

4. For equation-matching questions:

   • Graph answer choices one by one and compare center/intercepts/size.

Digital SAT Tips

1. Geometry and trig are usually fewer questions, but these are highly score-efficient if mastered.
2. Draw your own clean sketch when diagram is crowded.
3. Label known values directly on the diagram.
4. Check whether answers need exact form (like $15\pi$) or decimal approximation.
5. For trig problems, confirm degree mode on calculator.
6. For circle equations, extract center and radius before doing anything else.
7. In word problems, convert units before formula substitution.
8. When possible, eliminate answer choices by estimation before full computation.

Strategy tip: If the question asks for an expression (for example, $x+y$), do not stop at finding just one variable.

Practice Problems

Problem 1 (composite area)

A shape is made of a $12\times9$ rectangle with a $4\times3$ rectangle removed from one corner. What is its area?

A) $84$
B) $90$
C) $96$
D) $108$

Problem 2 (parallel lines)

Two parallel lines are cut by a transversal. Corresponding angles are labeled $(3x+12)^\circ$ and $(5x-18)^\circ$. What is $x$?

A) $9$
B) $12$
C) $15$
D) $18$

Problem 3 (special triangle)

In a $45$-$45$-$90$ triangle, the hypotenuse is $10\sqrt2$. What is each leg?

A) $5$
B) $10$
C) $10\sqrt2$
D) $20$

Problem 4 (trig)

In a right triangle, angle $\theta=50^\circ$ and adjacent side to $\theta$ is 7. Which expression gives the hypotenuse $h$?

A) $h=7\sin50^\circ$
B) $h=7\cos50^\circ$
C) $h=\frac{7}{\cos50^\circ}$
D) $h=\frac{7}{\tan50^\circ}$

Problem 5 (circle sector)

A circle has radius 12 cm. What is the area of a $60^\circ$ sector?

A) $12\pi$
B) $24\pi$
C) $36\pi$
D) $48\pi$

Geometry and trigonometry on the SAT are predictable when you keep diagrams organized, formulas deliberate, and units consistent. Aim to turn these 5-7 questions into reliable points by practicing setup and interpretation, not just arithmetic.