SAT Geometry Formulas & Short Notes 2025: Complete Guide

Complete SAT geometry formula reference with short notes. All circle, triangle, 3D figure, coordinate geometry, and trigonometry formulas properly explained.

SAT Geometry Short Notes & Formulas 2025

Master SAT geometry with this comprehensive formula reference and quick notes guide. Geometry comprises approximately 15% of the SAT Math section (5-7 questions), making it essential to memorize key formulas and understand fundamental geometric principles. This complete reference covers all geometry concepts tested on the digital SAT, including circles, triangles, rectangles, 3D figures, coordinate geometry, and trigonometry—with every formula properly formatted and explained for quick review before test day.

Important Note: The SAT provides 12 geometry formulas in the reference section at the beginning of each math module. However, memorizing these formulas saves valuable time during the test and ensures accuracy. This guide includes both the provided formulas and additional geometry concepts you must know that are NOT given on the test.

Formulas Provided on the SAT

These formulas appear in the digital SAT's built-in reference sheet. While you can access them during the test, memorizing them improves speed and reduces errors.

Circle Formulas

Area of a Circle
\[ A = \pi r^2 \]
Where: \(A\) = area, \(r\) = radius (distance from center to edge)
Note: Use \(\pi \approx 3.14\) or leave answer in terms of \(\pi\)
Circumference of a Circle
\[ C = 2\pi r \quad \text{or} \quad C = \pi d \]
Where: \(C\) = circumference (distance around), \(r\) = radius, \(d\) = diameter
Note: Diameter = 2 × radius, so \(d = 2r\)

Triangle & Rectangle Formulas

Area of a Rectangle
\[ A = lw \]
Where: \(A\) = area, \(l\) = length, \(w\) = width
Note: Also works for squares where \(l = w\)
Area of a Triangle
\[ A = \frac{1}{2}bh \]
Where: \(A\) = area, \(b\) = base, \(h\) = height (perpendicular to base)
Note: Works for ALL triangles, not just right triangles
Pythagorean Theorem
\[ a^2 + b^2 = c^2 \]
Where: \(a\) and \(b\) = legs of right triangle, \(c\) = hypotenuse (longest side)
Note: Only applies to RIGHT triangles (one 90° angle)

Special Right Triangles

45-45-90 Triangle
\[ \text{Side ratios: } x : x : x\sqrt{2} \]
Pattern: Two legs equal (\(x\)), hypotenuse = leg × \(\sqrt{2}\)
Example: If legs = 5, then hypotenuse = \(5\sqrt{2}\)
30-60-90 Triangle
\[ \text{Side ratios: } x : x\sqrt{3} : 2x \]
Pattern: Short leg (\(x\)), long leg = short leg × \(\sqrt{3}\), hypotenuse = short leg × 2
Example: If short leg = 4, then long leg = \(4\sqrt{3}\), hypotenuse = 8

3D Figure Volume Formulas

Volume of a Rectangular Prism (Box)
\[ V = lwh \]
Where: \(V\) = volume, \(l\) = length, \(w\) = width, \(h\) = height
Volume of a Cylinder
\[ V = \pi r^2 h \]
Where: \(V\) = volume, \(r\) = radius of circular base, \(h\) = height
Think: Area of circle × height
Volume of a Sphere
\[ V = \frac{4}{3}\pi r^3 \]
Where: \(V\) = volume, \(r\) = radius
Memorize: "Four-thirds pi r cubed"
Volume of a Cone
\[ V = \frac{1}{3}\pi r^2 h \]
Where: \(V\) = volume, \(r\) = radius of circular base, \(h\) = height
Think: One-third of cylinder volume
Volume of a Pyramid
\[ V = \frac{1}{3}lwh \]
Where: \(V\) = volume, \(l\) = length of base, \(w\) = width of base, \(h\) = height
Think: One-third of rectangular prism volume

Master SAT Geometry with Expert Instruction

Understanding formulas is just the first step—applying them correctly under timed conditions requires practice and strategy. SATHELP24x7 provides comprehensive geometry instruction ensuring you master every concept tested on the SAT Math section.

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Essential Formulas NOT Provided on SAT

These critical formulas and concepts do NOT appear in the SAT reference sheet. You MUST memorize them for test success.

Additional Triangle Formulas

Triangle Angle Sum
\[ \text{Sum of interior angles} = 180° \]
Rule: All three angles in any triangle always add to 180°
Example: If two angles are 60° and 80°, third angle = 180° - 60° - 80° = 40°
Triangle Inequality Theorem
\[ a + b > c \]
Rule: Sum of any two sides must be greater than the third side
Example: Can sides 3, 4, 8 form a triangle? No, because 3 + 4 = 7 < 8
Area of Trapezoid
\[ A = \frac{1}{2}(b_1 + b_2)h \]
Where: \(b_1\) and \(b_2\) = lengths of parallel bases, \(h\) = height
Think: Average of bases × height
Isosceles Triangle Property
\[ \text{If two sides equal, opposite angles equal} \]
Rule: In isosceles triangle, base angles are congruent
Example: If two sides are equal and vertex angle = 40°, base angles = 70° each

Circle Advanced Concepts

Arc Length
\[ \text{Arc Length} = \frac{\theta}{360°} \times 2\pi r \]
Where: \(\theta\) = central angle in degrees, \(r\) = radius
Think: Fraction of full circumference
Sector Area
\[ \text{Sector Area} = \frac{\theta}{360°} \times \pi r^2 \]
Where: \(\theta\) = central angle in degrees, \(r\) = radius
Think: Fraction of full circle area (like pizza slice)
Circle Equation (Standard Form)
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Where: Center = \((h, k)\), radius = \(r\)
Example: \((x - 3)^2 + (y + 2)^2 = 25\) has center (3, -2) and radius 5

Coordinate Geometry

Distance Formula
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Purpose: Find distance between two points \((x_1, y_1)\) and \((x_2, y_2)\)
Think: Pythagorean theorem on coordinate plane
Midpoint Formula
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Purpose: Find midpoint between two points
Think: Average of x-coordinates, average of y-coordinates
Slope Formula
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} \]
Purpose: Find slope of line through two points
Remember: Vertical lines have undefined slope, horizontal lines have slope = 0

Trigonometry Basics

SOH-CAH-TOA
\[ \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}} \]
Mnemonic: Some Old Hippie Caught Another Hippie Tripping On Acid
Only for: RIGHT triangles
Complementary Angle Relationships
\[ \sin\theta = \cos(90° - \theta) \quad \cos\theta = \sin(90° - \theta) \]
Rule: Sine of an angle equals cosine of its complement
Example: \(\sin 30° = \cos 60°\)
Key Trigonometric Values
\[ \sin 30° = \frac{1}{2} \quad \cos 30° = \frac{\sqrt{3}}{2} \quad \sin 45° = \frac{\sqrt{2}}{2} \quad \cos 45° = \frac{\sqrt{2}}{2} \quad \sin 60° = \frac{\sqrt{3}}{2} \quad \cos 60° = \frac{1}{2} \]
Memorize: These values appear frequently on SAT
Derive from: 30-60-90 and 45-45-90 special triangles

Additional Surface Area & Volume

Surface Area of Sphere
\[ SA = 4\pi r^2 \]
Where: \(SA\) = surface area, \(r\) = radius
Think: Four times the area of great circle
Surface Area of Cylinder
\[ SA = 2\pi r^2 + 2\pi rh \]
Where: First term = two circular ends, second term = curved side
Think: Top + bottom + wrapped rectangle
Volume of Right Prism
\[ V = Bh \]
Where: \(B\) = area of base, \(h\) = height
General rule: Works for ANY prism shape

Quick Geometry Tips for SAT Success

  • Draw diagrams: Always sketch what the problem describes, even if figure provided
  • Label everything: Mark all given information on your diagram immediately
  • Look for special triangles: Many problems involve 30-60-90 or 45-45-90 patterns
  • Use Pythagorean theorem liberally: When you see right triangle, check if you can apply \(a^2 + b^2 = c^2\)
  • Remember angle relationships: Vertical angles equal, supplementary angles = 180°, complementary = 90°
  • Check answer reasonableness: Does your calculated area/volume make sense given the dimensions?

Important Geometric Principles

Angle Relationships

  • Vertical Angles: Opposite angles formed by intersecting lines are equal
  • Supplementary Angles: Two angles that sum to 180° (straight line)
  • Complementary Angles: Two angles that sum to 90° (right angle)
  • Linear Pair: Adjacent angles on straight line sum to 180°
  • Exterior Angle Theorem: Exterior angle of triangle equals sum of two non-adjacent interior angles

Parallel Lines Cut by Transversal

  • Corresponding Angles: Equal (same position on different parallel lines)
  • Alternate Interior Angles: Equal (inside parallel lines, opposite sides of transversal)
  • Alternate Exterior Angles: Equal (outside parallel lines, opposite sides of transversal)
  • Consecutive Interior Angles: Supplementary (sum to 180°)

Similar Triangles

  • Definition: Triangles with same shape but different size
  • Property: Corresponding angles equal, corresponding sides proportional
  • Ratio: If sides in ratio 2:3, then areas in ratio 4:9 (ratio squared)
  • Application: Used for indirect measurement and shadow problems

Common Geometry Mistakes to Avoid

  • Confusing radius and diameter: Remember diameter = 2 × radius
  • Using wrong height: Height must be perpendicular to base
  • Forgetting to square radius: Area formulas use \(r^2\), not just \(r\)
  • Applying Pythagorean to non-right triangles: Only works with 90° angle
  • Mixing up special triangle ratios: Memorize both 30-60-90 and 45-45-90
  • Not converting units: Ensure all measurements in same units before calculating

Complete Your SAT Math Preparation

Geometry is just one part of SAT Math success. For comprehensive formula references covering all SAT Math topics, explore our complete resources:

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Practice Application Strategy

Understanding formulas is essential, but applying them correctly under timed conditions requires systematic practice. Follow this strategic approach to master SAT geometry:

5-Step Geometry Problem-Solving Method

  1. Read Carefully: Identify what the question asks—area, volume, perimeter, angle measure?
  2. Draw and Label: Create clear diagram with all given information marked
  3. Identify Formula: Determine which formula(s) apply to this specific situation
  4. Substitute and Solve: Plug values into formula, simplify carefully
  5. Check Reasonableness: Does answer make sense given context?

When to Use Calculator

The digital SAT allows calculator use throughout Math section. Use strategically for:

  • Square roots: Calculate \(\sqrt{2}\), \(\sqrt{3}\) when decimal answer needed
  • Pi calculations: Use calculator's π button for accuracy
  • Complex arithmetic: Multi-step calculations with decimals or fractions
  • Graphing functions: Bluebook has graphing calculator for visualization

Don't over-rely: Simple calculations often faster by hand. Practice both methods.

Test Day Geometry Checklist

Before SAT Test Day - Ensure You Know:

  • ✓ All circle formulas (area, circumference, arc length, sector area)
  • ✓ Triangle basics (angle sum, Pythagorean theorem, area formula)
  • ✓ Special right triangle ratios (30-60-90 and 45-45-90)
  • ✓ Rectangle and parallelogram area formulas
  • ✓ All 3D volume formulas (prism, cylinder, sphere, cone, pyramid)
  • ✓ Coordinate geometry (distance, midpoint, slope)
  • ✓ Basic trigonometry (SOH-CAH-TOA)
  • ✓ Circle equations in coordinate plane
  • ✓ Angle relationships (vertical, supplementary, complementary)

Quick Review Resources

One week before your SAT, review these essential resources:

Beyond Geometry: Complete SAT Preparation

While geometry comprises 15% of SAT Math, comprehensive preparation requires mastering all mathematical domains. Geometry connects closely with algebra, particularly in coordinate geometry questions combining both disciplines. Ensure your overall math preparation is complete by reviewing algebra, advanced math, and problem-solving topics alongside geometry.

Additional SAT Resources

Master SAT Math with Expert Instruction

Memorizing formulas is essential, but true mastery comes from understanding when and how to apply them under timed conditions. SATHELP24x7 provides comprehensive SAT Math preparation covering all geometry concepts plus algebra, advanced math, and problem-solving strategies that maximize your score.

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