SAT Geometry Cheatsheet: Complete Formula & Theorem Reference Guide
Geometry accounts for 15% of SAT Math questions—and these are often the easiest points to secure. This comprehensive geometry cheatsheet from SATHELP24x7.com compiles every essential formula, theorem, and relationship you need to master for the Digital SAT. Unlike abstract algebra concepts, geometry formulas follow logical visual patterns that make them easier to remember and apply.
What makes this cheatsheet different: We've organized formulas by geometric shape and concept, included memory aids for complex relationships, clearly marked which formulas ARE and AREN'T on the SAT reference sheet, and highlighted high-frequency concepts that appear most often. Print this guide, study it systematically, and watch geometry questions transform from intimidating to routine.
KEY NOTATION: Formulas marked ✓ ARE provided on the SAT reference sheet but should be memorized for speed. Formulas marked ⚠️ are NOT provided and MUST be memorized. All angle relationships and geometric theorems must be memorized—none appear on the reference sheet.
Triangles
Basic Triangle Properties
⚠️ Triangle Angle Sum
\( A + B + C = 180° \)
Sum of all interior angles equals 180°
⚠️ Exterior Angle Theorem
Exterior angle = sum of two remote interior angles
The angle outside equals the sum of the two angles inside and opposite
⚠️ Triangle Inequality
Any two sides > third side
Example: If sides are 5 and 7, third side must be between 2 and 12
Triangle Area Formulas
✓ Basic Triangle Area
\( A = \frac{1}{2}bh \)
Provided on reference sheet; b = base, h = height
⚠️ Area Using Two Sides & Angle
\( A = \frac{1}{2}ab\sin C \)
a, b are sides; C is angle between them
⚠️ Heron's Formula
\( A = \sqrt{s(s-a)(s-b)(s-c)} \)
Where \( s = \frac{a+b+c}{2} \) (semi-perimeter)
Right Triangle Formulas
✓ Pythagorean Theorem
\( a^2 + b^2 = c^2 \)
Provided on reference sheet; a, b = legs, c = hypotenuse
⚠️ Finding a Leg
\( a = \sqrt{c^2 - b^2} \)
When you know hypotenuse and one leg
⚠️ Altitude to Hypotenuse
\( h^2 = xy \)
Altitude squared = product of segments it creates on hypotenuse
⚠️ Common Pythagorean Triples (Memorize These)
3-4-5 | 5-12-13 | 8-15-17 | 7-24-25
And all their multiples: 6-8-10, 9-12-15, 10-24-26, etc.
Pythagorean Theorem Quick Check: For any right triangle, the hypotenuse (longest side) must be LESS than the sum of the two legs. If you get c > a + b, you made a calculation error.
Special Right Triangles
✓ Critical Triangle Ratios (Memorize for Speed)
45-45-90 Triangle
\( x : x : x\sqrt{2} \)
Legs: Equal length
Hypotenuse: Leg × √2
Hypotenuse: Leg × √2
If leg = 5, hypotenuse = 5√2 ≈ 7.07
30-60-90 Triangle
\( x : x\sqrt{3} : 2x \)
Short leg (30°): x
Long leg (60°): x√3
Hypotenuse: 2x
Long leg (60°): x√3
Hypotenuse: 2x
If short leg = 3, then long = 3√3, hyp = 6
45-45-90 Memory: "Isosceles right triangle—legs equal, hypotenuse gets √2"
30-60-90 Memory: "1, 2, root 3" (short, hypotenuse, long) or "Half of equilateral triangle"
30-60-90 Memory: "1, 2, root 3" (short, hypotenuse, long) or "Half of equilateral triangle"
Triangle Similarity & Congruence
⚠️ Similar Triangle Properties
• Corresponding angles are equal
• Corresponding sides are proportional
• If \( \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \), triangles are similar
• Areas ratio = (side ratio)²
• Corresponding sides are proportional
• If \( \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \), triangles are similar
• Areas ratio = (side ratio)²
Similar vs. Congruent: Similar = same shape, different size (proportional). Congruent = same shape AND same size (identical).
Circles
Basic Circle Formulas
✓ Circle Area
\( A = \pi r^2 \)
Provided on reference sheet; r = radius
✓ Circumference
\( C = 2\pi r = \pi d \)
Provided on reference sheet; d = diameter = 2r
⚠️ Diameter from Radius
\( d = 2r \)
Diameter is twice the radius
Don't Mix These Up: Area uses r² (squared). Circumference uses r (not squared). Area is space inside; circumference is distance around.
Arcs and Sectors
⚠️ Arc Length
\( L = \frac{\theta}{360°} \times 2\pi r \)
θ = central angle in degrees; fraction of circumference
⚠️ Sector Area
\( A = \frac{\theta}{360°} \times \pi r^2 \)
"Pizza slice" area; fraction of total circle area
⚠️ Arc Length (Radians)
\( L = r\theta \)
When angle θ is in radians (rare on SAT)
Arc Length Quick Trick
If central angle is 90° (quarter circle): Arc = ¼ circumference
If central angle is 60° (⅙ circle): Arc = ⅙ circumference
Pattern: Arc fraction = Angle fraction
Circle Equations
⚠️ Circle Equation (Standard Form)
\( (x - h)^2 + (y - k)^2 = r^2 \)
Center at (h, k); radius = r
⚠️ Circle at Origin
\( x^2 + y^2 = r^2 \)
Special case where center is (0, 0)
⚠️ Finding Radius from Equation
\( r = \sqrt{\text{number on right side}} \)
Example: \( x^2 + y^2 = 25 \) → r = 5
Circle Theorems
⚠️ Essential Circle Properties
• Inscribed Angle: Half of central angle that subtends same arc
• Angle in Semicircle: Always 90° (inscribed in semicircle)
• Tangent ⊥ Radius: Tangent line perpendicular to radius at point of contact
• Chord Properties: Perpendicular from center bisects chord
• Angle in Semicircle: Always 90° (inscribed in semicircle)
• Tangent ⊥ Radius: Tangent line perpendicular to radius at point of contact
• Chord Properties: Perpendicular from center bisects chord
Quadrilaterals
Rectangle & Square
✓ Rectangle Area
\( A = lw \)
Provided on reference sheet; length × width
⚠️ Rectangle Perimeter
\( P = 2l + 2w \)
Sum of all four sides
⚠️ Square Area
\( A = s^2 \)
s = side length; all sides equal
⚠️ Square Perimeter
\( P = 4s \)
Four equal sides
⚠️ Square Diagonal
\( d = s\sqrt{2} \)
From 45-45-90 triangle property
⚠️ Rectangle Diagonal
\( d = \sqrt{l^2 + w^2} \)
Pythagorean theorem application
Parallelogram, Rhombus & Trapezoid
⚠️ Parallelogram Area
\( A = bh \)
Base × perpendicular height (not slant height)
⚠️ Rhombus Area (Method 1)
\( A = bh \)
Same as parallelogram
⚠️ Rhombus Area (Method 2)
\( A = \frac{1}{2}d_1 d_2 \)
Half product of diagonals
⚠️ Trapezoid Area
\( A = \frac{1}{2}(b_1 + b_2)h \)
Average of parallel bases × height
Trapezoid Memory: "Average the bases, multiply by height, divide by 2" or think: (base₁ + base₂)/2 gives you the average base, then multiply by height.
Quadrilateral Properties
⚠️ Quadrilateral Angle Sum
Sum of interior angles = 360°
Applies to all four-sided figures: rectangles, squares, parallelograms, trapezoids, rhombuses, and irregular quadrilaterals
Polygons
General Polygon Formulas
⚠️ Sum of Interior Angles
\( S = (n - 2) \times 180° \)
n = number of sides
⚠️ Each Interior Angle (Regular Polygon)
\( \text{Angle} = \frac{(n-2) \times 180°}{n} \)
All angles equal in regular polygon
⚠️ Sum of Exterior Angles
Always = 360°
True for ALL polygons regardless of number of sides
⚠️ Each Exterior Angle (Regular)
\( \text{Angle} = \frac{360°}{n} \)
n = number of sides
Quick Polygon Angle Reference
Triangle: 180° | Quadrilateral: 360°
Pentagon: 540° | Hexagon: 720°
Octagon: 1080°
Pentagon: 540° | Hexagon: 720°
Octagon: 1080°
3D Geometry (Solid Figures)
Volume Formulas
✓ Rectangular Solid (Box)
\( V = lwh \)
Provided on reference sheet; length × width × height
✓ Cube Volume
\( V = s^3 \)
Special case where l = w = h = s
✓ Cylinder Volume
\( V = \pi r^2 h \)
Provided on reference sheet; like stacked circles
✓ Sphere Volume
\( V = \frac{4}{3}\pi r^3 \)
Provided on reference sheet
✓ Cone Volume
\( V = \frac{1}{3}\pi r^2 h \)
Provided; ⅓ of cylinder with same base/height
✓ Pyramid Volume
\( V = \frac{1}{3}Bh \)
Provided; B = area of base, h = height
Volume Pattern: Cone and Pyramid both use ⅓ (remember: pointy tops get ⅓). Cylinder and Box don't have pointy tops—no ⅓ needed.
Surface Area Formulas
⚠️ Rectangular Solid Surface Area
\( SA = 2lw + 2lh + 2wh \)
Sum of all six faces
⚠️ Cube Surface Area
\( SA = 6s^2 \)
Six equal square faces
✓ Sphere Surface Area
\( SA = 4\pi r^2 \)
Provided on reference sheet
⚠️ Cylinder Surface Area
\( SA = 2\pi r^2 + 2\pi rh \)
Two circles + rectangle (unrolled side)
Coordinate Geometry
Distance and Midpoint
⚠️ Distance Formula
\( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)
Distance between points (x₁,y₁) and (x₂,y₂)
⚠️ Midpoint Formula
\( M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \)
Average the x-coordinates and y-coordinates
Distance Formula: It's just the Pythagorean theorem in disguise! The differences (x₂-x₁) and (y₂-y₁) are the legs of a right triangle, and distance is the hypotenuse.
Slope and Lines
⚠️ Slope
\( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} \)
Steepness of line between two points
⚠️ Parallel Lines
\( m_1 = m_2 \)
Same slope means parallel
⚠️ Perpendicular Lines
\( m_1 \cdot m_2 = -1 \)
Slopes are negative reciprocals
Angles
Angle Relationships
⚠️ Complementary Angles
Sum = 90°
Form a right angle together
⚠️ Supplementary Angles
Sum = 180°
Form a straight line together
⚠️ Vertical Angles
Equal
Opposite angles when two lines cross
⚠️ Linear Pair
Sum = 180°
Adjacent angles on a straight line
Parallel Lines Cut by Transversal
⚠️ Angle Relationships with Parallel Lines
• Corresponding Angles: Equal (same position at each intersection)
• Alternate Interior Angles: Equal (between parallel lines, opposite sides of transversal)
• Alternate Exterior Angles: Equal (outside parallel lines, opposite sides of transversal)
• Consecutive Interior Angles: Supplementary (sum = 180°)
• Alternate Interior Angles: Equal (between parallel lines, opposite sides of transversal)
• Alternate Exterior Angles: Equal (outside parallel lines, opposite sides of transversal)
• Consecutive Interior Angles: Supplementary (sum = 180°)
Parallel Lines Memory: "Corresponding = C-shape = Congruent" and "Alternate interior looks like a Z or backward Z, angles are equal"
Trigonometry (Right Triangle Trig)
SOHCAHTOA
⚠️ Sine
\( \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
SOH: Sine = Opposite/Hypotenuse
⚠️ Cosine
\( \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)
CAH: Cosine = Adjacent/Hypotenuse
⚠️ Tangent
\( \tan\theta = \frac{\text{opposite}}{\text{adjacent}} \)
TOA: Tangent = Opposite/Adjacent
SOHCAHTOA Memory: "Some Old Hippie Caught Another Hippie Trippin' On Acid" or "Silly Old Harry Caught A Herring Trawling Off America"
Important Trig Values
⚠️ Common Angle Values (Memorize)
sin 30° = ½ | cos 30° = √3/2 | tan 30° = 1/√3
sin 45° = √2/2 | cos 45° = √2/2 | tan 45° = 1
sin 60° = √3/2 | cos 60° = ½ | tan 60° = √3
sin 45° = √2/2 | cos 45° = √2/2 | tan 45° = 1
sin 60° = √3/2 | cos 60° = ½ | tan 60° = √3
⚠️ Pythagorean Identity
\( \sin^2\theta + \cos^2\theta = 1 \)
Fundamental trig identity
⚠️ Complementary Angles
\( \sin(90° - \theta) = \cos\theta \)
And vice versa: cos(90° - θ) = sin θ
Quick Problem-Solving Tips
Geometry Strategy Checklist
- Draw It: If no diagram provided, sketch one immediately
- Label Everything: Mark all given measurements on your diagram
- Look for Right Triangles: They unlock Pythagorean theorem and trig
- Check for Special Triangles: 45-45-90 and 30-60-90 save time
- Identify Shape Properties: Use what you know about that specific shape
- Work Backwards: Sometimes easier to plug in answer choices
- Use Desmos: Graph coordinate geometry problems to visualize
- Check Reasonableness: Does your answer make visual sense?
⚠️ Common Geometry Mistakes to Avoid
- Confusing area formulas with perimeter/circumference
- Using diameter instead of radius (or vice versa)
- Forgetting to square the radius in circle area
- Mixing up leg and hypotenuse in right triangles
- Not checking if diagram is drawn to scale
- Assuming angles look like 90° when not marked
- Forgetting to multiply by ½ in triangle and trapezoid areas
Transform Geometry Knowledge Into Perfect Scores
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