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13.3 Geometric Properties, Two-Dimensional Polygons, and Three-Dimensional Solids
California Subject Examinations for Teachers CSET
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1 Chapter 1: The CSET Program and the Multiple Subjects Examination
1.1 The CSET Program: Which Tests a California Teacher Actually Needs 1.2 CSET Multiple Subjects: Test Structure, Scoring, and Test-Day Mechanics
2 Chapter 2: Language Structure, Linguistics, and Non-Written Communication
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3 Chapter 3: Literacy Development, Reading Comprehension, and Text Complexity
3.1 Foundations of Early Literacy: Print Concepts, Phonemic Awareness, and Phonics 3.2 Reading Fluency and Vocabulary Acquisition 3.3 Reading Comprehension of Literary and Informational Texts 3.4 Text Complexity and Matching Readers to Texts and Tasks
4 Chapter 4: Writing, Conventions, Research, and Literacy Assessment
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5 Chapter 5: Ancient Civilizations and Medieval World History
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6 Chapter 6: United States History and Government
6.1 Indigenous North America, Colonial Settlement, and the American Revolution 6.2 Constitutional Foundations, Federalism, and the American System of Government 6.3 Westward Expansion, Sectional Conflict, the Civil War, and Reconstruction 6.4 Industrialization, the Progressive Era, World Wars, and the Civil Rights Movement
7 Chapter 7: California History and Geography
7.1 California Geography, Indigenous Nations, and the Spanish Mission Era 7.2 The Mexican Rancho Period, the Bear Flag Revolt, and the 1849 Gold Rush 7.3 California Statehood, Governance, Water Infrastructure, and Modern Economic Growth
8 Chapter 8: Social Science Skills: Geography, Civics, Economics, and Source Analysis
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9 Chapter 9: Scientific Practices and Physical Science
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10 Chapter 10: Life Science and Ecology
10.1 Cell Biology, Cellular Organization, and Metabolic Processes 10.2 Genetics, Heredity, Molecular Biology, and Evolution 10.3 Ecosystems, Ecological Interactions, Energy Flow, and Biogeochemical Cycles
11 Chapter 11: Earth, Water, and Space Science
11.1 Earth's Interior, Plate Tectonics, and Geological Processes 11.2 Atmospheric Structure, Hydrology, Weather Systems, and Climate 11.3 Earth's Water: Oceans, Estuaries, Freshwater Systems, and Tidal Dynamics 11.4 Earth-Sun-Moon Dynamics, the Solar System, and the Universe
12 Chapter 12: Number Sense, Operations, and Fractions
12.1 Number Classifications, Place Value, Properties of Operations, and Integers 12.2 Number Theory, Prime Factorization, GCF/LCM, and Exponents 12.3 Fractions, Decimals, Percentages, and Proportional Problem Solving
13 Chapter 13: Algebra, Functions, and Geometry
13.1 Ratios, Rates, Proportions, and Scaling Relationships 13.2 Algebraic Expressions, Linear Equations, Systems, and Functions 13.3 Geometric Properties, Two-Dimensional Polygons, and Three-Dimensional Solids
14 Chapter 14: Coordinate Geometry, Measurement, Statistics, and Mathematical Practices
14.1 Coordinate Geometry, Linear Graphs, and Geometric Transformations 14.2 Measurement Systems, Dimensional Analysis, and Precision 14.3 Descriptive Statistics, Data Displays, and Probability Theory 14.4 Mathematical Practices: Reasoning, Modeling, Estimation, and Justification
15 Chapter 15: Physical Education: Movement, Fitness, Social Development, and Inclusion
15.1 Motor Development, Fundamental Movement Skills, and Movement Concepts 15.2 Fitness Principles, Exercise Physiology, Nutrition, and Safety 15.3 Self-Image, Social Development, and Personal and Social Responsibility 15.4 Movement Forms, Cultural Traditions, and Inclusive Adapted Physical Education
16 Chapter 16: Human Development: Cognition, Intelligence, Social and Physical Growth
16.1 Cognitive and Linguistic Development: Piaget, Vygotsky, and Information Processing 16.2 Perspectives on Intelligence, Play, and Individual Differences 16.3 Social-Emotional and Moral Development: Erikson, Kohlberg, and Attachment Theory 16.4 Physical Development, Ecological Systems, and Mandated Reporting
17 Chapter 17: Visual and Performing Arts: Dance, Music, and Theatre
17.1 Fundamental Elements of Dance, Choreographic Principles, and Cultural Styles 17.2 Core Elements of Music, Staff Notation, Rhythm, and Auditory Analysis 17.3 Music History, World Traditions, and Elementary Music Pedagogy 17.4 Elements of Theatre, Dramatic Structures, Acting Techniques, and Stagecraft
18 Chapter 18: Visual Art, Arts Integration, and Constructed-Response Mastery
18.1 Elements of Art, Principles of Design, and Visual Analysis 18.2 Art Media, Techniques, World Art History, and Art Criticism 18.3 California Arts Standards and Cross-Curricular Arts Integration 18.4 CSET Constructed-Response Strategy, Scoring Scale, and Domain-Specific Exemplars
13.3 Geometric Properties, Two-Dimensional Polygons, and Three-Dimensional Solids
Key Takeaways Explain This Topic AI
Axiomatic geometric foundations establish that intersecting lines form congruent vertical angles and supplementary linear pairs ( 180 ∘ 180^\circ 18 0 ∘ ); parallel lines cut by a transversal generate congruent alternate interior, alternate exterior, and corresponding angles, while consecutive interior angles are supplementary.
Triangle classification is governed by side lengths (scalene, isosceles, equilateral) and interior angle measures (acute, right, obtuse); fundamental theorems include the Triangle Angle Sum Theorem ( ∑ = 180 ∘ \sum = 180^\circ ∑ = 18 0 ∘ ), Exterior Angle Theorem, and Triangle Inequality Theorem ( a + b > c a + b > c a + b > c ).
Right triangle relationships encompass the Pythagorean Theorem ( a 2 + b 2 = c 2 a^2 + b^2 = c^2 a 2 + b 2 = c 2 ), canonical integer triples (e.g., 3 - 4 - 5 , 5 - 12 - 13 , 8 - 15 - 17 3\text{-}4\text{-}5, 5\text{-}12\text{-}13, 8\text{-}15\text{-}17 3 - 4 - 5 , 5 - 12 - 13 , 8 - 15 - 17 ), and Special Right Triangles ( 45 ∘ - 45 ∘ - 90 ∘ 45^\circ\text{-}45^\circ\text{-}90^\circ 4 5 ∘ - 4 5 ∘ - 9 0 ∘ with ratio 1 : 1 : 2 1:1:\sqrt{2} 1 : 1 : 2 and 30 ∘ - 60 ∘ - 90 ∘ 30^\circ\text{-}60^\circ\text{-}90^\circ 3 0 ∘ - 6 0 ∘ - 9 0 ∘ with ratio 1 : 3 : 2 1:\sqrt{3}:2 1 : 3 : 2 ).
The Quadrilateral Hierarchy defines inclusive structural classifications (a square is simultaneously a rectangle, rhombus, parallelogram, and trapezoid); polygon interior angle sums equal ( n − 2 ) 180 ∘ (n-2)180^\circ ( n − 2 ) 18 0 ∘ , and exterior angle sums universally total 360 ∘ 360^\circ 36 0 ∘ .
Circle geometry connects radius, diameter, circumference ( C = 2 π r C = 2\pi r C = 2 π r ), area ( A = π r 2 A = \pi r^2 A = π r 2 ), sector areas, and arc lengths; three-dimensional polyhedra adhere to Euler's Formula ( V − E + F = 2 V - E + F = 2 V − E + F = 2 ) and standardized volume formulas ( V = B h V = Bh V = B h for prisms/cylinders, V = 1 3 B h V = \frac{1}{3}Bh V = 3 1 B h for pyramids/cones, V = 4 3 π r 3 V = \frac{4}{3}\pi r^3 V = 3 4 π r 3 for spheres).
Last updated: August 2026
13.3 Geometric Properties, Two-Dimensional Polygons, and Three-Dimensional Solids
CSET Focus: Geometry on CSET Multiple Subjects Subtest II tests your knowledge of spatial reasoning, axiomatic proofs, 2D polygon properties, and 3D spatial solids. You must master angle pair relationships formed by parallel transversals, triangle classifications and existence conditions (Triangle Inequality), Pythagorean triples and special right triangle ratios, the inclusive quadrilateral hierarchy, polygon angle sum formulas, circle perimeter/area/arc/sector calculations, Euler's polyhedral formula ( V − E + F = 2 V - E + F = 2 V − E + F = 2 ), and composite volume and surface area computations.
1. Geometric Foundations, Angles, and Parallel Transversals
Euclidean geometry is built upon undefined terms ( Point , Line , Plane ) from which definitions, postulates, and theorems are rigorously derived.
Line Segment ( A B ‾ \overline{AB} A B ): A bounded portion of a line consisting of endpoints A A A and B B B and all points between them.
Ray ( A B → \overrightarrow{AB} A B ): A portion of a line starting at initial point A A A and extending infinitely in the direction of point B B B .
Angle Classifications and Pair Relationships
Angles are formed by two rays sharing a common vertex. They are categorized by degree measure θ \theta θ :
- Acute Angle: 0 ∘ < θ < 90 ∘ 0^\circ < \theta < 90^\circ 0 ∘ < θ < 9 0 ∘
- Right Angle: θ = 90 ∘ \theta = 90^\circ θ = 9 0 ∘ (rays are perpendicular, denoted ⊥ \perp ⊥ )
- Obtuse Angle: 90 ∘ < θ < 180 ∘ 90^\circ < \theta < 180^\circ 9 0 ∘ < θ < 18 0 ∘
- Straight Angle: θ = 180 ∘ \theta = 180^\circ θ = 18 0 ∘ (forms a collinear straight line)
- Reflex Angle: 180 ∘ < θ < 360 ∘ 180^\circ < \theta < 360^\circ 18 0 ∘ < θ < 36 0 ∘
- Fundamental Angle Pair Theorems
Complementary Angles: Two angles whose measures sum to 90 ∘ 90^\circ 9 0 ∘ ( ∠ 1 + ∠ 2 = 90 ∘ \angle 1 + \angle 2 = 90^\circ ∠1 + ∠2 = 9 0 ∘ ).
Supplementary Angles: Two angles whose measures sum to 180 ∘ 180^\circ 18 0 ∘ ( ∠ 1 + ∠ 2 = 180 ∘ \angle 1 + \angle 2 = 180^\circ ∠1 + ∠2 = 18 0 ∘ ).
Linear Pair: Two adjacent angles whose non-common sides form a straight line; linear pairs are always supplementary.
Vertical Angles: Non-adjacent opposite angles formed by two intersecting lines; vertical angles are always congruent ( ∠ 1 ≅ ∠ 3 \angle 1 \cong \angle 3 ∠1 ≅ ∠3 and ∠ 2 ≅ ∠ 4 \angle 2 \cong \angle 4 ∠2 ≅ ∠4 ). Lines L1 and L2 Cut by Transversal T
- T
- /
- 1 / 2 / <-- Line L1
- ─────/──────/──────
- 3 / 4 /
- / /
- 5 / 6 / <-- Line L2 (L1 || L2)
- ───/──────/──────
- 7 / 8
- /
- Parallel Lines Cut by a Transversal ( L 1 ∥ L 2 L_1 \parallel L_2 L 1 ∥ L 2 )
When two coplanar parallel lines are intersected by a transversal line T T T , eight distinct angles are formed, yielding specific congruence and supplementary relationships:
Angle Relationship Identifying Angle Pairs Geometric Property Algebraic Relationship
Corresponding Angles ∠ 1 & ∠ 5 , ∠ 2 & ∠ 6 , ∠ 3 & ∠ 7 , ∠ 4 & ∠ 8 \angle 1 \& \angle 5, \; \angle 2 \& \angle 6, \; \angle 3 \& \angle 7, \; \angle 4 \& \angle 8 ∠1&∠5 , ∠2&∠6 , ∠3&∠7 , ∠4&∠8 Congruent ∠ 1 = ∠ 5 \angle 1 = \angle 5 ∠1 = ∠5
Alternate Interior Angles ∠ 3 & ∠ 6 , ∠ 4 & ∠ 5 \angle 3 \& \angle 6, \; \angle 4 \& \angle 5 ∠3&∠6 , ∠4&∠5 Congruent ∠ 3 = ∠ 6 \angle 3 = \angle 6 ∠3 = ∠6
Alternate Exterior Angles ∠ 1 & ∠ 8 , ∠ 2 & ∠ 7 \angle 1 \& \angle 8, \; \angle 2 \& \angle 7 ∠1&∠8 , ∠2&∠7 Congruent ∠ 1 = ∠ 8 \angle 1 = \angle 8 ∠1 = ∠8
Consecutive Interior Angles ∠ 3 & ∠ 5 , ∠ 4 & ∠ 6 \angle 3 \& \angle 5, \; \angle 4 \& \angle 6 ∠3&∠5 , ∠4&∠6 Supplementary ∠ 3 + ∠ 5 = 180 ∘ \angle 3 + \angle 5 = 180^\circ ∠3 + ∠5 = 18 0 ∘
Consecutive Exterior Angles ∠ 1 & ∠ 7 , ∠ 2 & ∠ 8 \angle 1 \& \angle 7, \; \angle 2 \& \angle 8 ∠1&∠7 , ∠2&∠8 Supplementary ∠ 1 + ∠ 7 = 180 ∘ \angle 1 + \angle 7 = 180^\circ ∠1 + ∠7 = 18 0 ∘
- 2. Triangles: Theorems, Pythagorean Relationships, and Special Triangles
- Triangle Classification Schema
Triangles are three-sided polygons classified simultaneously by side lengths and interior angle measures:
By Side Lengths:
Scalene: All three sides have different lengths; all three angles have different measures.
Isosceles: At least two sides are congruent; base angles opposite the congruent sides are congruent.
Equilateral: All three sides are congruent; all three interior angles measure exactly 60 ∘ 60^\circ 6 0 ∘ (equiangular).
By Interior Angle Measures:
Acute: All three interior angles are strictly acute ( < 90 ∘ <90^\circ < 9 0 ∘ ).
Right: Exactly one angle is a right angle ( 90 ∘ 90^\circ 9 0 ∘ ).
Obtuse: Exactly one angle is an obtuse angle ( > 90 ∘ >90^\circ > 9 0 ∘ ).
Core Triangle Theorems
Triangle Angle Sum Theorem: The sum of the interior angle measures of any triangle in the Euclidean plane is always 180 ∘ 180^\circ 18 0 ∘ : ∠ A + ∠ B + ∠ C = 180 ∘ \angle A + \angle B + \angle C = 180^\circ ∠ A + ∠ B + ∠ C = 18 0 ∘
Exterior Angle Theorem: The measure of an exterior angle of a triangle equals the sum of the measures of its two non-adjacent (remote) interior angles: ∠ ext = ∠ A + ∠ B \angle \text{ext} = \angle A + \angle B ∠ ext = ∠ A + ∠ B
The Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side: a + b > c , a + c > b , b + c > a a + b > c, \qquad a + c > b, \qquad b + c > a a + b > c , a + c > b , b + c > a
Practical Test: For three given side lengths a ≤ b ≤ c a \le b \le c a ≤ b ≤ c , a triangle exists if and only if the sum of the two shorter sides exceeds the longest side: a + b > c a + b > c a + b > c .
Allowable Third Side Range: If two side lengths a a a and b b b are known ( a ≤ b a \le b a ≤ b ), the third side c c c must satisfy: ∣ a − b ∣ < c < a + b |a - b| < c < a + b ∣ a − b ∣ < c < a + b .
The Pythagorean Theorem and Right Triangle Triples
For any right triangle with perpendicular legs a a a and b b b and hypotenuse c c c : a 2 + b 2 = c 2 ⟺ c = a 2 + b 2 a^2 + b^2 = c^2 \iff c = \sqrt{a^2 + b^2} a 2 + b 2 = c 2 ⟺ c = a 2 + b 2
Pythagorean Converse (Classification of Triangles by Sides): For side lengths a ≤ b ≤ c a \le b \le c a ≤ b ≤ c :
If a 2 + b 2 = c 2 ⟹ a^2 + b^2 = c^2 \implies a 2 + b 2 = c 2 ⟹ Right Triangle .
If a 2 + b 2 > c 2 ⟹ a^2 + b^2 > c^2 \implies a 2 + b 2 > c 2 ⟹ Acute Triangle .
If a 2 + b 2 < c 2 ⟹ a^2 + b^2 < c^2 \implies a 2 + b 2 < c 2 ⟹ Obtuse Triangle .
Canonical Primitive Pythagorean Triples: Common integer side length sets tested on CSET:
( 3 , 4 , 5 ) ⟹ (3, 4, 5) \implies ( 3 , 4 , 5 ) ⟹ Scaled: ( 6 , 8 , 10 ) , ( 9 , 12 , 15 ) , ( 12 , 16 , 20 ) , ( 30 , 40 , 50 ) (6, 8, 10), (9, 12, 15), (12, 16, 20), (30, 40, 50) ( 6 , 8 , 10 ) , ( 9 , 12 , 15 ) , ( 12 , 16 , 20 ) , ( 30 , 40 , 50 )
( 5 , 12 , 13 ) ⟹ (5, 12, 13) \implies ( 5 , 12 , 13 ) ⟹ Scaled: ( 10 , 24 , 26 ) , ( 15 , 36 , 39 ) (10, 24, 26), (15, 36, 39) ( 10 , 24 , 26 ) , ( 15 , 36 , 39 )
( 8 , 15 , 17 ) ⟹ (8, 15, 17) \implies ( 8 , 15 , 17 ) ⟹ Scaled: ( 16 , 30 , 34 ) (16, 30, 34) ( 16 , 30 , 34 )
( 7 , 24 , 25 ) ⟹ (7, 24, 25) \implies ( 7 , 24 , 25 ) ⟹ Scaled: ( 14 , 48 , 50 ) (14, 48, 50) ( 14 , 48 , 50 )
- ( 9 , 40 , 41 ) (9, 40, 41) ( 9 , 40 , 41 )
- Special Right Triangles
Two special right triangles occur frequently across standard mathematics curricula: 45°-45°-90° (Isosceles Right) 30°-60°-90° Triangle
- /| /|
- / | / |
- x√2 / | x 2x / | x (opposite 30°)
- / | / |
- /45° | /30° |
- └─────┘ └─────┘
- x x√3 (opposite 60°)
45 ∘ − 45 ∘ − 90 ∘ 45^\circ-45^\circ-90^\circ 4 5 ∘ − 4 5 ∘ − 9 0 ∘ Triangle (Isosceles Right):
Side Ratio: 1 : 1 : 2 ⟹ x : x : x 2 1 : 1 : \sqrt{2} \implies x : x : x\sqrt{2} 1 : 1 : 2 ⟹ x : x : x 2
Legs are congruent ( a = b = x a = b = x a = b = x ); Hypotenuse c = x 2 c = x\sqrt{2} c = x 2 .
If hypotenuse c c c is known, leg x = c 2 = c 2 2 x = \frac{c}{\sqrt{2}} = \frac{c\sqrt{2}}{2} x = 2 c = 2 c 2 .
30 ∘ − 60 ∘ − 90 ∘ 30^\circ-60^\circ-90^\circ 3 0 ∘ − 6 0 ∘ − 9 0 ∘ Triangle (Half-Equilateral):
Side Ratio: 1 : 3 : 2 ⟹ x : x 3 : 2 x 1 : \sqrt{3} : 2 \implies x : x\sqrt{3} : 2x 1 : 3 : 2 ⟹ x : x 3 : 2 x
- Short leg (opposite 30 ∘ 30^\circ 3 0 ∘ ) = x = x = x
- Long leg (opposite 60 ∘ 60^\circ 6 0 ∘ ) = x 3 = x\sqrt{3} = x 3
- Hypotenuse (opposite 90 ∘ 90^\circ 9 0 ∘ ) = 2 x = 2x = 2 x
- 3. Polygons and the Inclusive Quadrilateral Hierarchy
A polygon is a closed 2D plane figure bounded by three or more straight line segments intersecting only at their endpoints (vertices).
Polygon Angle Formulas ( n n n -sided polygon)
Sum of Interior Angles ( S int S_{\text{int}} S int ): Partitioning an n n n -gon into ( n − 2 ) (n - 2) ( n − 2 ) non-overlapping triangles from a single vertex proves: S int = ( n − 2 ) × 180 ∘ S_{\text{int}} = (n - 2) \times 180^\circ S int = ( n − 2 ) × 18 0 ∘
Each Interior Angle of a Regular (Equiangular) n n n -gon: θ int = ( n − 2 ) × 180 ∘ n = 180 ∘ − 360 ∘ n \theta_{\text{int}} = \frac{(n - 2) \times 180^\circ}{n} = 180^\circ - \frac{360^\circ}{n} θ int = n ( n − 2 ) × 18 0 ∘ = 18 0 ∘ − n 36 0 ∘
Sum of Exterior Angles ( S ext S_{\text{ext}} S ext ): For any convex polygon regardless of the number of sides n n n , the sum of one set of exterior angles is always 360 ∘ 360^\circ 36 0 ∘ : S ext = 360 ∘ ⟹ Each Exterior Angle of Regular n -gon = 360 ∘ n S_{\text{ext}} = 360^\circ \implies \text{Each Exterior Angle of Regular } n\text{-gon} = \frac{360^\circ}{n} S ext = 36 0 ∘ ⟹ Each Exterior Angle of Regular n -gon = n 36 0 ∘
The Quadrilateral Hierarchy (Inclusive CCSS Definition)
The California Mathematics Standards follow the inclusive classification system for quadrilaterals (four-sided polygons): QUADRILATERALS (4-sided polygon)
- │
- TRAPEZOIDS (≥ 1 pair parallel sides)
- │
- PARALLELOGRAMS (2 pairs parallel sides)
- ┌────────┴────────┐
- RECTANGLES RHOMBUSES
- (4 Right Angles) (4 Equal Sides)
- └────────┬────────┘
- SQUARES
- (4 Right Angles AND 4 Equal Sides)
- Quadrilateral Hierarchy Properties Matrix
- Shape Minimum Defining Properties Diagonals Properties Inclusive Relationships
- Trapezoid At least one pair of parallel opposite sides Diagonals intersect Superclass of parallelograms
Parallelogram Two pairs of parallel opposite sides; opposite sides and angles congruent Diagonals bisect each other Is a trapezoid
Rectangle Parallelogram with four congruent right angles ( 90 ∘ 90^\circ 9 0 ∘ ) Diagonals are congruent and bisect each other Is a parallelogram and a trapezoid
Rhombus Parallelogram with four congruent sides Diagonals are perpendicular bisectors and bisect vertex angles Is a parallelogram and a trapezoid
Square Regular quadrilateral: four congruent sides AND four right angles Diagonals are congruent, perpendicular bisectors Is simultaneously a rectangle, rhombus, parallelogram, and trapezoid
Kite Quadrilateral with two distinct pairs of adjacent congruent sides Diagonals are perpendicular; one diagonal bisects the other Non-parallelogram (unless all 4 sides equal)
4. Circle Geometry: Radii, Chords, Tangents, Arcs, and Sectors
A circle is the locus of all points in a 2D plane equidistant from a fixed center point.
Circle Elements and Key Relationships
Radius ( r r r ): Distance from center to any point on the boundary ( d = 2 r d = 2r d = 2 r ).
Diameter ( d d d ): Segment connecting two boundary points passing through the center ( d = 2 r d = 2r d = 2 r ).
Chord: Any line segment connecting two points on the circle circumference (diameter is the longest chord).
Secant Line: A line that intersects a circle at exactly two points.
Tangent Line: A coplanar line that touches the circle at exactly one point of tangency . A tangent line is always perpendicular ( ⊥ \perp ⊥ ) to the radius drawn to the point of tangency ( 90 ∘ 90^\circ 9 0 ∘ ).
- Circumference and Area Formulas
- Circumference (Perimeter of Circle): C = 2 π r = π d C = 2\pi r = \pi d C = 2 π r = π d
- Area of Circle: A = π r 2 A = \pi r^2 A = π r 2
- Arc Length and Sector Area Formulas
For a central angle θ \theta θ measured in degrees:
Arc Length ( L L L ): The fractional curved boundary distance subtended by θ \theta θ : L = ( θ 360 ∘ ) × 2 π r = θ π r 180 ∘ L = \left(\frac{\theta}{360^\circ}\right) \times 2\pi r = \frac{\theta \pi r}{180^\circ} L = ( 36 0 ∘ θ ) × 2 π r = 18 0 ∘ θ π r
Sector Area ( A sector A_{\text{sector}} A sector ): The "pie slice" region enclosed by two radii and the intercepted arc: A sector = ( θ 360 ∘ ) × π r 2 A_{\text{sector}} = \left(\frac{\theta}{360^\circ}\right) \times \pi r^2 A sector = ( 36 0 ∘ θ ) × π r 2
- 5. 2D Area / Perimeter and 3D Polyhedra / Volume Formulas Matrix
- Two-Dimensional Geometric Formulas
2D Shape Perimeter ( P P P ) Area ( A A A ) Formula Geometric Variables Notation
Triangle P = a + b + c P = a + b + c P = a + b + c A = 1 2 b h A = \frac{1}{2} b h A = 2 1 bh b = base , h = perpendicular height b = \text{base}, h = \text{perpendicular height} b = base , h = perpendicular height
Equilateral Triangle P = 3 s P = 3s P = 3 s A = s 2 3 4 A = \frac{s^2\sqrt{3}}{4} A = 4 s 2 3 s = side length s = \text{side length} s = side length
Rectangle P = 2 l + 2 w P = 2l + 2w P = 2 l + 2 w A = l ⋅ w A = l \cdot w A = l ⋅ w l = length , w = width l = \text{length}, w = \text{width} l = length , w = width
Parallelogram P = 2 a + 2 b P = 2a + 2b P = 2 a + 2 b A = b ⋅ h A = b \cdot h A = b ⋅ h b = base , h = perpendicular height b = \text{base}, h = \text{perpendicular height} b = base , h = perpendicular height
Trapezoid P = a + b 1 + c + b 2 P = a + b_1 + c + b_2 P = a + b 1 + c + b 2 A = b 1 + b 2 2 ⋅ h A = \frac{b_1 + b_2}{2} \cdot h A = 2 b 1 + b 2 ⋅ h b 1 , b 2 = parallel bases , h = height b_1, b_2 = \text{parallel bases}, h = \text{height} b 1 , b 2 = parallel bases , h = height
Rhombus / Kite P = 4 s P = 4s P = 4 s (Rhombus) A = 1 2 d 1 d 2 A = \frac{1}{2} d_1 d_2 A = 2 1 d 1 d 2 d 1 , d 2 = lengths of perpendicular diagonals d_1, d_2 = \text{lengths of perpendicular diagonals} d 1 , d 2 = lengths of perpendicular diagonals
Regular Polygon P = n ⋅ s P = n \cdot s P = n ⋅ s A = 1 2 a P A = \frac{1}{2} a P A = 2 1 a P a = apothem (center to side midpoint) , P = perimeter a = \text{apothem (center to side midpoint)}, P = \text{perimeter} a = apothem (center to side midpoint) , P = perimeter
- Circle C = 2 π r C = 2\pi r C = 2 π r A = π r 2 A = \pi r^2 A = π r 2 r = radius r = \text{radius} r = radius
- Three-Dimensional Solids: Euler's Formula and Spatial Measurements
- Euler's Polyhedral Formula
For any convex three-dimensional polyhedron with V V V vertices, E E E edges, and F F F faces: V − E + F = 2 ⟺ V + F = E + 2 V - E + F = 2 \iff V + F = E + 2 V − E + F = 2 ⟺ V + F = E + 2
Example (Cube / Rectangular Prism): V = 8 V = 8 V = 8 vertices, F = 6 F = 6 F = 6 faces ⟹ 8 − E + 6 = 2 ⟹ 14 − E = 2 ⟹ E = 12 \implies 8 - E + 6 = 2 \implies 14 - E = 2 \implies E = 12 ⟹ 8 − E + 6 = 2 ⟹ 14 − E = 2 ⟹ E = 12 edges.
Comprehensive 3D Volume and Surface Area Formulas
3D Solid Type Volume ( V V V ) Formula Surface Area ( S A SA S A ) Formula Spatial Dimensional Variables
Right Rectangular Prism V = l w h = B h V = lwh = Bh V = lw h = B h S A = 2 l w + 2 l h + 2 w h SA = 2lw + 2lh + 2wh S A = 2 lw + 2 l h + 2 w h l , w , h = length, width, height ; B = l w l, w, h = \text{length, width, height}; B = lw l , w , h = length, width, height ; B = lw
General Right Prism V = B h V = Bh V = B h S A = 2 B + P base h SA = 2B + P_{\text{base}} h S A = 2 B + P base h B = base area , P base = base perimeter , h = height B = \text{base area}, P_{\text{base}} = \text{base perimeter}, h = \text{height} B = base area , P base = base perimeter , h = height
Right Circular Cylinder V = π r 2 h = B h V = \pi r^2 h = Bh V = π r 2 h = B h S A = 2 π r 2 + 2 π r h SA = 2\pi r^2 + 2\pi rh S A = 2 π r 2 + 2 π r h r = radius of circular base , h = cylinder height r = \text{radius of circular base}, h = \text{cylinder height} r = radius of circular base , h = cylinder height
Right Regular Pyramid V = 1 3 B h V = \frac{1}{3}Bh V = 3 1 B h S A = B + 1 2 P base l SA = B + \frac{1}{2} P_{\text{base}} l S A = B + 2 1 P base l B = base area , h = vertical height , l = slant height B = \text{base area}, h = \text{vertical height}, l = \text{slant height} B = base area , h = vertical height , l = slant height
Right Circular Cone V = 1 3 π r 2 h = 1 3 B h V = \frac{1}{3}\pi r^2 h = \frac{1}{3}Bh V = 3 1 π r 2 h = 3 1 B h S A = π r 2 + π r l SA = \pi r^2 + \pi r l S A = π r 2 + π r l r = radius , h = height , l = r 2 + h 2 = slant height r = \text{radius}, h = \text{height}, l = \sqrt{r^2 + h^2} = \text{slant height} r = radius , h = height , l = r 2 + h 2 = slant height
Sphere V = 4 3 π r 3 V = \frac{4}{3}\pi r^3 V = 3 4 π r 3 S A = 4 π r 2 SA = 4\pi r^2 S A = 4 π r 2 r = radius of sphere r = \text{radius of sphere} r = radius of sphere