Study guide
14.1 Coordinate Geometry, Linear Graphs, and Geometric Transformations
California Subject Examinations for Teachers CSET
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14.1 Coordinate Geometry, Linear Graphs, and Geometric Transformations
Key Takeaways Explain This Topic AI
The Cartesian coordinate plane provides an analytical framework uniting algebra and Euclidean geometry through the Distance Formula ( d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 ), Midpoint Formula ( M = ( x 1 + x 2 2 , y 1 + y 2 2 ) M = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}) M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) ), and Slope ( m = Δ y Δ x m = \frac{\Delta y}{\Delta x} m = Δ x Δ y ).
Linear relations can be expressed in Slope-Intercept ( y = m x + b y=mx+b y = m x + b ), Point-Slope ( y − y 1 = m ( x − x 1 ) y-y_1=m(x-x_1) y − y 1 = m ( x − x 1 ) ), or Standard Form ( A x + B y = C Ax+By=C A x + B y = C ); parallel lines share identical slopes ( m 1 = m 2 m_1=m_2 m 1 = m 2 ), whereas perpendicular lines have negative reciprocal slopes ( m 1 ⋅ m 2 = − 1 m_1 \cdot m_2 = -1 m 1 ⋅ m 2 = − 1 ).
Rigid motions (isometries) preserve distance, angle measure, collinearity, and area: translations shift points along a vector ⟨ a , b ⟩ \langle a, b \rangle ⟨ a , b ⟩ , reflections flip figures across a line of symmetry (reversing orientation), and rotations turn figures about a fixed center by angle θ \theta θ (preserving orientation).
Dilations are non-rigid similarity transformations with center C C C and scale factor k > 0 k > 0 k > 0 that map ( x , y ) → ( k x , k y ) (x,y) \to (kx, ky) ( x , y ) → ( k x , k y ) from the origin, preserving angle measures while scaling perimeters by k k k and areas by k 2 k^2 k 2 .
A regular tessellation is a monohedral tiling of congruent regular polygons where vertex interior angles sum to exactly 360 ∘ 360^\circ 36 0 ∘ ; in the Euclidean plane, only equilateral triangles ( 60 ∘ 60^\circ 6 0 ∘ ), squares ( 90 ∘ 90^\circ 9 0 ∘ ), and regular hexagons ( 120 ∘ 120^\circ 12 0 ∘ ) satisfy this criterion.
Last updated: August 2026
14.1 Coordinate Geometry, Linear Graphs, and Geometric Transformations
CSET Focus: The California Subject Examinations for Teachers (CSET) Multiple Subjects Subtest II requires candidates to demonstrate mastery of coordinate geometry and geometric transformations. You must be prepared to calculate distances, midpoints, and slopes on the Cartesian plane; convert between linear equation forms; determine perpendicular and parallel relationships; apply algebraic mapping rules for rigid motions (translations, reflections, rotations) and dilations; and explain the mathematical principles underlying line symmetry, rotational symmetry, and Euclidean tessellations.
1. The Cartesian Coordinate Plane & Analytical Line Formulas
The Cartesian coordinate plane is formed by the perpendicular intersection of two real number lines at the origin ( 0 , 0 ) (0,0) ( 0 , 0 ) . The horizontal axis is designated as the x x x -axis (abscissa) and the vertical axis as the y y y -axis (ordinate). These axes partition the infinite two-dimensional plane into four distinct quadrants numbered counterclockwise: y-axis
- │
- Quadrant II │ Quadrant I
- (-x, +y) │ (+x, +y)
- │
- ─────────────────────────┼───────────────────────── x-axis
- │ (0,0) Origin
- Quadrant III │ Quadrant IV
- (-x, -y) │ (+x, -y)
- │
- Analytical Coordinate Formulas
- 1. The Distance Formula
The Euclidean distance d d d between two points P 1 ( x 1 , y 1 ) P_1(x_1, y_1) P 1 ( x 1 , y 1 ) and P 2 ( x 2 , y 2 ) P_2(x_2, y_2) P 2 ( x 2 , y 2 ) is directly derived from the Pythagorean Theorem ( a 2 + b 2 = c 2 a^2 + b^2 = c^2 a 2 + b 2 = c 2 ) by constructing a right triangle with legs parallel to the coordinate axes: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
Example: Find the distance between A ( − 3 , 5 ) A(-3, 5) A ( − 3 , 5 ) and B ( 3 , − 3 ) B(3, -3) B ( 3 , − 3 ) : d = ( 3 − ( − 3 ) ) 2 + ( − 3 − 5 ) 2 = 6 2 + ( − 8 ) 2 = 36 + 64 = 100 = 10 d = \sqrt{(3 - (-3))^2 + (-3 - 5)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 d = ( 3 − ( − 3 ) ) 2 + ( − 3 − 5 ) 2 = 6 2 + ( − 8 ) 2 = 36 + 64 = 100 = 10
2. The Midpoint Formula
The midpoint M M M of the line segment connecting P 1 ( x 1 , y 1 ) P_1(x_1, y_1) P 1 ( x 1 , y 1 ) and P 2 ( x 2 , y 2 ) P_2(x_2, y_2) P 2 ( x 2 , y 2 ) represents the coordinate-wise arithmetic mean of the endpoints: M = ( x 1 + x 2 2 , y 1 + y 2 2 ) M = \left( \frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2} \right) M = ( 2 x 1 + x 2 , 2 y 1 + y 2 )
Example: The midpoint of segment A B AB A B connecting ( − 4 , 7 ) (-4, 7) ( − 4 , 7 ) and ( 8 , 1 ) (8, 1) ( 8 , 1 ) is: M = ( − 4 + 8 2 , 7 + 1 2 ) = ( 4 2 , 8 2 ) = ( 2 , 4 ) M = \left( \frac{-4 + 8}{2}, \; \frac{7 + 1}{2} \right) = \left( \frac{4}{2}, \; \frac{8}{2} \right) = (2, 4) M = ( 2 − 4 + 8 , 2 7 + 1 ) = ( 2 4 , 2 8 ) = ( 2 , 4 )
3. Slope and Rate of Change
The slope ( m m m ) of a non-vertical line measures its steepness and direction, defined as the ratio of vertical change (rise, Δ y \Delta y Δ y ) to horizontal change (run, Δ x \Delta x Δ x ): m = Δ y Δ x = y 2 − y 1 x 2 − x 1 ( x 1 ≠ x 2 ) m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \quad (x_1 \ne x_2) m = Δ x Δ y = x 2 − x 1 y 2 − y 1 ( x 1 = x 2 )
Positive Slope ( m > 0 m > 0 m > 0 ): Line rises from left to right.
Negative Slope ( m < 0 m < 0 m < 0 ): Line falls from left to right.
Zero Slope ( m = 0 m = 0 m = 0 ): Horizontal line ( y = k y = k y = k ), where Δ y = 0 \Delta y = 0 Δ y = 0 and Δ x ≠ 0 \Delta x \ne 0 Δ x = 0 .
Undefined Slope: Vertical line ( x = h x = h x = h ), where Δ x = 0 \Delta x = 0 Δ x = 0 and division by zero is undefined.
Forms of Linear Equations
Slope-Intercept Form: y = m x + b y = mx + b y = m x + b , where m m m is the slope and b b b is the y y y -intercept ( 0 , b ) (0, b) ( 0 , b ) .
Point-Slope Form: y − y 1 = m ( x − x 1 ) y - y_1 = m(x - x_1) y − y 1 = m ( x − x 1 ) , where m m m is the slope and ( x 1 , y 1 ) (x_1, y_1) ( x 1 , y 1 ) is any specific point on the line.
Standard Form: A x + B y = C Ax + By = C A x + B y = C , where A , B , C ∈ Z A, B, C \in \mathbb{Z} A , B , C ∈ Z , A ≥ 0 A \ge 0 A ≥ 0 , and A A A and B B B are not both zero. The slope of a standard form line is m = − A B m = -\frac{A}{B} m = − B A and the y y y -intercept is ( 0 , C B ) (0, \frac{C}{B}) ( 0 , B C ) .
Parallel and Perpendicular Line Relationships
Parallel Lines ( L 1 ∥ L 2 L_1 \parallel L_2 L 1 ∥ L 2 ): Two non-vertical lines are parallel if and only if their slopes are equal ( m 1 = m 2 m_1 = m_2 m 1 = m 2 ) and their y y y -intercepts are distinct ( b 1 ≠ b 2 b_1 \ne b_2 b 1 = b 2 ). Coincident lines share both identical slopes and identical intercepts.
Perpendicular Lines ( L 1 ⊥ L 2 L_1 \perp L_2 L 1 ⊥ L 2 ): Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals (opposite reciprocals): m 1 ⋅ m 2 = − 1 ⟺ m 2 = − 1 m 1 m_1 \cdot m_2 = -1 \iff m_2 = -\frac{1}{m_1} m 1 ⋅ m 2 = − 1 ⟺ m 2 = − m 1 1
Horizontal and Vertical Special Case: Any horizontal line ( y = k y = k y = k , m = 0 m = 0 m = 0 ) is perpendicular to any vertical line ( x = h x = h x = h , undefined slope).
Slope Relationships & Linear Forms Matrix
Geometric Relationship / Form Algebraic Formulation Defining Characteristic Visual / Analytical Exemplar
Slope-Intercept Form y = m x + b y = mx + b y = m x + b Explicit slope m m m and y y y -intercept ( 0 , b ) (0,b) ( 0 , b ) y = − 3 4 x + 6 y = -\frac{3}{4}x + 6 y = − 4 3 x + 6 ( m = − 3 4 , b = 6 m = -\frac{3}{4}, b = 6 m = − 4 3 , b = 6 )
Point-Slope Form y − y 1 = m ( x − x 1 ) y - y_1 = m(x - x_1) y − y 1 = m ( x − x 1 ) Anchored at point ( x 1 , y 1 ) (x_1, y_1) ( x 1 , y 1 ) with slope m m m y − 4 = 2 ( x + 3 ) y - 4 = 2(x + 3) y − 4 = 2 ( x + 3 ) (passes through ( − 3 , 4 ) (-3, 4) ( − 3 , 4 ) )
Standard Form A x + B y = C Ax + By = C A x + B y = C A , B , C ∈ Z , A ≥ 0 A, B, C \in \mathbb{Z}, A \ge 0 A , B , C ∈ Z , A ≥ 0 3 x − 4 y = 12 3x - 4y = 12 3 x − 4 y = 12 ( m = 3 4 , x -int = ( 4 , 0 ) , y -int = ( 0 , − 3 ) m = \frac{3}{4}, x\text{-int} = (4,0), y\text{-int} = (0,-3) m = 4 3 , x -int = ( 4 , 0 ) , y -int = ( 0 , − 3 ) )
Parallel Lines m 1 = m 2 , b 1 ≠ b 2 m_1 = m_2, \; b_1 \ne b_2 m 1 = m 2 , b 1 = b 2 Equal rates of change; never intersect y = 3 x − 5 y = 3x - 5 y = 3 x − 5 and y = 3 x + 8 y = 3x + 8 y = 3 x + 8
Perpendicular Lines m 1 ⋅ m 2 = − 1 m_1 \cdot m_2 = -1 m 1 ⋅ m 2 = − 1 Negative reciprocal slopes; meet at 90 ∘ 90^\circ 9 0 ∘ y = 2 5 x + 1 y = \frac{2}{5}x + 1 y = 5 2 x + 1 and y = − 5 2 x − 4 y = -\frac{5}{2}x - 4 y = − 2 5 x − 4
Horizontal Line y = k y = k y = k Slope m = 0 m = 0 m = 0 ; parallel to x x x -axis y = − 7 y = -7 y = − 7 (passes through all points with y = − 7 y=-7 y = − 7 )
Vertical Line x = h x = h x = h Slope is undefined; parallel to y y y -axis x = 5 x = 5 x = 5 (passes through all points with x = 5 x=5 x = 5 )
2. Geometric Transformations: Rigid Motions (Isometries)
A transformation is a bijection (one-to-one and onto mapping) that assigns each pre-image point P ( x , y ) P(x, y) P ( x , y ) in the plane to a unique image point P ′ ( x ′ , y ′ ) P'(x', y') P ′ ( x ′ , y ′ ) . An isometry (or rigid motion ) is a transformation that strictly preserves Euclidean distance : d ( P , Q ) = d ( P ′ , Q ′ ) for all points P , Q d(P, Q) = d(P', Q') \quad \text{for all points } P, Q d ( P , Q ) = d ( P ′ , Q ′ ) for all points P , Q
Rigid motions also preserve angle measure, collinearity, betweenness, and area. The image figure is congruent to the pre-image figure ( Δ A B C ≅ Δ A ′ B ′ C ′ \Delta ABC \cong \Delta A'B'C' Δ A BC ≅ Δ A ′ B ′ C ′ ). GEOMETRIC TRANSFORMATIONS
- ┌──────────────────┴──────────────────┐
- RIGID MOTIONS (Isometries) NON-RIGID (Similarity)
- (Preserve distance, angle, area) (Preserves angle, shape)
- ┌───────────┼───────────┐ │
- TRANSLATIONS REFLECTIONS ROTATIONS DILATIONS
- (Slide: v) (Flip: Line) (Turn: Center θ) (Scale: k)
- [Direct] [Opposite] [Direct] [Direct]
- 1. Translations (Slides)
A translation shifts every point in the plane by a constant horizontal distance a a a and vertical distance b b b , represented by the translation vector v ⃗ = ⟨ a , b ⟩ \vec{v} = \langle a, b \rangle v = ⟨ a , b ⟩ : ( x , y ) ⟶ ( x + a , y + b ) (x, y) \longrightarrow (x + a, \; y + b) ( x , y ) ⟶ ( x + a , y + b )
Translations are direct isometries because they preserve the geometric orientation (handedness) of vertices.
2. Reflections (Flips)
A reflection flips a figure over a specified line of reflection L L L . The line L L L acts as the perpendicular bisector of the line segment connecting each pre-image point P P P to its image point P ′ P' P ′ . Reflections are opposite isometries because they reverse the cyclic clockwise/counterclockwise orientation of vertices.
Standard Coordinate Reflection Rules:
Reflection across the x x x -axis ( y = 0 y = 0 y = 0 ): ( x , y ) ⟶ ( x , − y ) (x, y) \longrightarrow (x, -y) ( x , y ) ⟶ ( x , − y )
Reflection across the y y y -axis ( x = 0 x = 0 x = 0 ): ( x , y ) ⟶ ( − x , y ) (x, y) \longrightarrow (-x, y) ( x , y ) ⟶ ( − x , y )
Reflection across the line y = x y = x y = x : ( x , y ) ⟶ ( y , x ) (x, y) \longrightarrow (y, x) ( x , y ) ⟶ ( y , x )
Reflection across the line y = − x y = -x y = − x : ( x , y ) ⟶ ( − y , − x ) (x, y) \longrightarrow (-y, -x) ( x , y ) ⟶ ( − y , − x )
Reflection across a vertical line x = h x = h x = h : ( x , y ) ⟶ ( 2 h − x , y ) (x, y) \longrightarrow (2h - x, y) ( x , y ) ⟶ ( 2 h − x , y )
Reflection across a horizontal line y = k y = k y = k : ( x , y ) ⟶ ( x , 2 k − y ) (x, y) \longrightarrow (x, 2k - y) ( x , y ) ⟶ ( x , 2 k − y )
3. Rotations (Turns)
A rotation turns every point around a fixed point called the center of rotation by a specified directed angle θ \theta θ . In standard coordinate geometry, positive angles represent counterclockwise (CCW) rotations about the origin ( 0 , 0 ) (0,0) ( 0 , 0 ) . Rotations are direct isometries .
Standard Origin-Centered Rotation Rules:
90 ∘ 90^\circ 9 0 ∘ CCW Rotation (or 270 ∘ 270^\circ 27 0 ∘ CW): ( x , y ) ⟶ ( − y , x ) (x, y) \longrightarrow (-y, x) ( x , y ) ⟶ ( − y , x )
180 ∘ 180^\circ 18 0 ∘ Rotation (Half-Turn): ( x , y ) ⟶ ( − x , − y ) (x, y) \longrightarrow (-x, -y) ( x , y ) ⟶ ( − x , − y )
270 ∘ 270^\circ 27 0 ∘ CCW Rotation (or 90 ∘ 90^\circ 9 0 ∘ CW): ( x , y ) ⟶ ( y , − x ) (x, y) \longrightarrow (y, -x) ( x , y ) ⟶ ( y , − x )
360 ∘ 360^\circ 36 0 ∘ Rotation (Full Turn / Identity): ( x , y ) ⟶ ( x , y ) (x, y) \longrightarrow (x, y) ( x , y ) ⟶ ( x , y )
Coordinate Transformation Mapping Rules Table
Transformation Type Parameter / Line / Center Coordinate Algebraic Rule Isometry Category Orientation Effect
Translation Vector v ⃗ = ⟨ a , b ⟩ \vec{v} = \langle a, b \rangle v = ⟨ a , b ⟩ ( x , y ) → ( x + a , y + b ) (x, y) \to (x + a, \; y + b) ( x , y ) → ( x + a , y + b ) Rigid (Isometry) Preserved (Direct)
Reflection x x x -axis ( y = 0 y=0 y = 0 ) ( x , y ) → ( x , − y ) (x, y) \to (x, -y) ( x , y ) → ( x , − y ) Rigid (Isometry) Reversed (Opposite)
Reflection y y y -axis ( x = 0 x=0 x = 0 ) ( x , y ) → ( − x , y ) (x, y) \to (-x, y) ( x , y ) → ( − x , y ) Rigid (Isometry) Reversed (Opposite)
Reflection Diagonal line y = x y = x y = x ( x , y ) → ( y , x ) (x, y) \to (y, x) ( x , y ) → ( y , x ) Rigid (Isometry) Reversed (Opposite)
Reflection Diagonal line y = − x y = -x y = − x ( x , y ) → ( − y , − x ) (x, y) \to (-y, -x) ( x , y ) → ( − y , − x ) Rigid (Isometry) Reversed (Opposite)
Rotation 90 ∘ 90^\circ 9 0 ∘ CCW about ( 0 , 0 ) (0,0) ( 0 , 0 ) ( x , y ) → ( − y , x ) (x, y) \to (-y, x) ( x , y ) → ( − y , x ) Rigid (Isometry) Preserved (Direct)
Rotation 180 ∘ 180^\circ 18 0 ∘ about ( 0 , 0 ) (0,0) ( 0 , 0 ) ( x , y ) → ( − x , − y ) (x, y) \to (-x, -y) ( x , y ) → ( − x , − y ) Rigid (Isometry) Preserved (Direct)
Rotation 270 ∘ 270^\circ 27 0 ∘ CCW about ( 0 , 0 ) (0,0) ( 0 , 0 ) ( x , y ) → ( y , − x ) (x, y) \to (y, -x) ( x , y ) → ( y , − x ) Rigid (Isometry) Preserved (Direct)
Dilation Origin ( 0 , 0 ) (0,0) ( 0 , 0 ) , scale factor k k k ( x , y ) → ( k x , k y ) (x, y) \to (kx, ky) ( x , y ) → ( k x , k y ) Non-Rigid (Similarity) Preserved (Direct for k > 0 k > 0 k > 0 )
CSET Diagnostic Trap: When performing compositions of transformations, order matters! A reflection followed by a translation generally does not yield the same resulting coordinates as the translation followed by the reflection. Always evaluate transformations step-by-step in the sequence specified.
3. Non-Rigid Transformations: Dilations & Similarity
A dilation is a non-rigid transformation that expands or compresses a geometric figure relative to a fixed center of dilation C C C by a positive scalar factor k k k ( k > 0 k > 0 k > 0 ).
Properties of Dilations
Origin-Centered Coordinate Mapping: ( x , y ) ⟶ ( k x , k y ) (x, y) \longrightarrow (kx, ky) ( x , y ) ⟶ ( k x , k y )
Scale Factor Regimes:
Enlargement ( k > 1 k > 1 k > 1 ): The image is strictly larger than the pre-image ( d ( C , P ′ ) = k ⋅ d ( C , P ) d(C, P') = k \cdot d(C, P) d ( C , P ′ ) = k ⋅ d ( C , P ) ).
Reduction ( 0 < k < 1 0 < k < 1 0 < k < 1 ): The image is strictly smaller than the pre-image.
Identity ( k = 1 k = 1 k = 1 ): The image is congruent to the pre-image.
Preservation of Invariants: Dilations preserve angle measure, collinearity, betweenness, and parallelism of lines. Consequently, dilations produce similar figures ( Δ A B C ∼ Δ A ′ B ′ C ′ \Delta ABC \sim \Delta A'B'C' Δ A BC ∼ Δ A ′ B ′ C ′ ).
Dimensional Scaling Ratios:
Linear Dimensions (Side Lengths, Perimeter, Altitude): Scaled by factor k k k . Perimeter ′ = k × Perimeter \text{Perimeter}' = k \times \text{Perimeter} Perimeter ′ = k × Perimeter
Two-Dimensional Quantities (Surface Area, Base Area): Scaled by factor k 2 k^2 k 2 . Area ′ = k 2 × Area \text{Area}' = k^2 \times \text{Area} Area ′ = k 2 × Area
Three-Dimensional Quantities (Volume, Capacity): Scaled by factor k 3 k^3 k 3 . Volume ′ = k 3 × Volume \text{Volume}' = k^3 \times \text{Volume} Volume ′ = k 3 × Volume
Worked Example: A triangle with an area of 24 cm 2 24\text{ cm}^2 24 cm 2 is dilated by a scale factor of k = 3 k = 3 k = 3 . The perimeter of the new triangle is 3 3 3 times the original, but its area becomes 3 2 × 24 = 9 × 24 = 216 cm 2 3^2 \times 24 = 9 \times 24 = 216\text{ cm}^2 3 2 × 24 = 9 × 24 = 216 cm 2 .
- 4. Symmetry and Euclidean Tessellations
- Symmetry Classifications
Line Symmetry (Reflectional / Bilateral): A figure possesses line symmetry if there exists a line L L L such that reflecting the figure over L L L maps the figure entirely onto itself. A regular n n n -sided polygon has exactly n n n lines of symmetry.
Rotational Symmetry: A figure possesses rotational symmetry if it maps onto itself after a rotation of angle θ \theta θ where 0 ∘ < θ < 360 ∘ 0^\circ < \theta < 360^\circ 0 ∘ < θ < 36 0 ∘ about its geometric center.
Order of Rotational Symmetry ( n n n ): The number of distinct rotational positions in [ 0 ∘ , 360 ∘ ) [0^\circ, 360^\circ) [ 0 ∘ , 36 0 ∘ ) in which the figure looks identical.
Angle of Rotational Symmetry: The minimum positive angle of rotation: θ = 360 ∘ n \theta = \frac{360^\circ}{n} θ = n 36 0 ∘
Example: A regular hexagon has rotational symmetry of order 6 6 6 , with a fundamental angle of 360 ∘ 6 = 60 ∘ \frac{360^\circ}{6} = 60^\circ 6 36 0 ∘ = 6 0 ∘ (mapping onto itself at 60 ∘ , 120 ∘ , 180 ∘ , 240 ∘ , 300 ∘ , 360 ∘ 60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ, 360^\circ 6 0 ∘ , 12 0 ∘ , 18 0 ∘ , 24 0 ∘ , 30 0 ∘ , 36 0 ∘ ).
Point Symmetry: Equivalent to rotational symmetry of order 2 ( 180 ∘ 180^\circ 18 0 ∘ rotation), where every point ( x , y ) (x, y) ( x , y ) corresponds to a matching point ( − x , − y ) (-x, -y) ( − x , − y ) equidistant from the center.
Tessellations (Tilings) of the Plane
A tessellation (or tiling) is a pattern covering an infinite plane with repeating geometric shapes without any gaps or overlaps.
The Vertex Condition for Regular Tessellations
A regular tessellation uses only one type of regular polygon (monohedral tiling). For regular polygons to tile the plane, the sum of the interior angles of the polygons meeting at every shared vertex must equal exactly 360 ∘ 360^\circ 36 0 ∘ .
The interior angle θ \theta θ of a regular n n n -sided polygon is given by: θ = ( n − 2 ) × 180 ∘ n \theta = \frac{(n - 2) \times 180^\circ}{n} θ = n ( n − 2 ) × 18 0 ∘ Euclidean Tessellation Condition
- k × θ_interior = 360° (k ∈ ℤ⁺)
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- Equilateral Triangle Square Regular Hexagon
- n = 3, θ = 60° n = 4, θ = 90° n = 6, θ = 120°
- 6 × 60° = 360° 4 × 90° = 360° 3 × 120° = 360°
- [6 triangles at vertex] [4 squares at vertex] [3 hexagons at vertex]
- The Three Regular Tessellations in Euclidean Geometry
Equilateral Triangles ( n = 3 n = 3 n = 3 ): Interior angle = 60 ∘ = 60^\circ = 6 0 ∘ . Since 6 × 60 ∘ = 360 ∘ 6 \times 60^\circ = 360^\circ 6 × 6 0 ∘ = 36 0 ∘ , exactly 6 6 6 triangles meet at each vertex (vertex configuration 3.3.3.3.3.3 3.3.3.3.3.3 3.3.3.3.3.3 ).
Squares ( n = 4 n = 4 n = 4 ): Interior angle = 90 ∘ = 90^\circ = 9 0 ∘ . Since 4 × 90 ∘ = 360 ∘ 4 \times 90^\circ = 360^\circ 4 × 9 0 ∘ = 36 0 ∘ , exactly 4 4 4 squares meet at each vertex (vertex configuration 4.4.4.4 4.4.4.4 4.4.4.4 ).
Regular Hexagons ( n = 6 n = 6 n = 6 ): Interior angle = 120 ∘ = 120^\circ = 12 0 ∘ . Since 3 × 120 ∘ = 360 ∘ 3 \times 120^\circ = 360^\circ 3 × 12 0 ∘ = 36 0 ∘ , exactly 3 3 3 hexagons meet at each vertex (vertex configuration 6.6.6 6.6.6 6.6.6 ).
Why Regular Pentagons Cannot Tessellate: For a regular pentagon ( n = 5 n = 5 n = 5 ), the interior angle is ( 5 − 2 ) × 180 ∘ 5 = 108 ∘ \frac{(5-2) \times 180^\circ}{5} = 108^\circ 5 ( 5 − 2 ) × 18 0 ∘ = 10 8 ∘ . Three pentagons sum to 3 × 108 ∘ = 324 ∘ 3 \times 108^\circ = 324^\circ 3 × 10 8 ∘ = 32 4 ∘ (leaving a 36 ∘ 36^\circ 3 6 ∘ gap), while four pentagons sum to 432 ∘ 432^\circ 43 2 ∘ (overlapping). Since 108 ∘ 108^\circ 10 8 ∘ does not divide 360 ∘ 360^\circ 36 0 ∘ evenly, regular pentagons cannot form a regular tessellation.