Virginia Tech®home

MATH 1535 Course Page

Mathematics Department Banner

Geometry and the Mathematics of Design

MATH 1535 is offered in the Fall semester only.

A standard first-year mathematics sequence for architecture majors. Mathematical models of real-world problems, including discrete and continuous models, that address relevant global challenges in such areas as urban planning, building construction, and home design. Euclidean geometry, trigonometry, sequences and the golden ratio, graph theory, tilings, polygons and polyhedra, applications for 2- and 3-dimensional design and construction, use of geometric software.  (3H,3C)

Prerequistes: 2 units of high school algebra and 1 unit of high school geometry.

Required Online Textbook: Geometry and Mathematics of Design

Chapter   Section   Subject
1.1    Basic Geometry
  1.1.1 Introduction to Geogebra
  1.1.2 Lines
  1.1.3 Circles & Polygons
1.2   Basic Trigonometry
  1.2.1 Unit Circle Trigonometry
  1.2.2 Right Triangle Trigonometry
  1.2.3 Law of Cosines and Law of Sines
1.3   Properties of Geometric Figures
  1.3.1 Properties of Triangles
  1.3.2 Properties of Polygons
  1.3.3 Properties of Circles
  1.3.4 Geogebra and Geometric Figures
Chapter  Section  Topic
2.1    Geometric Constructions 
  2.1.1  Bisect a Line Segment
  2.1.2  Bisect an Angle
  2.1.3  Construct a Perpendicular Line 
  2.1.4  Construct a Parallel Line
  2.1.5  Geogebra Constructions
2.2    Isometries
  2.2.1  Reflections
  2.2.2  Translations & Glide Reflections
  2.2.3  Rotations
  2.2.4  Geogebra Isometries
2.3    Congruence
  2.3.1  Congruence of Two Figures
  2.3.2  Geogebra & Congruence
2.4    Similarity
  2.4.1  Geogebra & Similarity
  2.4.2  Relationships Among Similar Figures
2.5    Circles
  2.5.1  Tangents, Angles and Lines for a Circle
  2.5.2  Ellipses
Chapter Section Topic
3.1   Graphs & Maps
  3.1.1 Vertices & Edges
  3.1.2 Types of Graphs
  3.1.3 Isomorphic Graphs
  3.1.4 Subgraphs
  3.1.5 Hamiltonian Paths & Circuits
  3.1.6 Eulerian Paths and Circuits
3.2   Maps
  3.2.1 Vertices, Edges, & Faces
  3.2.2 Valences
  3.2.3 Dual Maps
3.3   Euler's Formula
  3.3.1 Vertex and Face Valence Formulas
  3.3.2 Euler's Formula
  3.3.3 Applying Euler's Formula
Chapter Section Topic
4.1   Introduction to Sequences
  4.1.1 Properties of Sequences
  4.1.2 Arithmetic Sequences
  4.1.3 Geometric Sequences
  4.1.4 Fibonacci Sequences
  4.1.5 Fractals
4.2   Golden Ratio
  4.2.1 Golden Ratio as a Number
  4.2.2 Golden Ratio and Sequences
  4.2.3 Golden Rectangles
  4.2.4 Other Golder Polygons
  4.2.5 Modulor System of LeCorbusier
Chapter Section Topic
5.1   Tilings
  5.1.1 Introduction to Tilings
  5.1.2 Tilings by One Regular Polygon
  5.1.3 Tilings by More Regular Polygons
  5.1.4 Dual Tilings and the Schlafli Symbol
5.2   Periodic Tilings
  5.2.1 Symmetries Within a Tiling
  5.2.2 Conway's Criterion
5.3   Non-Periodic Tilings
  5.3.1 Penrose Kites and Darts
  5.3.2 Penrose Tilings
Chapter Section Topic
6.1   Regular Polyhedrs
  6.1.1 Identifying Regular Polyhedra
  6.1.2 Spherical Deviation
  6.1.3 Creating Polyhedra
6.2   Semi-Regular Polyhedra
  6.2.1 Creating Semi-Regular Polyhedra
  6.2.2 Schlafli Symbol
  6.2.3 Descartes' Formula & Law
  6.2.4 Possible Semi-Regular Polyhedra
6.3   Polyhedra and the Cube
  6.3.1 Cube & Polyhedra Edge Length
  6.3.2 Cube & Polyhedra Volume

Check your instructor's Canvas course site for the date and time of your final exam.

During any Fall Semester, see the Timetable of classes for information on current offerings of MATH 1535.


  • The in-person version of MATH 1535 consists of two 75 minute lectures and one 50 minute recitation session. 
  • All quizzes can be taken at home and are unproctored.
  • The midterm exams and the final exam are  taken in the testing center at the Math Emporium. The quizzes and exams are multiple choice.

The Undergraduate Honor Code pledge that each member of the university community agrees to abide by states:

“As a Hokie, I will conduct myself with honor and integrity at all times. I will not lie, cheat, or steal, nor will I accept the actions of those who do.”

Students enrolled in this course are responsible for abiding by the Honor Code. A student who has doubts about how the Honor Code applies to any assignment is responsible for obtaining specific guidance from the course instructor before submitting the assignment for evaluation. Ignorance of the rules does not excuse any member of the University community from the requirements and expectations of the Honor Code.

For additional information about the Honor Code, please visit

If you need extra help with course materials:


If you are currently enrolled in MATH 1535, you can contact your instructor through your Canvas course website.