💯Math for Non-Math Majors Unit 10 – Geometry

Key Concepts and Definitions

• Geometry the branch of mathematics that deals with the properties, measurement, and relationships of points, lines, angles, surfaces, and solids
• Point a precise location in space, represented by a dot and named using a capital letter
• Line an infinite straight path, defined by two points and extending indefinitely in both directions
• Plane a flat, two-dimensional surface that extends infinitely in all directions
  • Contains an infinite number of points and lines
• Angle formed when two lines or line segments intersect at a point, measured in degrees
• Polygon a closed, two-dimensional shape with at least three straight sides
• Congruent figures or objects that have the same size and shape
• Similar figures or objects that have the same shape but different sizes, with proportional corresponding sides and equal corresponding angles

Basic Shapes and Their Properties

• Triangle a polygon with three sides and three angles
  • Sum of the angles in a triangle always equals 180°
  • Types of triangles: equilateral (all sides equal), isosceles (two sides equal), and scalene (no sides equal)
• Quadrilateral a polygon with four sides and four angles
  • Types of quadrilaterals: parallelogram, rectangle, square, rhombus, and trapezoid
• Circle a round plane figure whose boundary consists of points equidistant from a fixed center point
  • Radius the distance from the center of a circle to any point on its circumference
  • Diameter a line segment that passes through the center of a circle and has its endpoints on the circumference, equal to twice the radius
• Regular polygon a polygon with all sides equal in length and all angles equal in measure
• Parallel lines lines in a plane that never intersect, always maintaining a constant distance between them
• Perpendicular lines lines that intersect at a 90° angle

Angles and Measurements

• Acute angle an angle measuring less than 90°
• Right angle an angle measuring exactly 90°
• Obtuse angle an angle measuring greater than 90° but less than 180°
• Straight angle an angle measuring exactly 180°
• Complementary angles two angles whose sum equals 90°
• Supplementary angles two angles whose sum equals 180°
• Vertical angles opposite angles formed when two lines intersect, always congruent
• Angle bisector a line that divides an angle into two equal parts
• Protractor a tool used to measure and draw angles

Area and Perimeter Calculations

• Perimeter the distance around a closed figure, calculated by adding the lengths of all sides
  • Formula for the perimeter of a rectangle: $P = 2l + 2w$, where $l$ is length and $w$ is width
• Area the amount of space inside a closed figure, measured in square units
  • Formula for the area of a rectangle: $A = lw$, where $l$ is length and $w$ is width
  • Formula for the area of a triangle: $A = \frac{1}{2}bh$, where $b$ is base and $h$ is height
• Circumference the distance around a circle
  • Formula for circumference: $C = 2\pi r$ or $C = \pi d$, where $r$ is radius, $d$ is diameter, and $\pi \approx 3.14159$
• Area of a circle calculated using the formula $A = \pi r^2$, where $r$ is the radius
• Pythagorean theorem in a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides, expressed as $a^2 + b^2 = c^2$

3D Shapes and Volume

• Polyhedron a three-dimensional figure with flat polygonal faces, straight edges, and sharp corners or vertices
  • Examples: cube, pyramid, prism
• Sphere a three-dimensional figure shaped like a ball, with every point on its surface equidistant from its center
• Cylinder a three-dimensional figure with two identical circular bases connected by a curved surface
• Cone a three-dimensional figure with a circular base and a single vertex at the top
• Volume the amount of space occupied by a three-dimensional object, measured in cubic units
  • Formula for the volume of a rectangular prism: $V = lwh$, where $l$ is length, $w$ is width, and $h$ is height
  • Formula for the volume of a cylinder: $V = \pi r^2 h$, where $r$ is the radius of the base and $h$ is the height
• Surface area the total area of all surfaces of a three-dimensional object

Geometric Proofs and Logic

• Theorem a statement that can be proven true based on previously established statements, definitions, or axioms
• Postulate a statement that is assumed to be true without proof
• Deductive reasoning the process of reaching a conclusion based on a set of premises, moving from general statements to specific conclusions
• Inductive reasoning the process of drawing a general conclusion from a set of specific observations or examples
• Proof a logical argument that demonstrates the truth of a statement using accepted facts, definitions, and previously proven theorems
  • Two-column proof a format for writing proofs that consists of statements in one column and the corresponding reasons in the other column
• Congruence proofs proofs that demonstrate two figures are congruent by showing their corresponding parts are equal
• Similarity proofs proofs that demonstrate two figures are similar by showing their corresponding angles are equal and their corresponding sides are proportional

Real-World Applications

• Architecture geometric principles applied in the design and construction of buildings and structures
  • Examples: calculating roof pitch, determining the amount of materials needed, creating stable structures
• Engineering using geometric concepts to design and build machines, vehicles, and other technical objects
  • Examples: designing gears and pulleys, calculating stress and strain on components
• Navigation and transportation geometry used in mapping, determining shortest routes, and calculating distances and angles
• Computer graphics and design geometric algorithms and principles used to create realistic images, animations, and 3D models
• Art and design geometric shapes, patterns, and proportions used in various forms of art and visual design
  • Examples: perspective drawing, tessellations, golden ratio in composition
• Sports and games geometry applied in playing fields, court dimensions, and equipment design
  • Examples: angles in billiards, trajectory of a ball, symmetry in playing cards

Common Mistakes and How to Avoid Them

• Confusing perimeter and area make sure to use the appropriate formula for each calculation
• Misapplying the Pythagorean theorem only applies to right triangles, not all triangles
• Forgetting to include units in measurements always specify the unit of measurement (e.g., cm, m², ft³)
• Rounding errors be careful when rounding numbers in calculations, as it can lead to inaccurate results
  • Use a consistent number of decimal places throughout the problem
• Misinterpreting diagrams pay close attention to labels, measurements, and given information in geometric diagrams
• Neglecting to consider all cases or possibilities when solving problems, make sure to consider all relevant scenarios or conditions
• Rushing through steps take the time to carefully read the problem, plan your approach, and double-check your work
• Relying too heavily on memorization instead of understanding focus on understanding the underlying concepts and principles, rather than just memorizing formulas or procedures