The study of shapes, sizes, angles, and the properties of space. From ancient Greek constructions to modern computational geometry.
Foundations of Geometry
Geometry is one of the oldest branches of mathematics, dating back to ancient civilizations who needed to measure land, build structures, and navigate the seas. The word "geometry" literally means "earth measurement" (from Greek geo = earth, metron = measure).
Euclid, often called the "Father of Geometry," established geometry as a rigorous deductive system around 300 BCE. His famous work Elements remained the primary geometry textbook for over 2,000 years.
Basic Undefined Terms
Point: A location in space with no size or dimension, represented by a dot
Line: An infinite set of points extending endlessly in both directions
Plane: A flat surface extending infinitely in all directions
Euclid's Five Postulates
A straight line can be drawn between any two points.
A straight line segment can be extended indefinitely.
A circle can be drawn with any center and radius.
All right angles are equal.
If a line intersects two other lines such that the interior angles on one side sum to less than 180°, those two lines will eventually meet on that side (the Parallel Postulate).
Angles and Lines
An angle is formed by two rays sharing a common endpoint (vertex). Angles are measured in degrees (°) or radians.
Types of Angles
Acute angle: Less than 90°
Right angle: Exactly 90°
Obtuse angle: Between 90° and 180°
Straight angle: Exactly 180°
Reflex angle: Between 180° and 360°
Angle Relationships
Complementary angles: Two angles that sum to 90°
Supplementary angles: Two angles that sum to 180°
Vertical angles: Opposite angles formed by intersecting lines (always equal)
Adjacent angles: Angles that share a common side and vertex
Parallel Lines and Transversals
When a transversal crosses two parallel lines, it creates eight angles with special relationships:
Trapezoid (Trapezium): Exactly one pair of parallel sides
Kite: Two pairs of adjacent sides equal
Circles
A circle is the set of all points in a plane that are equidistant from a fixed point called the center.
Key Terms
Radius (r): Distance from center to any point on the circle
Diameter (d): Distance across the circle through the center; d = 2r
Circumference (C): The perimeter of the circle; C = 2πr = πd
Arc: A portion of the circumference
Chord: A line segment with both endpoints on the circle
Tangent: A line that touches the circle at exactly one point
Secant: A line that intersects the circle at two points
Circle Theorems
The angle at the center is twice the angle at the circumference (Inscribed Angle Theorem)
Angles in the same segment are equal
The angle in a semicircle is 90° (Thales' Theorem)
Opposite angles of a cyclic quadrilateral sum to 180°
A tangent to a circle is perpendicular to the radius at the point of tangency
Area and Perimeter
Perimeter is the total distance around a shape. Area is the amount of space enclosed by a shape.
Common Area Formulas
Rectangle: A = l × w
Triangle: A = ½ × b × h
Circle: A = πr²
Parallelogram: A = b × h
Trapezoid: A = ½(a + b) × h
Rhombus: A = ½ × d₁ × d₂
Example: Find the area of a triangle with base 10 cm and height 6 cm
A = ½ × b × h = ½ × 10 × 6 = 30 cm²
Heron's Formula
For a triangle with sides a, b, and c:
s = (a + b + c) / 2 (semi-perimeter)
A = √(s(s-a)(s-b)(s-c))
Volume and Surface Area
Volume measures the space inside a 3D object. Surface area is the total area of all the faces.
Common 3D Formulas
Cube: V = s³, SA = 6s²
Rectangular Prism: V = lwh, SA = 2(lw + lh + wh)
Cylinder: V = πr²h, SA = 2πr² + 2πrh
Sphere: V = (4/3)πr³, SA = 4πr²
Cone: V = (1/3)πr²h, SA = πr² + πrl
Pyramid: V = (1/3) × Base Area × h
Transformations
A geometric transformation changes the position, size, or orientation of a figure.
Types of Transformations
Translation: Slides every point the same distance in the same direction (preserves size and shape)
Reflection: Flips the figure over a line (mirror image)
Rotation: Turns the figure around a fixed point by a given angle
Dilation: Enlarges or reduces the figure by a scale factor from a center point (changes size but preserves shape)
Translations, reflections, and rotations are rigid transformations (isometries) — they preserve both size and shape. Dilations change the size but preserve the shape.
Coordinate Geometry
Coordinate geometry (analytic geometry) combines algebra and geometry using the coordinate plane.
Example: Find the distance between (1, 2) and (4, 6)
d = √((4-1)² + (6-2)²) = √(9 + 16) = √25 = 5
The Pythagorean Theorem
Perhaps the most famous theorem in all of mathematics:
a² + b² = c²
In a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b).
Pythagorean Triples
Sets of three positive integers that satisfy a² + b² = c²:
3, 4, 5 (and multiples: 6-8-10, 9-12-15, etc.)
5, 12, 13
8, 15, 17
7, 24, 25
Converse of the Pythagorean Theorem
If a² + b² = c² for the sides of a triangle, then the triangle is a right triangle. We can also determine:
If a² + b² > c²: the triangle is acute
If a² + b² < c²: the triangle is obtuse
Example: A ladder is 13 feet long and leans against a wall. If the base is 5 feet from the wall, how high does the ladder reach?
5² + h² = 13²
25 + h² = 169
h² = 144
h = 12 feet
Geometric Proofs
A geometric proof is a logical argument that uses definitions, postulates, and previously proven theorems to demonstrate that a geometric statement is true. Proofs are at the heart of mathematical thinking and develop rigorous reasoning skills.
Two-Column Proofs
The most common proof format in school geometry. Each step consists of a statement and a reason.
Example: Prove that vertical angles are equal
Given: Two lines intersect, forming angles 1, 2, 3, 4.
A proof written as a flowing paragraph, common in higher mathematics.
Example: Prove the exterior angle of a triangle equals the sum of the two non-adjacent interior angles
Let triangle ABC have an exterior angle at vertex C, call it ∠ACD, formed by extending side BC to point D. Since ∠ACB and ∠ACD form a linear pair, we have ∠ACB + ∠ACD = 180°. We also know the angle sum property: ∠A + ∠B + ∠ACB = 180°. Setting these equal: ∠A + ∠B + ∠ACB = ∠ACB + ∠ACD. Subtracting ∠ACB from both sides gives ∠ACD = ∠A + ∠B. ✓
Proof by Contradiction (Indirect Proof)
Assume the opposite of what you want to prove, then show this assumption leads to a contradiction.
Example: Prove that a triangle cannot have two right angles
Suppose a triangle has two right angles: ∠A = 90° and ∠B = 90°.
Then ∠A + ∠B = 180°. But the angle sum of a triangle is 180°, so ∠C = 180° − 180° = 0°.
An angle of 0° is impossible in a triangle — contradiction.
Therefore, a triangle cannot have two right angles. ✓
When writing proofs, every step must be justified. The main types of justifications are: definitions, postulates (accepted without proof), previously proven theorems, and algebraic properties (equality, inequality).
Important Theorems to Know
Triangle Angle Sum Theorem: Interior angles of a triangle sum to 180°
Exterior Angle Theorem: An exterior angle equals the sum of the two remote interior angles
Isosceles Triangle Theorem: Base angles of an isosceles triangle are equal (and converse)
Triangle Inequality Theorem: The sum of any two sides must be greater than the third side
Midsegment Theorem: A midsegment of a triangle is parallel to the third side and half its length
Angle Bisector Theorem: The bisector of an angle of a triangle divides the opposite side in the ratio of the adjacent sides
Compass and Straightedge Constructions
Classical constructions use only two tools: a compass (for drawing circles and arcs) and a straightedge (for drawing straight lines, without markings). These constructions date back to the ancient Greeks.
Fundamental Constructions
Copy a segment: Transfer the length of a given segment to a new location
Copy an angle: Reproduce a given angle at a new vertex
Bisect a segment: Find the exact midpoint using perpendicular bisector
Bisect an angle: Divide an angle into two equal parts
Perpendicular from a point to a line: Drop a perpendicular foot
Perpendicular at a point on a line: Erect a perpendicular
Parallel line through a point: Construct a line parallel to a given line
Constructing Regular Polygons
Some regular polygons can be constructed with compass and straightedge:
Equilateral triangle: Constructible (use two circles of equal radius)
Square: Constructible (perpendicular diameters of a circle)
Regular pentagon: Constructible (using the golden ratio)
Regular hexagon: Constructible (radius equals side length)
Regular heptagon (7-gon): NOT constructible!
Gauss proved in 1796 that a regular polygon with n sides is constructible if and only if n is a product of a power of 2 and distinct Fermat primes (primes of the form 2^(2^k) + 1). The known Fermat primes are 3, 5, 17, 257, and 65537.
A = (1/2) × perimeter × apothem = (1/2) × n × s × a
Where s is the side length, n is the number of sides, and a is the apothem (distance from center to the midpoint of a side).
Number of Diagonals
D = n(n − 3) / 2
Example: How many diagonals does a decagon (10 sides) have?
D = 10(10 − 3)/2 = 10 × 7/2 = 35 diagonals
Common Regular Polygons
Triangle (3): Interior angle 60°, 0 diagonals
Square (4): Interior angle 90°, 2 diagonals
Pentagon (5): Interior angle 108°, 5 diagonals
Hexagon (6): Interior angle 120°, 9 diagonals
Octagon (8): Interior angle 135°, 20 diagonals
Decagon (10): Interior angle 144°, 35 diagonals
Dodecagon (12): Interior angle 150°, 54 diagonals
Conic Sections
Conic sections are curves obtained by intersecting a cone with a plane at different angles. They are among the most important curves in mathematics and physics.
y² = 4px (opens right if p > 0, left if p < 0)
x² = 4py (opens up if p > 0, down if p < 0)
The focus is at distance p from the vertex, and the directrix is at distance p on the opposite side.
Parabolas have a remarkable reflective property: any signal coming parallel to the axis of symmetry reflects off the parabola and passes through the focus. This is why satellite dishes, telescope mirrors, and car headlights are parabolic.
For over 2,000 years, mathematicians attempted to prove Euclid's fifth postulate (the parallel postulate) from the other four axioms. In the 19th century, Nikolai Lobachevsky, János Bolyai, and Bernhard Riemann independently discovered that replacing the parallel postulate with different axioms produced perfectly consistent geometries — with startling properties.
Hyperbolic Geometry (Lobachevsky/Bolyai)
In hyperbolic geometry, through a point not on a given line, there are infinitely many lines parallel to the given line.
The angle sum of a triangle is less than 180°
The "defect" (180° minus the angle sum) is proportional to the triangle's area
Similar triangles do not exist — if two triangles have the same angles, they are congruent
The circumference of a circle grows exponentially with radius (not linearly)
Elliptic (Spherical) Geometry (Riemann)
In elliptic geometry, there are no parallel lines — every pair of lines intersects.
The angle sum of a triangle is greater than 180°
The "excess" (angle sum minus 180°) is proportional to the triangle's area
Lines are great circles on a sphere
This is the geometry of the surface of the Earth — airline routes follow great circles
Einstein's General Relativity showed that the geometry of spacetime is not Euclidean — massive objects curve spacetime, and the resulting geometry is Riemannian. Non-Euclidean geometry went from a curiosity to the mathematical foundation of our understanding of gravity.