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:
Corresponding angles are equal
Alternate interior angles are equal
Alternate exterior angles are equal
Co-interior (same-side) angles are supplementary (sum to 180°)
Triangles
A triangle is a polygon with three sides, three vertices, and three angles. The sum of the interior angles of any triangle is always 180°.
Classification by Sides
Equilateral: All three sides are equal (all angles are 60°)
Isosceles: At least two sides are equal (two angles are equal)
Scalene: No sides are equal (no angles are equal)
Classification by Angles
Acute triangle: All angles are less than 90°
Right triangle: One angle is exactly 90°
Obtuse triangle: One angle is greater than 90°
Triangle Congruence
Two triangles are congruent if they have exactly the same shape and size. The congruence criteria are:
SSS: Three sides are equal
SAS: Two sides and the included angle are equal
ASA: Two angles and the included side are equal
AAS: Two angles and a non-included side are equal
HL: Hypotenuse and leg of right triangles are equal
Triangle Similarity
Two triangles are similar if they have the same shape but not necessarily the same size. The similarity criteria are:
AA: Two pairs of corresponding angles are equal
SAS: Two pairs of corresponding sides are proportional with the included angle equal
SSS: All three pairs of corresponding sides are proportional
Quadrilaterals
A quadrilateral is a polygon with four sides. The sum of interior angles is always 360°.
Types of Quadrilaterals
Square: All sides equal, all angles 90°
Rectangle: Opposite sides equal, all angles 90°
Rhombus: All sides equal, opposite angles equal
Parallelogram: Opposite sides parallel and equal
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.
Distance Formula
d = √((x₂ - x₁)² + (y₂ - y₁)²)
Midpoint Formula
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Equation of a Circle
(x - h)² + (y - k)² = r²
Where (h, k) is the center and r is the radius.
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?