← Back to Math Roadmap

Geometry

Shapes, space, measurement, and spatial reasoning — the visual foundation of mathematics.

Fundamentals of Euclidean Geometry

▼
Geometry, as formalized by the Greek mathematician Euclid around 300 BCE, starts from a small set of axioms (self-evident truths) and builds up a vast logical structure of theorems. The most basic objects are points (zero-dimensional locations), lines (one-dimensional, extending infinitely in both directions), and planes (two-dimensional flat surfaces extending infinitely). A line segment is the portion of a line between two endpoints. A ray is a portion of a line extending infinitely in one direction. An angle is formed by two rays sharing a common endpoint called the vertex. Angles are measured in degrees or radians. A full circle is 360 degrees or two pi radians.

Types of Angles

▼
  • Acute angle: less than 90 degrees. Sharp and narrow.
  • Right angle: exactly 90 degrees. Formed by perpendicular lines. Marked with a small square.
  • Obtuse angle: between 90 and 180 degrees. Wide and open.
  • Straight angle: exactly 180 degrees. Forms a straight line.
  • Reflex angle: greater than 180 degrees but less than 360 degrees.
  • Complementary angles: two angles whose sum is 90 degrees.
  • Supplementary angles: two angles whose sum is 180 degrees.

The Pythagorean Theorem

▼
The Pythagorean theorem is perhaps the most famous result in all of mathematics. It states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. The hypotenuse is the longest side, opposite the right angle. The other two sides are called legs. There are hundreds of different proofs — algebraic, geometric, and even one by a U.S. president (James Garfield). The theorem has been known since Babylonian times (around 1800 BCE). It is the foundation of distance measurement in two and three dimensions and is essential for everything from construction to GPS calculations.
The Pythagorean theorem. Side c is the hypotenuse (longest side, opposite the right angle). a and b are the legs.

Perimeter, Area, and Volume

▼
The perimeter of a shape is the total distance around its boundary — think of the length of a fence enclosing a yard. The area is the amount of flat surface enclosed by the shape — think of the amount of carpet needed to cover a floor. The volume is the amount of three-dimensional space occupied by a solid — think of how much water fills a container. Each type of shape has its own formula. Circles are unique: they involve the constant pi (approximately 3.14159), which is the ratio of any circle's circumference to its diameter.

Key Area Formulas

▼
  • Square: side length squared.
  • Rectangle: length times width.
  • Triangle: one half times base times height. The height is the perpendicular distance from the base to the opposite vertex.
  • Circle: pi times radius squared.
  • Trapezoid: one half times the sum of the parallel bases times the height.
  • Parallelogram: base times height (perpendicular, not slanted side).

Theorems and Proofs

▼
Several important theorems govern the relationships in geometry. The triangle sum theorem states that the three interior angles of any triangle always add up to 180 degrees. If two lines are cut by a transversal, corresponding angles are equal when the lines are parallel. Similar triangles have the same shape but possibly different sizes — their corresponding angles are equal and their corresponding sides are proportional. For any circle, an inscribed angle (vertex on the circle) measures half the central angle (vertex at the center) subtending the same arc. Euler's formula for polyhedra states that the number of vertices minus the number of edges plus the number of faces equals two, for any convex polyhedron.
Euler's characteristic formula for convex polyhedra. V = vertices, E = edges, F = faces. A cube has 8−12+6 = 2.

Worked Examples

▼
Worked Examples
Example 1: Find the hypotenuse of a right triangle with legs 3 and 4.
c² = 3² + 4² = 9 + 16 = 25. So c = 5.

Example 2: Find the area of a circle with radius 5.
Area = pi × 5² = pi × 25 ≈ 78.54 square units.

Example 3: A triangle has angles of 45° and 75°. Find the third angle.
Triangle sum = 180°, so the third angle = 180° − 45° − 75° = 60°.

Pythagorean Theorem Visual Proof

▼
Adjust sides a and b. Watch the squares on each side. The sum of the areas of the smaller squares always equals the area of the largest square.