Geometry is one of the oldest mathematical sciences. The word geometry comes from the Greek words 'geo' meaning earth and 'metron' meaning measure. Originally, geometry was the practical art of measuring land, but it evolved into an abstract mathematical discipline that studies shapes, sizes, positions, and properties of space. From simple points and lines to complex three-dimensional forms, geometry provides the foundation for understanding spatial relationships in our world.
Geometry is built from fundamental elements that increase in dimension. We start with a point, which is dimensionless and represents only position. When points connect, they form lines, which are one-dimensional and extend infinitely in both directions. Multiple lines can define planes, which are two-dimensional flat surfaces extending infinitely. Finally, planes stack to create three-dimensional space, which contains all geometric objects. Each element builds upon the previous one, creating the foundation for all geometric study.
Plane geometry explores two-dimensional shapes and their properties. We begin with angles, which can be acute when less than 90 degrees, right when exactly 90 degrees, or obtuse when greater than 90 degrees. Triangles are classified by their sides: equilateral triangles have all sides equal, isosceles triangles have two equal sides, and scalene triangles have all different sides. Quadrilaterals include squares with four equal sides and right angles, rectangles with opposite sides equal, parallelograms, and trapezoids. Circles are defined by their radius, diameter, circumference, and area, forming the foundation of circular geometry.
Geometric theorems provide logical proofs and fundamental relationships. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides: a² + b² = c². The triangle angle sum theorem proves that the three interior angles of any triangle always add up to 180 degrees. When parallel lines are cut by a transversal, corresponding angles are equal. The inscribed angle theorem shows that an inscribed angle is always half the measure of the central angle that subtends the same arc. These theorems form the foundation of geometric reasoning.
Three-dimensional geometry extends plane geometry into spatial relationships. Polyhedra like cubes, pyramids, and prisms are bounded by flat faces, with specific numbers of edges and vertices following Euler's formula. Curved three-dimensional shapes include spheres, cylinders, and cones, each with unique volume and surface area formulas. Cross-sections reveal the two-dimensional profiles of three-dimensional objects, helping us understand their internal structure. We can visualize spatial relationships using both wireframe and solid representations, connecting back to plane geometry as two-dimensional shapes form the faces of three-dimensional objects.