Geometry Study Guide

Polygons are 2D shapes made of straight line segments. A regular polygon has all sides equal in length and all interior angles equal. The sum of the interior angles of an n-sided polygon is given by the formula (n-2) × 180°. Each interior angle of a regular n-sided polygon is then [(n-2) × 180°] / n. The sum of the exterior angles of any convex polygon is always 360°. Therefore, each exterior angle of a regular n-sided polygon is 360° / n. An important relationship is that an interior angle and its adjacent exterior angle always sum to 180°. Diagonals connect non-adjacent vertices. The number of diagonals in an n-sided polygon is given by n(n-3)/2. Understanding these formulae is crucial for solving complex problems involving polygons, such as finding the number of sides given an angle, or determining angles in composite shapes. For example, if you know the sum of interior angles is 1080°, (n-2) × 180° = 1080°, so n-2 = 6, meaning n = 8 (an octagon). If an exterior angle is 30°, then 360°/n = 30°, so n = 360°/30° = 12 (a dodecagon).

Three-dimensional (3D) solids are objects that have length, width, and height. They occupy space. A special type of 3D solid is a polyhedron. A polyhedron is a 3D solid whose surfaces are all flat faces. These faces are polygons, and they meet at straight lines called edges. The points where edges meet are called vertices. Examples of polyhedra include cubes, cuboids, prisms, and pyramids. Non-polyhedral solids are 3D shapes that have at least one curved surface. Examples include spheres, cylinders, and cones. Understanding the properties of these shapes, such as their number of faces, edges, and vertices, is fundamental in geometry. Euler's formula, for example, connects the number of faces (F), vertices (V), and edges (E) for all simple polyhedra: F + V - E = 2. This formula is a powerful tool for checking the properties of a polyhedron. Identifying the type of faces (e.g., triangles, squares, rectangles) is also crucial for describing and classifying polyhedra. For instance, a triangular prism has two triangular faces as its bases and three rectangular faces connecting them, while a square-based pyramid has one square base and four triangular faces meeting at an apex.